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+{"text":"\\section{Introduction} \n\n\nA cornerstone of fractal fluids is \nthe non-integer dimension \\cite{Fractal1,Fractal2,CM}. \nThe mass of fractal fluid  \nsatisfies a power law relation $M \\sim R^{D}$, \nwhere $M$ is the mass of the ball region with radius $R$,\nand $D$ is the mass dimension \\cite{TarasovSpringer}. \nFractal fluid can be described by four different approaches:\n(1) Using the methods of \"Analysis on fractals\" \n\\cite{Kugami,Strichartz-1,Strichartz-2,Harrison,Kumagai,DGV} \nit is possible to describe fractal media;\n(2) An application of fractional-differential continuum models \nsuggested in \\cite{CCC2001,CCC2002}, and then developed in \n\\cite{CCC2004a,CCC2004c,CCSPZ2009,CCC2009,Yang2013a,Yang2013b},\nwhere so-called local fractional derivatives \\cite{Yang2013c} \nare used; \n(3) Applying fractional-integral continuum models suggested in \n\\cite{PLA2005-1,AP2005-2,IJMPB2005-2,MPLB2005-1,TarasovSpringer} \n(see also \\cite{MOS-3}-\\cite{MOS-5} and \\cite{Bal-1}-\\cite{Bal-4}), where\nintegrations of non-integer orders and \na notion of density of states \\cite{TarasovSpringer} are used;  \n(4) Fractal media can be described by using\nthe theory of integration and differentiation for\na non-integer dimensional space \\cite{Collins,Stillinger,PS2004}\n(see also \\cite{CNSNS2015,JMP2014}.\n\n\nLet us note that main difference of \nthe continuum models with non-integer dimensional spaces\nform the fractional continuum models suggested in \n\\cite{PLA2005-1,AP2005-2,IJMPB2005-2,MPLB2005-1,TarasovSpringer}\nmay be reduced to the following.\n(a) Arbitrariness in the choice of the numerical factor\nin the density of states is fixed by the equation\nof the volume of non-integer dimensional ball region.\n(b) In the fractional continuum models suggested \nin \\cite{PLA2005-1,AP2005-2,TarasovSpringer}, \nthe differentiations are integer orders whereas\nthe integrations are non-integer orders.\nIn the continuum models with non-integer dimensional spaces\nthe integrations and differentiations are defined \nfor the spaces with non-integer dimensions.\n\n\nIn this paper, we consider approach based on\nthe non-integer dimensional space.\nThe power law $M \\sim R^{D}$ can be naturally \nderived by using the integrations \nin non-integer dimensional space \\cite{Collins},\nwhere the mass dimension of fractal fluid\nis connected with the dimension of this space.\nA vector calculus for non-integer dimensional space \nproposed in this paper\nallows us to use continuum models with\nnon-integer dimensional spaces to describe\nfor fractal fluids.\nThis is due to the fact that\nalthough the non-integer dimension  \ndoes not reflect completely the geometric\nproperties of the fractal media, it nevertheless permits\na number of important conclusions about the behavior\nof fractal structures.\nTherefore continuum models with non-integer dimensional spaces\ncan be successfully used to describe fractal fluids.\n\nIntegration over non-integer dimensional spaces \nare actively used in the theory of critical phenomena and \nphase transitions in statistical physics \n\\cite{WilsonFisher,WK1974}, and \nin the dimensional regularization of\nultraviolet divergences \nin quantum field theory \\cite{HV1972,Leibbrandt,Collins}. \nThe axioms for integrations in non-integer dimensional space are proposed in \\cite{Wilson,Stillinger} and this type of integration is considered in the book by Collins \\cite{Collins}\nfor rotationally covariant functions. \nIn the paper \\cite{Stillinger} a mathematical basis of integration on non-integer dimensional space is given. \nStillinger \\cite{Stillinger} suggested a generalization of \nthe Laplace operator for non-integer dimensional spaces also. \nUsing a product measure approach,  \nthe Stillinger's methods \\cite{Stillinger}\nhas been generalized by Palmer and Stavrinou \\cite{PS2004} \nfor multiple variables case\nwith different degrees of confinement in orthogonal directions. \nThe scalar Laplace operators suggested \nby Stillinger in \\cite{Stillinger} \nand Palmer, Stavrinou in \\cite{PS2004}\nfor non-integer dimensional spaces, \nhave successfully been used for \neffective descriptions in physics and mechanics. \nThe Stillinger's form of Laplacian \nfor the Schr\\\"odinger equation in non-integer dimensional space \nis used by He \\cite{XFHe1,XFHe2,XFHe3} to describe\na measure of the anisotropy and \nconfinement by the effective non-integer dimensions. \nQuantum mechanical models with non-integer (fractional) \ndimensional space have been discussed in \n\\cite{Stillinger,PS2004,Thilagam1997b,MA2001a,MA2001b,QM1,QM5}\nand \\cite{Muslih2010,MA2012,QM8,QM9}.\nRecent progress in non-integer dimensional space approach\nalso includes description of\nthe fractional diffusion processes in\nnon-integer dimensional space in \\cite{LSTLRL}, and\nthe electromagnetic fields in non-integer dimensional space\nin \\cite{MB2007,BGG2010,MSBR2010}\nand \\cite{ZMN2010,ZMN2011a,ZMN2011b,ZMN2011c,ZMN}. \n\n\nUnfortunately, \\cite{Stillinger,PS2004} proposed \nonly the second order differential operators \nfor scalar fields in the form of the scalar Laplacian \nin the non-integer dimensional space. \nA generalization of the vector Laplacian \\cite{VLap}\nfor the non-integer dimensional space is not suggested\nin \\cite{Stillinger,PS2004}. \nThe first order operators such as gradient, \ndivergence, curl operators  \nare not considered in \\cite{Stillinger,PS2004} also.\nIn the work \\cite{ZMN} \nthe gradient, divergence, and curl operators\nare suggested only as approximations of the square of \nthe Laplace operator.\nConsideration only the scalar Laplacian \nin the non-integer dimensional space approach\ngreatly restricts us in application of continuum models\nwith non-integer dimensional spaces for fractal fluids and material.\nFor example, we cannot use the Stillinger's form of Laplacian\nfor vector field ${\\bf v}({\\bf r},t)$ in hydrodynamics of fractal fluids, in fractal theory of elasticity and thermoelasticity, \nin electromagnetic theory of fractal media\nto describe processes\nin the framework non-integer dimensional space approach.\n\nIn this paper, we propose to use a vector calculus\nfor non-integer dimensional space, and we define\nthe first and second orders differential vector\noperations such as gradient, divergence, \nthe scalar and vector Laplace operators \nfor non-integer dimensional space.\nIn order to derive the vector differential operators \nin non-integer dimensional space\nwe use the method of analytic continuation in dimension.\nFor simplification we consider rotationally covariant \nscalar and vector functions that are independent of angles.\nIt allows us to reduce differential equations \nin non-integer dimensional space to\nordinary differential equations with respect to $r$. \nThe proposed operators allow us to describe fractal media\nto describe processes in the framework of continuum models\nwith non-integer dimensional spaces. \nIn this paper we describe a Poiseuille flow of \nan incompressible viscous fractal fluid in the pipe. \nA generalization of the Navier-Stokes equation \nfor non-integer dimensional space to describe \nfor fractal fluid are suggested. \nA solution of this equation for steady flow of \nfractal fluid in a pipe and corresponding \nfractal fluid discharge are derived.\n\n\n\n\\section{Fractal fluids} \n\n\nA basic characteristic of fractal fluids\nis the non-integer dimensions such as mass \nor \"particle\" dimensions \\cite{TarasovSpringer}. \nFor fractal fluids \nthe number of particles $N_D(W)$ or mass $M_D(W)$ \nin any region $W \\subset \\mathbb{R}^3$ of this fluid \nincrease more slowly \nthan the 3-dimensional volume $V_3(W)$ of this region.\nFor the ball region $W$ with radius $R$ in \nan isotropic fractal fluid, \nthis property can be described by the relation\nbetween the number of particles $N_D(W)$\nin the region $W$ of fractal fluid, and \nthe radius $R$ in the form\n\\be \\label{NDW} \nN_D(W) = N_0 (R\/ R_0)^D , \\quad  R\/R_0 \\gg 1 , \n\\ee\nwhere $R_0$ is the characteristic size of fractal fluid\nsuch as a minimal scale of self-similarity \nof a considered fractal fluid. \nThe number $D$ is called the \"particle\" dimension. \nIt is a measure of how the fluid particles\nfill the space.\nThe parameter $D$ does not depend on the shape \nof the region $W$.\nTherefore fractal fluids can be considered \nas fluid with non-integer \"particle\" or mass dimension. \n\nIf the fractal fluid consists of particles \nwith identical masses $m_0$, then relation (\\ref{NDW}) gives\n\\be \\label{MDW} \nM_D(W) = M_0 (R\/ R_0)^D , \\quad  R\/R_0 \\gg 1 , \n\\ee\nwhere $M_0=m_0 \\, N_0$. \nIn this case, the mass dimension coincides\nwith the \"particle\" dimension.\n\nAs the basic mathematical tool for\ncontinuum models of fractal fluids,\nwe propose to use the integration and differentiation\nin non-integer dimensional spaces.\nIn Section 7, we will show that \nthe power-law relation (\\ref{MDW}) for \nan isotropic fractal fluid  \ncan be naturally derived by using the integration over \nnon-integer dimensional space, where\nthe space dimension is equal to\nthe mass dimension of fractal fluid. \n\n\nIn order to describe fractal fluid by \ncontinuum models with non-integer dimensional spaces,\nwe use the concepts of \ndensity of states $c_3(D,{\\bf r})$\nthat describes how closely packed permitted \nplaces (states) in the space $\\mathbb{R}^3$, \nwhere the fractal fluid is distributed.\nThe expression $dV_D({\\bf r})=c_3(D,{\\bf r})dV_3$ \nis equal to the number of permitted places (states) \nbetween $V_3$ and $V_3 +dV_3$ in $\\mathbb{R}^3$. \nThe notation $d^D {\\bf r}$ \nalso will be used instead of $ d V_D ({\\bf r})$.\nNote that density of states and \ndistribution function are different concepts, \nand it is impossible to describe \nall properties of fractal fluids\nby the distribution function only.\n\nFor fractal fluids, we can use the equation\n\\be \ndN_D(W)= n({\\bf r}) \\, d V_D ({\\bf r}) , \n\\ee\nwhere $n({\\bf r})$ is a concentration of particles\nthat describes a distribution of number of particles\non a set of permitted places (possible states).\nThe density of states is chosen such that\n$d V_D ({\\bf r}) = c_3(D,{\\bf r}) \\, dV_3$\ndescribes the number of permitted states in $dV_3$.\n\nThe form of the function $c_3(D,{\\bf r})$ \nis defined by symmetries of \nconsidered problem and properties of \nthe described fractal fluid.\nA general property of density of states\nfor fractal fluids is a power-law type \nof these functions that reflects a scaling property \n(fractality) of the fractal fluid. \nTo simplify our consideration in this paper \nwe will consider only isotropic fractal fluids \nwith density of states that is independent of angles.\nIn this case, the form of density of states is\ndefined such that $dV_D$ is an elementary volume\nof the non-integer dimensional space.\n\nIn the continuum models of fractal fluids,\nwe should work with the dimensionless variables\n$x\/R_0 \\to x$, $y\/R_0 \\to x$, $z\/R_0 \\to x$, \n${\\bf r}\/R_0 \\to {\\bf r}$,\nin order to physical quantities of \nfractal fluids have correct physical dimensions. \n\n\n\n\n\n\n\\section{Vector differential operators \nin non-integer dimensional space}\n\nTo derive equations for vector differential operators \nin non-integer dimensional space,\nwe use equations for the differential operators \nin the spherical (and cylindrical) coordinates in $\\mathbb{R}^n$ \nfor arbitrary $n$ to highlight the explicit \nrelations with dimension $n$.\nThen the vector differential operators  \nfor non-integer dimension $D$ \ncan be defined by continuation in \ndimension from integer $n$ to non-integer $D$. \nTo simplify we will consider only scalar \nfields $\\varphi$ and vector fields ${\\bf v}$ \nthat are independent of angles \n\\[ \\varphi({\\bf r}) =\\varphi(r) ,\n\\quad {\\bf v}({\\bf r})={\\bf v} (r) = v_r\\, {\\bf e}_r , \\]\nwhere $r=|{\\bf r}|$ is the radial distance,\n${\\bf e}_r={\\bf r} \/r$ is the local orthogonal unit \nvector in the directions of increasing $r$,  \nand $v_r=v_r(r)$ is the radial component of ${\\bf v}$.\nWe will work with rotationally covariant functions only. \nThis simplification is analogous to the simplification\nfor definition of integration over non-integer dimensional space\ndescribed in Section 4 of the book \\cite{Collins}.\n\n\n\\subsection{Vector differential operators \nfor spherical and cylindrical cases}\n\n\n\nUsing the continuation from integer $n$ \nto arbitrary non-integer $D$,  \nwe can get explicit definitions of  differential operators\nfor non-integer dimensional space in the following forms.\nNote that the same expressions can be obtained by using \nthe integration in non-integer dimensional space\nand the correspondent Gauss's theorem \\cite{CNSNS2015}\n\nLet us define the differential vector operations\nsuch as  gradient, divergence, the scalar and \nvector Laplacian for non-integer dimensional space. \nFor simplifications, we assume that\nthe vector field ${\\bf v}={\\bf v}({\\bf r})$ \nbe radially directed and the scalar and vector \nfields $\\varphi({\\bf r})$, ${\\bf v}({\\bf r})$ \nare not dependent on the angles.\n\nThe divergence in non-integer dimensional space \nfor the vector field ${\\bf v}={\\bf v}(r)$ is\n\\be \\label{Div-D}\n\\operatorname{Div}^{D}_{r} {\\bf v} = \n\\frac{\\partial v_r}{\\partial r} + \\frac{D-1}{r} \\, v_r.\n\\ee\n\nThe gradient in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{Grad-D}\n\\operatorname{Grad}^{D}_r \\varphi = \\frac{\\partial \\varphi}{\\partial r} \\, {\\bf e}_r .\n\\ee\n\nThe scalar Laplacian in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-D}\n^S\\Delta^{D}_r \\varphi= \n\\operatorname{Div}^{D}_r \\operatorname{Grad}^{D}_{r} \\varphi =\n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \\frac{D-1}{r} \\, \n\\frac{\\partial \\varphi}{\\partial r} .\n\\ee\n\nThe vector Laplacian in non-integer dimensional space \nfor the vector field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-D}\n^V\\Delta^{D}_r {\\bf v} = \n\\operatorname{Grad}^{D}_r \\operatorname{Div}^{D}_{r} {\\bf v} =\n\\Bigl(\n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{D-1}{r} \\, \n\\frac{\\partial v_r}{\\partial r}  -  \\frac{D-1}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r.\n\\ee\n\nIf $D=n$, equations (\\ref{Div-D}-\\ref{V-Delta-D})\ngive the well-known formulas for\ninteger dimensional space $\\mathbb{R}^n$.\n\n\nLet us consider a case of axial symmetry \nof the fluid, where the fields \n$\\varphi(r)$ and ${\\bf v}(r)=v_r(r) \\, {\\bf e}_r$\nare also axially symmetric.\nWe will direct the $Z$-axis along the axis of symmetry. \nTherefore we use a cylindrical coordinate system.\n\nThe divergence in non-integer dimensional space \nfor the vector field ${\\bf v}={\\bf v}(r)$ is\n\\be \\label{Div-DC}\n\\operatorname{Div}^{D}_{r} {\\bf v} = \n\\frac{\\partial v_r}{\\partial r} + \\frac{D-2}{r} \\, v_r.\n\\ee\n\nThe gradient in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{Grad-DC}\n\\operatorname{Grad}^{D}_r \\varphi = \n\\frac{\\partial \\varphi}{\\partial r} \\, {\\bf e}_r .\n\\ee\n\nThe scalar Laplacian in non-integer dimensional space \nfor the scalar field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-DC}\n^S\\Delta^{D}_r \\varphi = \n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial \\varphi}{\\partial r} .\n\\ee\n\nThe vector Laplacian in non-integer dimensional space \nfor the vector field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-DC}\n^V\\Delta^{D}_r {\\bf v} = \n\\Bigl(\n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial v_r}{\\partial r}  -  \\frac{D-2}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r .\n\\ee\n\n\n\nEquations (\\ref{Div-DC}-\\ref{V-Delta-DC}) can be\neasy generalized for the case $\\varphi=\\varphi(r,z)$ and \n${\\bf v}(r,z)=v_r(r,z) \\, {\\bf e}_r+ v_r(r,z) \\, {\\bf e}_z$.\nIn this case the curl operator for ${\\bf v}(r,z)$\nis different from zero, and\n\\be \\label{Curl-DC}\n\\operatorname{Curl}^{D}_r {\\bf v} = \n\\left( \\frac{\\partial v_r}{\\partial z} -\n\\frac{\\partial v_z}{\\partial r} \\right) \\, {\\bf e}_{\\theta} .\n\\ee\n\n\nFor $D=3$ equations (\\ref{Div-D}) - (\\ref{V-Delta-DC}) and \n(\\ref{Curl-DC}) give the well-known\nexpressions for the gradient, divergence, curl operator,\nscalar and vector Laplacian operators\n\nThe suggested operators for $0<D<3$ allow us to reduce \n$D$-dimensional vector differentiations \n(\\ref{Div-D}) - (\\ref{V-Delta-D}) and \n(\\ref{Div-DC}) - (\\ref{V-Delta-DC})\nto derivatives with respect to $r=|{\\bf r}|$. \nIt allows us to reduce partial differential equations for\nfields in non-integer dimensional space to\nordinary differential equations with respect to $r$.\n\n\n\\subsection{Stillinger's Laplacian for \nnon-integer dimensional space}\n\n\nFor a function $\\varphi=\\varphi(r,\\theta)$ of radial distance $r$ \nand related angle $\\theta$ measured relative \nto an axis passing through the origin, \nthe scalar Laplacian in a non-integer dimensional space\nproposed by Stillinger \\cite{Stillinger} is\n\\be \\label{NI-1}\n^{St}\\Delta^{D} = \\frac{1}{r^{D-1}} \\frac{\\partial}{\\partial r} \\left( r^{D-1} \\,\\frac{\\partial}{\\partial r} \\right) +\n\\frac{1}{r^2 \\, \\sin^{D-2} \\theta} \n\\frac{\\partial}{\\partial \\theta} \n\\left(  \\sin^{D-2} \\theta \n\\frac{\\partial}{\\partial \\theta} \\right) ,\n\\ee\nwhere $D$ is the dimension of space ($0<D <3$),\nand the variables $r \\ge 0$, $0\\le \\theta \\le \\pi$.\nNote that\n$(\\, ^{St}\\Delta^{D} )^2 \\ne \\, ^{St}\\Delta^{2D}$.\nIf the function depends on radial distance $r$ \nonly ($\\varphi=\\varphi(r)$), then\n\\be \\label{NI-R}\n^{St}\\Delta^{D} \\varphi (r) = \\frac{1}{r^{D-1}} \\frac{\\partial}{\\partial r} \\left( r^{D-1} \\,\\frac{\\partial \\varphi (r)}{\\partial r} \\right) =\n\\frac{\\partial^2 \\varphi (r)}{\\partial r^2} +\n\\frac{D-1}{r} \\, \\frac{\\partial \\varphi (r)}{\\partial r} .\n\\ee\nIt is easy to see that \nthe Stillinger's form of Laplacian $\\, ^{St}\\Delta^{D}$\nfor radial scalar functions $\\varphi ({\\bf r})=\\varphi (r)$\ncoincides with the scalar Laplacian \n$\\, ^S\\Delta^{D}_r$ defined by (\\ref{S-Delta-D}), i.e.,\n\\be \\label{NI-R-2}\n^{St}\\Delta^{D} \\varphi (r) = \\, ^S\\Delta^{D} \\varphi (r) .\n\\ee\nThe Stillinger's Laplacian can be applied \nfor scalar fields only. It cannot be used \nto describe vector fields ${\\bf v}=v_r(r) \\, {\\bf e}_r$ \nbecause this Laplacian for $D=3$ is not equal to \nthe usual vector Laplacian for $\\mathbb{R}^3$,\n\\be\n^{St}\\Delta^{3} {\\bf v}(r) \\ne \\, \\Delta {\\bf v}(r) =\n\\Bigl( \n\\frac{\\partial^2 v_r}{\\partial r^2} + \\frac{2}{r} \\, \n\\frac{\\partial v_r}{\\partial r} - \\frac{2}{r^2} \\, v_r\n\\Bigr) \\, {\\bf e}_r  .\n\\ee\nThe gradient, divergence, curl operator and vector Laplacian are not considered by Stillinger in paper \\cite{Stillinger}.\n\n\n\n\n\\subsection{Differential operators for $d \\ne D-1$}\n\nLet us consider a ball region $B_D$ in the fractal fluids\nwith the boundary $S_d= \\partial B_D$ with dimensions\n\\be\n\\operatorname{dim} (B_D)=D, \\quad \n\\operatorname{dim} (S_d) = d .\n\\ee \nEquations (\\ref{Div-D}) - (\\ref{V-Delta-DC}) \ndefine differential operators for spaces\nwith non-integer dimension $D$ and \nboundary dimensions $d=D-1$.\nIn general, the dimension $D$ of the region of a fractal fluid \nand the dimension $d$ of boundary of this region\nare not related by the equation $d = D-1$, i.e.,\n\\be\n\\operatorname{dim} (\\partial B_D) \\ne D-1 . \n\\ee\n\nUsing the integration in non-integer dimensional space\nand the correspondent Gauss's theorem, we can define \nthe divergence for the case $d \\ne D-1$ by the equation\n\\be \\label{Div-Dd}\n\\operatorname{Div}^{D,d}_{r} {\\bf v} =\\frac{\\pi^{(d+1-D)\/2} \\, \\Gamma (D\/2) }{\\Gamma ((d+1)\/2)}\n\\left( \\frac{1}{r^{D-1-d}}\n\\frac{\\partial v_r(r)}{\\partial r} + \\frac{d}{r^{D-d}} \\, v_r(r) \n\\right) .\n\\ee\nFor $d=D-1$, we get (\\ref{Div-D}).\nWe can define the parameter\n\\be \\label{ar}\n\\alpha_r = D-d ,\n\\ee\nthat can be interpreted as a dimension of fractal fluid  \nalong the radial direction ${\\bf e}_r$. Using (\\ref{ar}), \nequation (\\ref{Div-Dd}) can be rewritten in the form\n\\be \\label{Div-Dd2}\n\\operatorname{Div}^{D,d}_{r} {\\bf v} = \n\\pi^{(1-\\alpha_r)\/2} \\, \n\\frac{\\Gamma ((d+\\alpha_r)\/2) }{ \\Gamma ((d+1)\/2)}\n\\left( \\frac{1}{r^{\\alpha_r-1}}\n\\frac{\\partial v_r(r)}{\\partial r} + \n\\frac{d}{r^{\\alpha_r}} \\, v_r(r) \\right) .\n\\ee\nThis is divergence operator for non-integer boundary dimension\n$d \\ne D-1$.\nFor $\\alpha_r=1$, we have $d=D-1$, and then equations \n(\\ref{Div-Dd}), (\\ref{Div-Dd2}) give (\\ref{Div-D}).\n\nThe gradient for the scalar field $\\varphi({\\bf r})= \\varphi(r)$ \nand the radial dimension $\\alpha_r \\ne 1$ is defined by\n\\be \\label{Grad-Dd}\n\\operatorname{Grad}^{D,d}_{r} \\varphi = \n\\frac{\\Gamma (\\alpha_r\/2)}{ \\pi^{\\alpha_r\/2} \\, r^{\\alpha_r-1}} \\,\n\\frac{\\partial \\varphi(r)}{\\partial r} \\, {\\bf e}_r .\n\\ee\nFor $\\alpha_r=1$, equation (\\ref{Grad-Dd}) gives (\\ref{Grad-D}).\n\nUsing the operators (\\ref{Grad-Dd}) and (\\ref{Div-Dd})\nfor the fields $\\varphi=\\varphi (r)$ and \n${\\bf v}=v(r) \\, {\\bf e}_r$,\nwe can get the scalar and vector Laplace operators \nfor the case $d \\ne D-1$ by\n\\be\n^S\\Delta^{D,d}_r \\varphi= \n\\operatorname{Div}^{D,d}_r \n\\operatorname{Grad}^{D,d}_{r} \\varphi , \\quad\n^V\\Delta^{D,d}_r {\\bf v} = \n\\operatorname{Grad}^{D,d}_r \n\\operatorname{Div}^{D,d}_{r} {\\bf v} .\n\\ee\n\nThe scalar Laplacian for $d \\ne D-1$ \nfor the field $\\varphi=\\varphi (r)$ is\n\\be \\label{S-Delta-Dd}\n^S\\Delta^{D,d}_r \\varphi= A(d,\\alpha_r)\n\\left(  \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 \\varphi}{\\partial r^2} + \n\\frac{d+1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial \\varphi}{\\partial r} \\right) ,\n\\ee\nwhere\n\\be \\label{Ada}\nA(d,\\alpha_r) = \n\\frac{\\Gamma ((d+\\alpha_r)\/2) \\, \\Gamma (\\alpha_r\/2)}{ \n\\pi^{\\alpha_r-1\/2} \\, \\Gamma ((d+1)\/2)} .\n\\ee\n\nThe vector Laplacian for $d \\ne D-1$  \nfor the field ${\\bf v}=v(r) \\, {\\bf e}_r$ is\n\\be \\label{V-Delta-Dd}\n^V\\Delta^{D,d}_r {\\bf v} = A(d,\\alpha_r) \n\\left( \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 v_r}{\\partial r^2} \n+ \\frac{d+1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial  v_r}{\\partial r} \n- \\frac{d \\, \\alpha_r}{r^{2\\alpha_r}} \\,  v_r\n\\right) \\, {\\bf e}_r.\n\\ee\n\nThe vector differential operators (\\ref{Grad-Dd}), \n(\\ref{Div-Dd}), (\\ref{S-Delta-Dd}) and (\\ref{V-Delta-Dd})\nallow us to describe complex fractal fluids with \nthe boundary dimensions $d \\ne D-1$\nby continuum models with non-integer dimensional spaces.\n\n\n\n\\section{Navier-Stokes equations in non-integer \ndimensional space for fractal fluid} \n\n\n\nA motion of an incompressible viscous fractal fluid\nin the framework of continuum model with \nnon-integer dimensional space is described by the equations\n\\be \\label{nse-1}\n\\operatorname{Div}^D_r {\\bf v} =0 ,\n\\ee\n\\be \\label{NSE}\n\\frac{d {\\bf v}}{dt} = {\\bf f}\n- \\frac{1}{\\rho} \\operatorname{Grad}^D_r p + \n\\nu \\, ^V\\Delta^D_r {\\bf v} , \n\\ee\nwhere ${\\bf f}$ is the vector field of mass forces,\n$\\nu$ is the kinematic viscosity is the ratio of the dynamic viscosity $\\mu$ to the density of the fluid $\\rho$, and\n$d\/dt$ is the material derivative\n\\be \\label{nse-2}\n\\frac{d {\\bf v}}{dt} = \\frac{\\partial {\\bf v}}{\\partial t} +\n\\Bigl({\\bf v},\\operatorname{Grad}^D_r {\\bf v}\\Bigr) .\n\\ee\nIn equations (\\ref{nse-1})-(\\ref{nse-2}) \nthe gradient $\\operatorname{Grad}^{D}_r$, \nthe divergence $\\operatorname{Div}^{D}_{r}$,\nand the vector Laplacian $^V\\Delta^{D}_r$\nare defined by equations\n(\\ref{Grad-D}), (\\ref{Div-D}), (\\ref{V-Delta-D})\nfor spherical symmetry.\nFor cylindrical symmetry,\nthese operators are defined by equations\n(\\ref{Grad-DC}), (\\ref{Div-DC}), (\\ref{V-Delta-DC}).\n\nIf the dimension $D$ of the region of a fractal fluid\nand the dimension $d$ of boundary of this region\nare not related by the relation $d = D-1$, i.e.,\n$\\alpha_r = D-d \\ne 1$, then we should use the equations\n\\be \\label{nse-1d}\n\\operatorname{Div}^{D,d}_r {\\bf v} =0 ,\n\\ee\n\\be \\label{NSEd}\n\\frac{d {\\bf v}}{dt} = {\\bf f}\n- \\frac{1}{\\rho} \\operatorname{Grad}^{D,d}_r p + \n\\nu \\, ^V\\Delta^{D,d}_r {\\bf v} , \n\\ee\n\\be \\label{nse-2d}\n\\frac{d {\\bf v}}{dt} = \\frac{\\partial {\\bf v}}{\\partial t} +\n\\Bigl({\\bf v},\\operatorname{Grad}^{D,d}_r {\\bf v}\\Bigr) ,\n\\ee\nwhere the gradient $\\operatorname{Grad}^{D,d}_r$, \nthe divergence $\\operatorname{Div}^{D,d}_{r}$,\nand the vector Laplacian $^V\\Delta^{D,d}_r$\nare defined by equations\n(\\ref{Grad-Dd}), (\\ref{Div-Dd2}), (\\ref{V-Delta-Dd}).\n\nEquations (\\ref{NSE}) and (\\ref{NSEd}) \ncan be called the Navier-Stokes equations\nfor non-integer dimensional space.\n\n\nIt is convenient to work in the dimensionless space variables\n$x\/R_0 \\to x$, $y\/R_0 \\to x$, $z\/R_0 \\to x$, $r\/R_0 \\to r$,\nthat yields dimensionless integration and dimensionless \ndifferentiation in non-integer dimensional space.\nHere $R_0$ is the characteristic size of a \nfractal fluid, which is always finite. \nFor example, $R_0$ can be the minimal scale of self-similarity \nof a considered fractal fluid. \nThen the density is properly scaled such that\nthe mass $Q$ of fractal fluid and the fields ${\\bf v}$, $p$, $f$\nhave correct physical dimensions. \n\n\n\nThe Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}) \ndescribe dynamics of fractal fluids in the framework of\ncontinuum models with non-integer dimensional spaces.\nThese equations allow us to describe the isotropic fractal fluid \nonly when the presence of spherical or cylindrical symmetry.\n\nEquations (\\ref{NSE}) and (\\ref{NSEd}) can be used, when\nthe fields $p$, ${\\bf v}$, ${\\bf v}$ have the form\n$p  =p(r)$ and ${\\bf v}=v_r(r) \\, {\\bf e}_r$,\n${\\bf f})=f_r(r) \\, {\\bf e}_r$\ndoes not depend on the angles.\n\nAs an example of application of the \nNavier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}),\nwe consider a steady flow of fractal fluid in a pipe, \nand fractal fluid discharge in the next sections.\nIn the next sections, we derive the Poiseuille equation \nfor fractal fluids from the Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}). \nEquations can be used for any other problems of\nhydrostatics and hydrodynamics of fractal fluids\nwithin the case of spherical and cylindrical symmetries\n\nTo consider anisotropic fractal fluids,\nand problems without spherical and cylindrical symmetries,\nwe cannot use the Navier-Stokes equations \n(\\ref{NSE}) and (\\ref{NSEd}).\nIn this case, we should apply a product measure approach\n\\cite{JMP2014}.\nThe product measure approach to describe fractal\nproperties of space-time has been considered in \\cite{Svozil}.\nThe product measure approach for the fractional spaces\nhas been suggested in \\cite{PRE2005,JPCS2005},\nwhere fractional phase space is considered with its \ninterpretation as a non-integer (fractional) dimensional space.\nFor non-integer dimensional spaces \nthe product measure approach is suggested in \\cite{PS2004},\nwhere each orthogonal coordinates has own dimension. \nThe product measure approach for \nthe fractional-integral continuum models\nhas been considered in \\cite{MOS-4,MOS-4b,MOS-4c,MOS-5}.\n\n\n\n\\section{Steady flow of fractal fluid in a pipe} \n\nIn this section, we derive the Poiseuille equation \nfor fractal fluids from the Navier-Stokes equations (\\ref{NSE}). \nLet us consider a simple problem of motion \nof an incompressible viscous fractal fluid.\nUsing the continuum models with non-integer dimensional space,\nwe describe a steady flow of fractal fluid \nin a pipe with circular cross-section. \nWe take the axis of the pipe as the $X$-axis. \nThe velocity for laminar of fractal fluid\nis along the $X$-axis at all points, \nand is a function of $r$ only\n\\be \n{\\bf v}={\\bf v}(r)=v_x(r) \\, {\\bf e}_x .\n\\ee\nThe equation of continuity is satisfied identically. \nThe components of the Navier-Stokes equation \nfor $Y$-axis and $Z$-axis \ngive that the pressure is constant over \nthe cross-section of the pipe. \n\nWe shall solve the equation for a pipe \nwith circular cross-section. \nTaking the origin at the center of the circle \nwe can use cylindrical symmetry\n${\\bf v}={\\bf v}(r)=v_x(r) \\, {\\bf e}_x$. \nUsing the Navier-Stokes equations (\\ref{NSE}), we have \n\\be \\label{NSE-X}\n^S\\Delta^D_r v_x(r)= \\frac{1}{ \\mu} \\,\\frac{dp}{dx} ,\n\\ee\nwhere $\\mu= \\rho \\, \\nu$, and $dp\/dx$ is a constant. \nThe pressure gradient $dp\/dx$ may be written $ - \\Delta p \/l$, \nwhere $\\Delta p$ is the pressure difference between \nthe ends of the pipe and $l$ is its length. \n\n\nUsing the scalar Laplacian \nfor non-integer dimensional space\nthe Navier-Stokes equation (\\ref{NSE-X}) takes the form\n\\be \\label{Cyl-1}\n\\frac{\\partial^2 v_x(r)}{\\partial r^2} + \\frac{D-2}{r} \\, \n\\frac{\\partial v_x}{\\partial r} - \\frac{1}{ \\mu} \\,\\frac{dp}{dx} = 0 .\n\\ee\nFor $1<D<3$ and $0<D<1$, \nthe general solution of (\\ref{Cyl-1}) is\n\\be \\label{vxD}\nv_x(r) =C_1 \\, r^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, r^2 \\quad (0<D<3, \\quad D \\ne 1).\n\\ee\nFor $D=3$, we have\n\\be\nv_x(r) =C_1 \\, \\ln(r) + C_2 + \\frac{1}{4 \\, \\mu} \n\\frac{dp}{dx} \\, r^2 .\n\\ee\nFor $D=1$, we get the general solution\n\\be\nv_x(r) =C_1 \\, r^2 + C_2 + \\frac{1}{4 \\, \\mu} \n\\frac{dp}{dx} \\, r^2 \\, \\left(2 \\, \\ln(r) - 1 \\right) .\n\\ee\nIt should be noted that dimensions $D=1$ of the fractal fluid \ndo not correspond to the distribution of particles \nalong the line. \nThe fractal media with $D=1$\ndescribe a distribution of fluid particles \nin 3-dimensional space \nsuch that the mass dimension of the distribution \nis equal to $D=1$.\n\n\nLet us determine a flow of fractal fluid in a pipe \nof annular cross-section with \nthe internal radius $R_1$ and external radius $R_2$. \nThe constants $C_1$ and $C_2$ \nin the general solution (\\ref{vxD}) \nare determined from the boundary conditions \n\\be \\label{vR1vR2}\nv_x(R_1) = v_x(R_2) = 0 .\n\\ee\nUsing (\\ref{vxD}), these conditions have the form\n\\be\nC_1 \\, R_1^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, R_1^2 =0 ,\n\\ee\n\\be\nC_1 \\, R_2^{3-D} + C_2 + \\frac{1}{2 \\, (D-1) \\, \\mu} \n\\frac{dp}{dx} \\, R_2^2 =0 .\n\\ee\nThen the constants are\n\\be \\label{C1}\nC_1 = \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \\, \n\\frac{R_2^2- R_1^2}{R_1^{3-D}-R_2^{3-D}} ,\n\\ee\n\\be \\label{C2}\nC_2 = \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \n\\, \\frac{ R_1^{3-D}\\, R_2^2- R_2^{3-D} \\, R_1^2}{R_1^{3-D}-R_2^{3-D}} .\n\\ee\nSubstitution of (\\ref{C1}) and (\\ref{C2}) into (\\ref{vxD}) gives\n\\be \\label{vxr}\nv_x(r) = \\frac{1}{2 \\, (D-1) \\, \\mu} \\, \\frac{dp}{dx}\n\\left( \\frac{ R_2^2- R_1^2}{R_1^{3-D}-R_2^{3-D}} \\, r^{3-D} + \n\\frac{R_1^{3-D}\\, R_2^2- R_2^{3-D} \\, R_1^2}{R_1^{3-D}-R_2^{3-D}} +  r^2 \\right) ,\n\\ee\nwhere $0<D<1$ and $1<D <3$.\n\nUsing the variables\n\\[ x=\\frac{R_2}{R_1} , \\quad  y=\\frac{r}{R_1} , \\]\nequation (\\ref{vxr}) can be represented in the form\n\\be \\label{vxr2}\nv(x,y) = \\frac{R^2_1}{2 \\, (D-1) \\, \\mu} \\, \\frac{dp}{dx}\n\\left( - \\frac{1-x^2}{1-x^{3-D}} \\, y^{3-D} + \n\\frac{x^2 - x^{3-D}}{1-x^{3-D}} +  y^2 \\right) ,\n\\ee\nTo demonstrate some properties of the velocity\n$v_x(r)$ defined by (\\ref{vxr}), \nwe can visualize the function (\\ref{vxr2}), \nfor $x \\in [1;100]$, $y \\in [1;100]$ \nand different values of dimensions \n$D=2.9$, $D=2.7$, $D=2.0$, $D=1.1$.\nThe plots of function (\\ref{vxr2}) are presented by Figures 1-4, \nwhere $\\mu=1$, $R_1=1$ and ${dp}\/{dx}=-1$.\n\n\n\n\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{101.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.9$.} \n\\label{Plot1}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{102.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.7$.} \n\\label{Plot2}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{103.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=2.0$.} \n\\label{Plot3}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{104.eps}} \n\\end{minipage}\n\\caption{Plot of the velocity function $z=v(x,y)$ \ndefined by (\\ref{vxr2}) for the ranges $x \\in [1;100]$,  \n$y \\in [1;100]$, and $D=1.1$.} \n\\label{Plot4}\n\\end{figure}\n\n\n\n\nThe flow in a pipe of annular cross-section  \nwith the radius $R$, i.e. $R_1=0$ and $R_2=R$, we have \n\\be \\label{PE-FF}\nv_x(r) = - \\frac{1}{2 \\, (D-1) \\, \\mu} \\frac{dp}{dx} \\, R^2\n\\left( \\left(\\frac{r}{R}\\right)^{3-D} \n- \\left(\\frac{r}{R}\\right)^2 \\right) \\quad\n(0<D \\le 3, \\quad D \\ne 1).\n\\ee\nEquation (\\ref{PE-FF}) can be called \nthe Poiseuille equation for flow of fractal fluid. \nFor the case of non-fractal fluid ($D=3$), \nequation (\\ref{PE-FF}) gives the well-known Poiseuille equation\n\\be \\label{PE-FF-D3}\nv_x(r) = - \\frac{1}{4 \\, \\mu} \\frac{dp}{dx} \\, R^2\n\\left( 1 - \\left(\\frac{r}{R}\\right)^2 \\right) .\n\\ee\nThus the velocity distribution \nacross the pipe is parabolic for the non-fractal fluids. \nFor the fractal fluids, we have \nnon-integer power-law (\\ref{PE-FF}). \n\nNote that suggested Poiseuille equation \nfor fractal fluids, which are derived \nfrom the proposed Navier-Stokes equation \nfor non-integer dimensional space \ncan be used only to describe \nflow of fractal fluid in pipes.\nTo describe flow of fractal fluid between parallel planes, we should define new differential operators \nand the Navier-Stokes equations (\\ref{NSE}) and (\\ref{NSEd}), \nshould be modified by using the product measure approach\n\\cite{Svozil,PS2004,PRE2005,JPCS2005}.\n\n\n\n\\section{Fractal fluid with $\\alpha_r \\ne 1$}\n\nLet us derive the Poiseuille equation for fractal fluids \nfrom the Navier-Stokes equations (\\ref{NSEd}). \nThe Navier-Stokes equations for \nfractal fluid with $\\alpha_r=D-d \\ne 1$ has the form\n\\be \\label{Eq-Dda}\nA(d_x,\\alpha_r) \\left(  \\frac{1}{r^{2 \\alpha_r-2}} \\, \n\\frac{\\partial^2 v_x(r)}{\\partial r^2} + \n\\frac{d_x +1-\\alpha_r}{r^{2\\alpha_r-1}} \\, \n\\frac{\\partial v_x(r)}{\\partial r} \\right) -\n\\frac{1}{ \\mu} \\,\\frac{dp}{dx} = 0 ,\n\\ee\nwhere $A(d_x,\\alpha_r)$ is defined by (\\ref{Ada}),\n$d_x=d- \\alpha_x$, and\n$\\alpha_x$ is dimension along the $X$-axis.\nUsing $v_x(r)$ as an effective scalar field\n$\\varphi_{eff}(r)=v_x(r)$, we can apply equations\n(\\ref{Grad-Dd}), (\\ref{Div-Dd2}) and (\\ref{S-Delta-Dd})\nwhere $D \\to D_x=D- \\alpha_x$ and $d \\to d_x=d- \\alpha_x$\nto get (\\ref{Eq-Dda}).\nEquation (\\ref{Eq-Dda}) with $\\alpha_r=\\alpha_x=1$ \ngives (\\ref{Cyl-1}).\n\nFor $1<D<3$ and $0<D<1$, \nthe general solution of (\\ref{Eq-Dda}) is\n\\be \\label{vxDda}\nv_x(r) =C_1 \\, r^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, r^{2 \\alpha_r} \n\\quad (0<D<3, \\quad D \\ne 1) .\n\\ee\nFor $\\alpha_r=\\alpha_x=1$ equation (\\ref{vxDda}) gives (\\ref{vxD}).\nThe constants $C_1$ and $C_2$ \nin the general solution (\\ref{vxDda}) \nare determined by the boundary conditions \n\\be \\label{vR1vR2b}\nv_x(R_1) = v_x(R_2) = 0 .\n\\ee\nThese conditions give the equations\n\\be \nC_1 \\, R_1^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, R_1^{2 \\alpha_r} =0 ,\n\\ee\n\\be \nC_1 \\, R_2^{\\alpha_r-d_x} + C_2 + \n\\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \n\\frac{dp}{dx} \\, R_2^{2 \\alpha_r} =0 .\n\\ee\nThen the coefficients are\n\\be \\label{C1da}\nC_1 = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \n\\frac{ R_1^{2\\alpha_r} - R_2^{2\\alpha_r} }{R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} ,\n\\ee\n\\be \\label{C2da}\nC_3 = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \n\\frac{ R_2^{2\\alpha_r} R_1^{\\alpha_r-d_x} - R_1^{2\\alpha_r} R_2^{\\alpha_r-d_x} }{ R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} ,\n\\ee\nSubstitution of (\\ref{C1da}) and (\\ref{C2da}) \ninto (\\ref{vxDda}) gives\n\\[\nv_x(r) = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \\Bigl(\n\\frac{ R_1^{2\\alpha_r} - R_2^{2\\alpha_r} }{R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} \\, r^{\\alpha_r-d_x} + \\]\n\\be \\label{vxDda2}\n+ \\frac{ R_2^{2\\alpha_r} R_1^{\\alpha_r-d_x} - R_1^{2\\alpha_r} R_2^{\\alpha_r-d_x} }{ R_1^{\\alpha_r-d_x} - R_2^{\\alpha_r-d_x}} \n- r^{2 \\alpha_r} \\Bigr) \n\\ee\nfor $0<D<3$, where $D \\ne 1$.\n\nIf $R_1=0$ and $R_2=R$, then equation (\\ref{vxDda2}) has the form\n\\be \\label{vxDda3}\nv_x(r) = - \\frac{1}{2 \\, (d_x+ \\alpha_r) \\, \n\\alpha_r \\, A(d_x,\\alpha_r) \\, \\mu} \\frac{dp}{dx} \\,\nR^{2\\alpha_r}  \\, \\left(\n\\left(\\frac{r}{R}\\right)^{\\alpha_r-d_x} -\n\\left( \\frac{r}{R}\\right)^{2 \\alpha_r} \\right) .\n\\ee\nFor $\\alpha_r=\\alpha_x=1$ \nequation (\\ref{vxDda3}) gives (\\ref{PE-FF}).\nThe cases $\\alpha_r <1$ and\/or $\\alpha_x <1$ \ncorresponds to fractal fluid.\n\nWe can assume that $\\alpha_x >1$ can be used \nto describe fractal turbulent flow in pipe.\nThis assumption is based on the fact\nthat trajectories of the fluid particles \nare fractal curve, then $\\alpha_x>1$\n(for example, the Koch curve with \n$\\alpha_x=\\ln(4) \/ \\ln(3) \\approx 1.262$).\n\n\n\n\n\\section{Fractal fluid discharge}\n\n\nIn general, fractal fluids cannot be considered \nas a fluid on fractal.\nReal fractal fluids have a characteristic smallest length scale\nsuch as the radius, $R_0$, of a particle \n(for example, an atom or molecule).\nIn real fluids the fractal structure cannot be observed on all\nscales but only those for which $R >R_0$, \nwhere $R_0$ is the characteristic scale of the particles.\nThe concept of non-integer mass dimension of fractal fluid\nis based on the idea of how the mass \nof a fluid region scales with the region size,\nif we assume unchanged density.\nFor many cases, we can write the asymptotic form for the relation between\nthe mass $M_D(W)$ of a ball region $W$ of fluid,\nand the radius $R$ containing this mass as follows:\n\\be \\label{1-1-MR} \nM_D(W)=M_0 \\left( \\frac{R}{R_0} \\right)^D  \\ee\nfor $R\/R_0 \\gg 1$.\nThe constant $M_0$ depends on how the spheres of radius $R_0$ \nare packed.\nThe parameter $D$, which is interpreted as a dimension, \ndoes not depend on the shape of the region $W$,\nor on whether the packing of spheres of radius $R_0$ is \nclose packing,\na random packing or a porous packing with a uniform distribution of holes.\nThe non-integer mass dimension $D$ of fractal fluid \nis a measure of how the fluid fills \nthe integer $n$-dimensional Euclidean space it occupies.\nNote that the fact that a fluid is random or contains cavities\ndoes not necessarily imply that the fluid is fractal.\n\n\nUsing the non-integer dimensional space approach,\nwe can calculate the mass of fractal homogeneous fluids. \nScaling law (\\ref{1-1-MR}) is obtained naturally \nin the framework of this approach.\nWe can use the integration in a non-integer dimensional space \n\\cite{Stillinger} that is described by the equation\n\\be \\label{NI-2}\n\\int_{R^D} d^D {\\bf r} \\, \\varphi ({\\bf r}) =\n\\frac{2 \\pi^{(D-1)\/2}}{\\Gamma((D-1)\/2)}\n\\int^{\\infty}_0 dr \\, r^{D-1} \\,\n\\int^{\\pi}_0 d \\theta \\, \\sin^{D-2}\\theta \\, \\varphi (r,\\theta) ,\n\\ee\nwhere $d^D {\\bf r}$ represent the volume element\nin the non-integer dimensional space.\nUsing (\\ref{NI-2}) with $\\varphi (r,\\theta)=1$, and\n\\be \\int^{\\pi}_0 d \\theta \\, sin^{D-2}\\theta = \n\\frac{\\pi^{1\/2}\\, \\Gamma (D\/2-1)}{\\Gamma(D\/2)} , \\ee\nwe get the volume of $D$-dimensional ball with radius $R$ in the form\n\\be\nV_D = \\frac{\\pi^{D\/2}}{\\Gamma(D\/2+1)} \\, R^{D} .\n\\ee\nThe mass of fluid in $W$ is described by the integral \n\\be \\label{1-MW3} \nM_D(W) = \\int_{W} \\rho({\\bf r}) \\, d^D {\\bf r} ,\n\\ee\nwhere ${\\bf r}$ is dimensionless vector variable. \nFor a ball with radius $R$ and constant density \n$\\rho({\\bf r})=\\rho =\\operatorname{const}$, we get \n\\be \\label{M-D}\nM_D(W) = \\rho \\, V_D =\n\\frac{\\pi^{D\/2} \\, \\rho}{\\Gamma(D\/2+1)} \\, R^{D} .\n\\ee\nThis equation define the mass of the \nfractal homogeneous ball region of fluid with volume $V_D$. \nFor $D=3$, equation (\\ref{M-D}) gives the well-known \nequation for mass of non-fractal ball region\n$M_3 =(4 \\rho \\pi \/3) \\, R^3$ because \n$\\Gamma(3\/2) =\\sqrt{\\pi}\/2$ and $\\Gamma(z+1)=z \\, \\Gamma(z)$. \n\n\nLet us determine the mass $Q$ of fluid passing per unit time \nthrough any cross-section of the pipe (called the discharge). \nNot all pipe volume is occupied by fractal fluid.\nThere are areas unoccupied by particles of fractal fluid.\nIn continuum model of fractal fluids, \nwe take into account this fact by using the integration \nin space with non-integer dimension\n\\be \\label{Q-Def}\nQ= \\rho \\, \\frac{2 \\, \\pi^{d\/2}}{\\Gamma(d\/2)} \n\\int^R_0 v_x(r) \\, r^{d-1} \\, dr ,\n\\ee\nwhere $d=D-1$ is non-integer dimension of \nthe cross-section, $\\rho$ is a constant density, \nand $v_x(r)$ is defined by equation (\\ref{PE-FF}).\nSubstitution of (\\ref{PE-FF}) into (\\ref{Q-Def}) gives\n\\be \\label{QR}\nQ= - \\frac{\\rho \\, \\pi^{(D-1)\/2}}{2 \\, (D+1) \\, \\Gamma((D-1)\/2)\\, \\mu} \n\\frac{dp}{dx} \\, R^{D+1} \\quad (0<D \\le 3 \\quad D \\ne 1).\n\\ee\nNote that $\\rho$ has physical dimension of mass \n(for example, kilogram).\nThe mass of fractal fluid is thus proportional \nto $(D+1)$-power of the radius of the pipe.\n\nFor $D=3$, equation (\\ref{QR}) gives the well-known equation\n\\be \\label{QRD3}\nQ= - \\frac{\\rho \\, \\pi}{8\\, \\mu} \\frac{dp}{dx} \\, R^{4} .\n\\ee\nThe mass of non-fractal fluid is proportional to \nthe fourth power of the radius of the pipe. \nThe dependence of $Q$ on $dp\/dx$ and $R$ given by formula \n(\\ref{QRD3}) was established empirically by G. Hagen in 1839 and J. L. M. Poiseuille in 1840, and theoretically justified by G. G. Stokes in 1845.\n\nEquation (\\ref{QR}) can be rewritten in the form\n\\be\nQ= - \\frac{\\rho \\, \\pi}{8\\, \\mu_{eff}} \\frac{dp}{dx} \\, R^{D+1} \n\\ee\nwith the effective dynamic viscosity \n\\be \\label{meff}\n\\mu_{eff} = \n\\frac{D+1}{4} \\, \\pi^{(3-D)\/2} \\ \\Gamma((D-1)\/2) \\ \\mu .\n\\ee\nThe mass (discharge) of fractal fluid is proportional to \nthe non-integer power $(D+1)<4$ of the radius of the pipe,\nand the dynamic viscosity is effectively changed. \n\nFor $D=3$, equation (\\ref{meff}) gives $\\mu_{eff}=\\mu$.\nFor $1<D<3$, we have $\\mu_{eff}>\\mu$.\nIf $0<D<1$, then $\\mu_{eff} <0$.\nThe effective dynamic viscosity of \nfractal fluid with $D \\in (1;3)$ \nincreases with increasing a deviation of the dimension $D$ \nfrom three. \nWe can see an interesting effect of \na negative effective dynamic viscosity for fractal fluid \nwith dimension $D \\in (0;1)$. \nThe strong fractality of the fluid, \nwhich is caused by small dimension, \nleads to an increased fluid flow \ncompared with conventional medium.\nThis is probably due to an increase in freedom of particles motion for the fractal fluid similar to 3D Cantor dust.\n\nThe discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) \nfor the different values of dimensions $0 < D < 3$ \nand the range $R\\in[0,1]$ are present on Figures 5-8, \nwhere $\\rho=1$, $\\mu=1$, and ${dp}\/{dx}=-1$.\n\n\n\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{201.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[2,3]$.} \n\\label{Plot5}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{202.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[1,2]$.} \n\\label{Plot6}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{203.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[0.5,1.5]$.} \n\\label{Plot7}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\begin{minipage}[h]{0.47\\linewidth}\n\\resizebox{11cm}{!}{\\includegraphics[angle=-90]{204.eps}} \n\\end{minipage}\n\\caption{Plot of the discharge function $Q=Q(R,D)$ defined by (\\ref{QR}) for the range $R\\in[0,1]$ and $D\\in[0,3]$.} \n\\label{Plot8}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we propose a generalization of the Navier-Stokes \nequations to describe fractal fluids in the framework of\ncontinuum models with non-integer dimensional spaces. \nThese equations contain generalized \ndifferential vector operators for\nnon-integer dimensional space \\cite{CNSNS2015,JMP2014}.\nAs an example of application of the suggested\nNavier-Stokes equations for fractal fluids, \nwe consider a Poiseuille flow of \nan incompressible viscous fractal fluid in the pipe. \nThe solution for steady flow of \nfractal fluid in a pipe and corresponding \nfractal fluid discharge have been derived.\n\nIn this paper fractal fluid is described as a continuum \nin non-integer dimensional space. \nWe assume that suggested continuum models with non-integer\ndimensional spaces and the correspondent\nNavier-Stokes equations for fractal fluids\nmay be important for fractal theory of different type of media.\n\n\nAs the main object for application of the proposed \ncontinuum models is a two-component medium, \nwhere distribution of one component  (gas, liquid, solid) \ninto another component (fluid, gas or empty space) \ncan be characterized by non-integer mass or \n\"particle\" dimension.\nThis non-integer dimensional component can be considered \nas a fractal fluid. \nOne of the possible experimental methods \nfor determining the presence of \nfractal properties of the two-component medium \nmay be to use labels with radioactive isotopes \nfor particles of component that is assumed a fractal.\n\n\nA basic idealized model of fractal fluid \nis a liquid distributed in empty space $\\mathbb{R}^3$\nwith non-integer mass dimension $D<3$.\nIn some sense the fractal fluid is considered \nas a liquid analog of fractal porous solid material.\nFractal fluid can also be viewed as a two-phase medium \nconsisting of a liquid and a discharged gas instead \nof empty space,\nwhere the liquid is characterized by fractal mass dimension.\n\n\nAs an object of study, we also can consider \nan emulsion, when both the dispersed and the continuous phase \nare liquids, and the dispersed phase is fractally distributed \nin continuous phase.\nAn emulsion that is a mixture of two immiscible liquids,\none of which (the dispersed phase) is fractally dispersed \nin the other (the continuous phase). \nIn this case the dispersed phase can be described \nas a fractal fluid by suggested continuum models \nwith non-integer dimensional space.\nThe proposed models can be used for\na solution that is a homogeneous mixture \ncomposed of only one (liquid) phase,\nwhere one phase has a fractal dimension. \nWe can consider a fractal distribution of a solute \ndissolved in a non-fractal solvent,\nthen the solute is considered as a fractal fluid.\nThe solvent that is fractally homogeneously mixed with solute \ncan be considered as a fractal homogeneous fluid.\nThe homogeneity property of the fractal fluid means \nthat two regions $W_1$ and $W_2$ \nwith the equal volumes $V_n(W_1)=V_n(W_2)$ \nhave equal number of particles $N_D(W_1)=N_D(W_2)$. \nIn other words the fractal fluid is called homogeneous \nif the power law $N_D(W) \\sim R^{D}$ \n(or $M_D(W) \\sim R^{D}$) does not depend on \nthe translation of the region $W$. \n\n\nWe can consider a fractal distribution of small solid particles \nin the suspension. In this case, we have \nan internal phase (solid) that is fractally distributed through \nthe external phase (fluid) by mechanical agitation. \n\nAn object of investigations can be a liquid mixed \nwith a solid particles, \nwhere the distribution of these particles in space\ncan be characterized by non-integer mass dimensions,\nwhich can be caused by a power law distribution \nof particles by size or mass.\n\n\nAs a complex medium which may exhibit \nfractal properties can be considered the blood\nthat is composed of\nproteins, glucose, mineral ions, hormones, \ncarbon dioxide, blood cells \nand other particles suspended in water. \nWe assume that the blood as a multi-phase medium\ncan have attributes of a fractal distribution for \nsome blood components including bacteria, \nviruses and medicinal substances getting into the blood.\n\n\nWe assume that the suggested approach to describe \nfractal fluids by continuum models with non-integer\ndimensional spaces may be important for\nfractal theory of blood flow in cardiovascular system, \ndynamics of fractal media in hydrologic modeling \n\\cite{FLU-1,FLU-2,FLU-3,FLU-4} and \nit allows to develop the fractal dynamics \nof multi-phase media \\cite{Soo,Nigmatulin}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nPortraits make up a large percentage of the photos on the web nowadays.  ``Selfies'' have become a phenomenon, and recent studies~\\cite{facesinstagram} show that images with faces are more popular (+38\\% ``likes\" on Instagram) than other pictures in online social networks. Portraits are also used in web user profiles, in news articles, to represent celebrities and public figures, and they are an essential part of all kinds of IDs.\n\nGiven the huge volume of digital portraits, their broad usage, and their importance for people identification, surfacing the best digital portraits in terms of  photographic quality is of crucial importance. A system able to automatically score the aesthetic value of portraits could be used to select good images for a variety of applications such as  journalism, photo sharing websites, web search, PhotoBoosts, and many others.\n\nShooting photos of people is not a trivial task: human faces convey emotions, stories, lifestyles, and a good photographer needs to be able to capture their essence and personality. As a matter of fact, portrait photography is a stand-alone branch of photography literature, with  its own rules and compositional techniques, and tons of dedicated books  \\cite{child2008studio, hurter2007portrait, weiser1999phototherapy}. Systems that automatically rate the quality of digital portraits should be therefore specifically designed for face photos, unlike traditional visual aesthetics works \\cite{datta, ke2006design}, based on general photographic rules .\n\nIn spite of its importance, there has been little work in the research community to specifically address computational aesthetics of portraits.  Preliminary works ~\\cite{khan2012evaluating, li2010aesthetic} leave out many of the aspects that are specific to portraits (e.g., illumination, landmark representation, affective properties, etc.), and have experimented only with small datasets (less than 500 images). \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth, natwidth=1362, natheight=921]{fig\/Dataset.pdf}\n\\caption{(a) Distribution of images per demographic category and aesthetic scores (b) Characteristics extracted from the portrayed subjects}\n\\label{Facefeatures}\n\\vspace{-0.5cm}\n\\end{figure}\nIn this paper, we try to fill this void and introduce a new framework to automatically evaluate portrait aesthetics\\footnote{The aim of this work is to estimate the photographic quality of the \\textit{representation} of the person, independent from the beauty of the subject represented. }. To do so, we design visual features to describe image quality and portrait-specific properties and present a large-scale analysis of a data set of  over 10,000 portraits.\nIn addition, we build predictive models that are able to determine the aesthetic score of digital portraits.\nMoreover, with such large scale study, we provide an analysis of what makes a portrait beautiful from a computational perspective. To our knowledge, this represents the first attempt in literature to understand the relevancy of features for portrait aesthetics. \n\nOur main contributions can be summarized as follows:\n\\\\\\textbf{(1) Dataset:} we build a large dataset of portraits annotated with physical characteristics (determined using facial analysis) by sampling the AVA~\\cite{murray2012ava} images.\n\\\\\\textbf{(2) Features:} we introduce new features to describe portrait composition, quality, illumination, memorability, emotions, and originality.\n\\\\\\textbf{(3) Feature analysis:} we perform analyses on a set of over 10,000 portraits and report observations. We find that race, gender, and age are largely uncorrelated with photographic beauty, but aesthetic score is related to sharpness of facial landmarks, image contrast, exposure, homogeneity, illumination pattern, uniqueness, and originality.\n\\\\\\textbf{(4) Aesethetic Prediction:} we develop predictive models to classify portraits as aesthetically beautiful or not.\n\nIn Sec. \\ref{related}, we describe related work, and explain our portrait dataset in Sec. \\ref{datasets}. Sec. \\ref{visual_features} presents the visual features and analyze their relations with portrait beauty in Sec. \\ref{analysis}, then present our classification experiments in Sec. \\ref{results}. %\n\n\\section{Related Work}\\label{related}\nOur work relates to research that applies image analysis techniques to  detect the visual presence of non-semantic, fuzzy concepts such as memorability~\\cite{isola2011}, emotions~\\cite{emotions,joshi2011aesthetics}, interestingness~\\cite{redi2012interestingness, interesting, gygli2013interestingness}, privacy~\\cite{zerr2012privacy}, and beauty~\\cite{datta}.\n In particular, this paper follows previous work on computational aesthetics~\\cite{datta,murray2012ava}, that explores the discriminative ability of visual features to automatically assess the beauty of images and videos. Pioneers in this field are~Datta et al. and Ke et al., \\cite{datta, ke2006design}, who built an aesthetic classification framework for images based on features inspired by photographic theory. In subsequent years, such works were improved by designing more discriminative features~\\cite{nishiyama2011aesthetic, interesting}, proving the effectiveness of generic features~\\cite{marchesotti2011assessing, murray2012ava}  and building more effective learning frameworks~\\cite{wu2011learning}. Similar frameworks were applied to automatic image composition and enhancement by Bhattacharya et al.~\\cite{bhattacharya2010framework}\n\nWhile these existing computational aesthetic works build general frameworks for photographs of any semantic category, we focus on a specific type of images, namely portraits, whose compositional  and aesthetic criteria constitute a separate subject of study in the photographic literature~\\cite{hurter2007portrait, weiser1999phototherapy, child2008studio}, and therefore need a separate computational framework for aesthetic assessment.\nThis aspect is also proven by our experiments: we show that our portrait-specific aesthetic framework performs much better than a general classifier  for portrait aesthetic assessment.\n\nA few works~\\cite{murray2012ava, luo2011content, interesting} perform topic-based aesthetic classification. They build category-specific subsets of images by sampling aesthetic databases according to given image tags (``city\", ``nature\", but also  ``humans\" or ``portraits\"), and then use general compositional features to build topic-specific models.  The framework in this paper differs from those works for two reasons. (1) \\textit{We build a rich large-scale portrait aesthetic database}. A dataset based on tag-based sampling as in \\cite{murray2012ava, luo2011content, interesting}, could ignore many face images without tags while including images with noisy tags (as shown in Section \\ref{datasets}).  In this paper, we adopt a \\textit{content-aware sampling strategy} based on detailed face analysis. We reduce a large scale aesthetic dataset \\cite{murray2012ava} to a subset of more than $10000$ face images annotated with information about the portrayed subject, useful for both analysis and feature extraction. (2) \\textit{We build portrait-specific aesthetic visual features}. The works in~\\cite{murray2012ava, luo2011content, interesting} use traditional aesthetic  features designed for a general case, and apply them to the topic-specific contexts. In our work, we design face-specific aesthetic features inspired by photographic literature, together with non-face features that describe crucial aspects of photographic portraiture, such as illumination, sharpness, manipulation detection, image quality, emotion and memorability. Moreover, we show their combined effectiveness for aesthetic assessment of face photographs compared to  traditional aesthetic features. \n\nThere are a few recent works that attempt to design portrait specific datasets and features. For example, Li et al.\\cite{li2010aesthetic} use  face expression, face pose and face position features to estimate the aesthetic value of the images in a dataset of $500$ face images annotated by micro-workers. This work was improved by the work in~\\cite{xuefeature}, that uses hand-crafted features together with low-level generic features, and by Khan et al.~\\cite{khan2012evaluating} using spatial composition rules specifically tailored for portrait photography, together with specific background contrast features and face brightness and size features. These works represent a first attempt towards portrait aesthetic classification. However, one major weak point of such works is that they rely on small datasets (<500 images), thus making the results less generalizable for large datasets like the one we consider. Moreover,  despite their focus on face analysis, the features proposed by those works miss many important aspects of portrait photography such as illumination, demographics, face landmark properties, affective dimension, semantics and post-processing. In our work, we use features that are able to capture these aspects, and prove their effectiveness  by showing that they outperform the features in \\cite{khan2012evaluating} when used in an aesthetic classification framework on the dataset used by Khan et al.~\\cite{khan2012evaluating}. Moreover, in this paper, we perform for the first time a deep analysis of the importance of each feature and each group of features for face photo aesthetics, giving interesting and probably unexpected insights about what makes a portrait beautiful.\n\n\\section{Large Scale Portrait Dataset}\\label{datasets}\nIn order to create a large scale corpus of face images annotated with beauty scores, we resort to the largest aesthetic database available in the literature, i.e. the AVA dataset \\cite{murray2012ava}, created from the photo challenge website dpchallenge.com, that contains more than $250,000$ images annotated with an aesthetic score, a challenge title, and semantic textual tags.\n\nAVA is a unique, rich dataset for visual aesthetics, and therefore a reliable source of data for our purposes.\nHowever, AVA images contain very diverse subjects other than faces.  Moreover, for analysis and classification purposes, we want to collect not only a reliable subset of portrait images, but also some rich information about the portrayed subject and its representation. With this in mind, we design a \\textit{content-aware sampling strategy} on the AVA dataset, based on both metadata-based filtering and face analysis:\n\\\\(1) \\textbf{Enhanched metadata-based filtering.} First, we select  from the AVA database not only the images tagged as ``Portrait'' but  also  all the images whose challenge title contains the words 'Portrait', 'Portraiture' or 'Portraits'. (e.g. \\textit{Portrait Of The Elderly}). A total of 21,719 images are collected at this stage.\n\\\\(2) \\textbf{Face detection-based filtering.}  We use  Face+\n~\\cite{face++} to filter the images collected after metadata-based filtering. We obtain  a subset of 10,141 images for which Face++  detected the presence of one or more faces (in case of multiple faces, we retain the information about the largest one only).\n\\\\(3) \\textbf{Subject properties.} We compute though Face++ basic information about the subject, such as position, orientation, demographics (race, gender, age), coordinates of facial landmarks (eyes, nose and mouth in relative coordinates), presence of smile, presence of glasses, etc. (for a complete list of features see Table \\ref{tab:features}). \n\nFor each of the resulting images, we assign the average aesthetic score (in a 1-10 range) according to the votes provided by the AVA dataset. Figure \\ref{Facefeatures} shows the composition of our dataset, highlighting the distribution, based on gender and other properties estimated by the Face++ detector. About 53\\% of the subjects are classified as female, and 1\/3 of the image corpus shows subjects between 14 and 26 years of age (Fig. \\ref{Facefeatures} (a)). Similar to the AVA dataset, the vast majority of the aesthetic scores lies between 4 and 6, with a peak around the mean, which stands at 5.5.\n\n\\section{Features for Portrait Aesthetic Assessment}\\label{visual_features}\n\\input{tables\/features}\nVisually stunning portrait photographs are often  the result of an artistic process that might not strictly follow  general rules of composition, or fulfill basic quality \trequirements. However, photographic portraiture literature\n \\cite{child2008studio, hurter2007portrait, weiser1999phototherapy} suggests that following some specific photographic principles can help making digital portraits more attractive, ensuring visual appeal and expressiveness. Among the various tips for good portraiture available in literature, we identified 5 main photographic dimensions, namely:\n\\\\\\textbf{Compositional Rules:} arrangement of lines, objects, lights and color, widely used in visual aesthetic literature \\cite{datta, luo2011content}.\n\\\\\\textbf{Scene Semantics:} where has the photo been shot? and which objects co-exist with the subject in the scene?\n\\\\\\textbf{Portrait-Specific Features:}  information about the subject (aspect, soft biometrics, demographics) and its representation (sharpness, illumination, etc.)\n\\\\\\textbf{Basic Quality Metrics:} principles that ensure the correct perception of the signal, without distorting the scene represented.  Rarely used in computational aesthetics, they can be fundamental for high-quality portraiture \\cite{hurter2007portrait}.\n\\\\\\textbf{Fuzzy Properties:}  portrait photographic beauty is related to non-objective properties such as emotions or uniqueness, which are unquantifiable with low level features. \n\nIn this work, we design 5 groups of features that aim at describing various aspects of each of these dimensions using computer vision techniques. \n\n\\subsection{Compositional Rules} \\label{sec:comp}\nAs highlighted in many previous works \\cite{datta, pere, bhattacharya2010framework}, the visual attractiveness of a  picture is strongly influenced by the arrangement of objects in the image, their lighting, their colors,  their perceptibility.\n\nSimilar compositional rules apply to portraits photography. However, since portraits generally focus on a single subject whose essence needs to be captured in the shot, two compositional aspects need particular consideration: lighting and sharpness. The correct illumination of the scene and the detailed representation of the subject ensures both  perceptibility and expressiveness. Given these observations, we design a set of new features that capture essential properties of image lighting and sharpness, and collect a set of existing features for image composition analysis.\n\\vspace{0.5cm}\n\\\\\\emph{Lighting  Features} \\\\\nThe lighting setup is crucial to determine the essence of the portrait. In previous works \\cite{datta,khan2012evaluating, luo2011content}, scene lighting is described using features based on overall image brightness. However, as proved by our results, the raw brightness channel information might not be enough to capture portrait lighting patterns. \n\nWe therefore design a new lighting feature to expose \\textbf{Lighting Patterns} based on  an illumination compensation algorithm originally created for face recognition \\cite{gross2003image}.  Such method considers an image $I$  as a product $I=R(I)\\cdot L(I)$, where $R(I)$ is the  'reflectance' of the image and $L(I)$ is its  ``illuminance'\" i.e. the perceived lighting distribution.\n\nIn order to infer the lighting  pattern of an image, we proceed as follows.\nFor each image, we calculate $L(I)$ and create an illuminance vector $V(I)$ by averaging its  illuminance $L(I)$ over local windows (25x25 subdivision).  Applying k-means clustering on the illuminance vectors of a set of training images, we group the illuminance vectors into 5 Lighting Patterns representing the most common lighting setups in our dataset (See Fig.\\ref{fig:otherfea}).\nFor a new image $J$, we assign its corresponding lighting pattern by looking at the closest cluster to its illuminance vector $V(J)$, and retain the cluster number as the \\textit{Lighting Pattern Feature}.\n\\vspace{0.2cm}\n\\\\\\emph{ Sharpness  Features} \\\\\nThe recognizability and sharpness of the subject is a basic requirement for good portraiture. To analyze the amount of sharpness in the image, we design two new features:\n\\\\\\textbf{Overall Sharpness}: \nSubject movements or camera defocus can affect the overall image sharpness, introducing disturbing blur in particular image regions. We compute the sharpness of a picture by calculating the strength of the edges after applying horizontal and vertical Sobel masks on the image, according to the Tenengrad method (as explained in \\cite{ng2001practical}).\n\\\\\\textbf{Camera Shake}: sometimes camera movements can create an overall blurriness in the image. In order to estimate this particular type of blur, we compute  the ratio between the number of pixels detected to be affected by camera shake and the total number of pixels, according to the camera motion estimation algorithm of Chakrabarti et al. \\cite{chakrabarti2010analyzing}.\n\\vspace{0.2cm}\n\\\\\\emph{Traditional Compoisitional Features}\\\\\nWe collect here a set of features from state-of-the art works that model compositional photographic rules using a computational approach.\n\\\\\\textbf{Color Features.} In order to capture color patterns and their relation with portrait aesthetics, we compute the following features extracted from literature: \\textit{Color names} \\cite{emotions}, \\textit{Hue, Saturation, Brightness (HSV)} \\cite{emotions, datta} , the \\textit{Pleasure, Arousal, Dominance} metrics \\cite{emotions}, the \\textit{Itten Color Histograms} \\cite{emotions}, and the corresponding \\textit{Itten Color Contrasts}: \\cite{emotions} Moreover, we compute 2 contrast metrics: \n\\textit{Contrast (Michelson)} \\cite{Michel}, and a traditional \\textit{Contrast} measure computed as the ratio between the difference of max-min values of the Y channel and the Y average.\n\\\\\\textbf{Spatial Arrangement Features.} The distribution of textures, lines and object in the image space is an important cue for aesthetic and affective image analysis, as proved in \\cite{datta, emotions, pere, interesting}. \nTo analyze spatial layout of objects and shapes in the scene, we  compute first two symmetry descriptors, namely \\textit{ Symmetry (Edges)}\\cite{interesting}, and \\textit{Symmetry (HOG)}, for which we retain the difference between the HOG \\cite{dalal2005histograms} descriptors from left half of the image, and  from the flipped right half. Moreover, we compute 2 new features that describe shapes and their distribution, namely the \\textit{Number of Circles}, and the \\textit{Rule of Thirds}, that, unlike previous works \\cite{datta, bhattacharya2010framework}, determines the rule of thirds by computing the amount of spectral saliency \\cite{hou2007saliency} in the 9 quadrants resulting from a 3x3 division of the image. \n\\\\\\textbf{Texture Features.} Textural features can help analyzing the overall smoothness, order and entropy of the image. We analyze image homogeneity by computing the \\textit{GLCM properties} \\cite{emotions}, the \\textit{Image Order} \\cite{redi2012interestingness}, and the \\textit{Level of Detail} \\cite{emotions}.\n\\subsection{ Semantics and Scene Content}\nAs proved by various works in visual aesthetics \\cite{redi2012interestingness, luo2011content, pere}, the content of the scene and the types of objects placed in the picture substantially influence the aesthetic assessment of pictures. In particular, in the portraiture context, it is important to analyze the setting where the photo has been shot, i.e. objects, scenery and overall harmony of subject with the scene. In order to estimate these properties, we compute an adapted version of the \\textbf{Object bank features} \\cite{li2010object} that retains the maximum probability of a pixel in the image to be part of one of the 208 objects in the Object Bank.\n\\subsection{Basic Quality Metrics}\nIn general, visually appealing portraits are also high-qiuality photographs, i.e. images where the degradation due to image registration or post-processing is not highly perceivable.\nIn order to deeply analyze this dimension, we design some rules to determine the perceived image degradation by looking at simple image metrics, independent of the composition, the content, or its artistic value, namely:\n\\\\\\textbf{Noise}: we compute the amount of camera noise by applying an image denoising algorithm \\cite{buades2005non}, and then computing the distance between the denoised image and the original one. \n\\\\\\textbf{Contrast Quality}: well-contrasted images, i.e. images where the contrast level allows to distinguish the picture shapes without  introducing disturbing over-saturated regions, can be recognized by the uniform distribution of the intensities on the image histogram.  We therefore compute the quality of the contrast  by negating \\footnote{We take the negative of the distance in order to have higher values of this features for higher contrast quality} of the distance between the original image and its contrast-equalized version. \n\\\\\\textbf{Exposure Quality}: the luminance histogram of an overexposed image is skewed towards the right part, while for an underexposed image it is skewed towards the left side. In order to capture this behavior, we convert the image to the YCbCr space,  we compute the skewness of the Y channel histogram over 255 bins. When the skewness is close to zero, the exposure is correct, when below or above zero, the image is under or over exposed. We negate the absolute value of the skewness as exposure balance metric.\n\\\\\\textbf{JPEG Quality}: when too strong, JPEG compression can cause disturbing effects such as blockiness or block smoothness. We implement the objective quality measure for JPEG images proposed by \\cite{wang2002no} and retain the JPEG quality score output by the algorithm.\n\\\\\\textbf{Image Manipulations}: more and more, digital pictures are post-processed after the shooting using editing tools. In order to understand the amount of post-processing applied on the image, we design 2 new quality metrics, inspired by blind image forensics techniques. First, we design a feature to compute the amount of  \\textit{Splicing Manipulation}: we retain the output of an SVM classifier trained with Markov Features \\cite{pevny2010steganalysis} computed on a training set of images annotated as spliced\/not spliced from the CASIA dataset \\cite{casia}  (85\\% accuracy on this set). \nNext, we build a feature to compute the amount of \\textit{Median Filtering Manipulation}, using the algorithm of Yuan et al. \\cite{yuan2011blind}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\linewidth,natwidth=1745,natheight=336]{fig\/patterns.pdf}\n\\caption{Illuminance Distribution of the 5 Lightning Patterns}\n\\label{fig:otherfea}\n\\vspace{-0.5cm}\n\\end{figure}\n\\subsection{Portrait-Specific Features}\\label{facefeatures}\nIn photographic portraiture, lot of effort should be spent on understanding the subject and its correct representation. Photographic portrait theory \\cite{hurter2007portrait} particularly stresses the importance of the focus, sharpness, lighting and position of the face landmarks (eyes,nose,mouth). \n\nIn order to describe the properties of the subject and its representation, we retain as candidate features all the values extracted automatically by the Face++ api, and we build on top of such values a set of features to deeply describe the face and landmark properties. Overall, the set of Face\/Subject features is as follows:\n\\\\\\textbf{Face++ description:} \\textit{Face Position}, namely x, y relative coordinates, plus relative width and height , \\textit{Face Orientation}, i.e.,  yaw, pitch and roll angle of the head, \\textit{Demographics  } like Race (white, black, asian), Age (in years) and Gender, \\textit{Landmark Coordinates}, namely Right\/Left Eye, Nose and Mouth position in relative coordinates, \\textit{Subject Expression  }, that  estimates wether the subject is smiling or not, and \\textit{Other Face Properties } such as  presence of glasses (none, sunglasses, normal glasses). \\\\\n\\textbf{Landmark Sharpness} for each landmark, we simply compute its sharpness by averaging the gradient magnitude over the landmark region.\n\\\\ \\textbf{Landmark Staitstics:}  for each landmark, we extract its average Hue and Brightness\n\\\\ \\textbf{Face\/Background Contrasts:} similar to the background contrast feature in \\cite{khan2012evaluating}, we analyze here the compositional differences between face region and background region. However,  while Khan et al.  \\cite{khan2012evaluating} simply retain the ratio between face region brightness and image brightness, we perform here a deeper analysis. We consider face ($F$) and background ($B$) as two separate sub-images. We then  compute the \\textit{Lighting Contrast} as the ratio between the average Lightning (see Sec. \\ref{sec:comp}) of F and the average Lightning of B,  the F\/B \\textit{Sharpness Contrast} (Sharpness is computed computed as for the Landmark Properties), and, similarly, the \\textit{Brightness Contrast}.\n\\subsection{ Fuzzy Properties}\nSome artistic traits of photographs cannot be directly captured by low-level features:\nmany times, photographic beauty is related to feelings vehiculated by the image, which not even words can describe. In our work, we try to model some of those 'fuzzy' properties using a computational approach, by re-using existing work on image memorability, originality and affective analysis.\n\\\\\\textbf{Emotion}, is the emotion aroused by the image positive or negative?  We address this question by training an emotion classifier (SVM, 75\\% accuracy) with traditional Compositional Features, using as a groundtruth a mixture of 3 affective dataset \\cite{dan2011geneva, borth2013large, emotions}. We binarize the annotation in order to reflect the positive\/negative trait of the emotion shown. For each image , we retain the emotion score predicted by such classifier as the image emotion feature.\n\\\\\\textbf{Originality} of the image composition is computed by retaining the output of an originality classifier trained with Compositional Features and the Photo.net database from \\cite{datta} (Support Vector Regression (SVR), 4,7\\% MSE ). \n\\\\\\textbf{Memorability} of the image content. We compute this by retaining the output of a memorability classifier trained with the Saliency Moments Features\\cite{sm}, and the  memorability database of Isola et al.  \\cite{isola2011} (SVR, 2\\% MSE). \n\\\\\\textbf{Uniqueness}: as in \\cite{redi2012interestingness}, we estimate the photo uniqueness as the euclidean distance between the average spectrum of the images in a database and the spectrum of each image.\n\n\\section{What Makes  a Portrait Beautiful?}\\label{analysis}\nAmong all the features in Section \\ref{visual_features}, which of them is more discriminative to identify beautiful  portraits in a computational framework? In this Section we explore the relations between the visual features extracted and portrait aesthetic scores, by first analyzing the importance of each feature group described in Sec \\ref{visual_features}, and by then looking at the relevance of each  single feature within dimensions defined.\n\\subsection{Feature Groups for Portrait Aesthetics}\\label{multicorr}\nTo measure the significance of the five feature sets, we perform regression analysis using LASSO~\\cite{tibshirani1996regression}\nfor the different groups of features (i.e. \\textit{Compositional Features}). Once the regression parameter vector is\nlearned, we use compute the Spearman correlation between the predicted scores and the original aesthetic scores. This gives us a multidimensional correlation\nmetric that indicates the relevance of feature group\nfor portrait aesthetic assessment.  \nWe split the data into 5 random partitions, using one of the partitions as the test set\nand the rest as training, and\nlearn regression coefficients to predict the aesthetic scores on the test set using the different groups of features.\n\nAs shown in Fig. \\ref{fig:analysis}(b) all the groups of features  correlate positively with aesthetic scores. As expected, given the importance face of representation for portraiture, the \\textit{Portrait-Specific Features} correlate the most among all the  groups of features proposed, with a correlation of $0.330 \\pm 0.029$.  \nDespite its rich semantic analysis, and the proved effectiveness for scene\nanalysis \\cite{li2010object}, the ObjectBank \\textit{Semantic Features}, with its\n$~190$ feature detectors, are not as predictive, achieving a correlation score of\n$0.211 \\pm 0.022$ in contrast to compositional features which achieve a\ncorrelation score of $0.290 \\pm 0.029$. In comparison to these large\nfeature sets, smaller sets of features such as \\textit{Basic Quality} and \\textit{Fuzzy\nProperties} with $6$ and $4$ dimensions respectively achieve a much lower correlation\nscore for portrait aesthetics assessment as a whole, despite the importance of\nsingle features within the groups.\n\nIn order to calculate the combined predictive power of the whole set of features proposed, we perform similar regression analysis on all features together , i.e. without logical grouping, and look at the behavior of the algorithm  as more and more features are taken into account.\nFigure \\ref{fig:analysis}(d) shows a plot of the Spearman correlation of the feature set as a function of the number of features used and chosen by LASSO.   Using one single feature (\\textit{Right\\_Eye\\_Sharpness}), the Spearman correlation between predicted aesthetic scores and original aesthetic scores is $0.252 \\pm 0.018$. The best correlation score of $0.398 \\pm 0.027$ is obtained taking into account all $~300$ features. However, adding more than $60$\nfeatures shows diminishing returns. The correlation with $60$ features stand at\n$0.37$. The smallest mean square error achieved on the test set stands at\n$0.430 \\pm 0.008$. \n\nTable~\\ref{tab:featuresrank} reports the weight of the features ranked by when they are first picked by LASSO. Also\nreported are the feature category and weights. Notice how all the feature groups appear in the top-10 features, thus confirming the importance of each dimension we consider for portrait aesthetic evaluation, with a predominance of face features. We can  also spot some first insights about the importance of single features: crucial for aesthetic prediction are  landmark sharpness (\\textit{Right\\_Eye} and \\textit{Left\\_Eye}), the Exposure Quality, and the high discriminative ability of the Fuzzy Properties \\textit{Uniqueness}.\n\\begin{table}\n\\scriptsize\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\textbf{Rank} & \\textbf{Feature Name} & \\textbf{Feature Group} &\\textbf{Weight} \\\\\\hline\n1 & Left\\_Eye\\_Sharpness & Portrait Features & \\bf{0.061894} \\\\\n2 & Right Eye\\_Sharpness & Portrait Features & \\bf{0.074302} \\\\\n3 & Exposure\\_Balance & Basic Quality & -0.031212 \\\\\n4 & Uniqueness & Fuzzy Properties & \\bf{0.14232} \\\\\n5 & Smiling & Portrait Features & -0.045702 \\\\\n6 & Cluster4\\_Lighnting & Compositional & 0.017803 \\\\\n7 & Fence & Semantics & -0.022525 \\\\\n8 & Hue\\_Inner\\_Quadrant & Compositional & -0.045009 \\\\\n9 & Nose\\_Hue & Portrait Features & -0.03898 \\\\\n10 & Flower & Semantics & 0.026438 \\\\\n\\hline\n\\end{tabular}\n\\caption{Feature ranks based on Lasso regression}\n\\label{tab:featuresrank}\n\\vspace{-1cm}\n\\end{table}\n\\subsection{Single Features for Portrait Aesthetics}\\label{singlecorr}\nTo analyze in a more detailed manner which features correlate most with beautiful portraits, we partition the dataset into 5  subsets, as in Sec. \\ref{multicorr} and  average the Spearman correlation coefficient $\\rho$  between the individual features values and the aesthetic scores of each partition. \n\nIn Figure \\ref{fig:analysis} (a), we report the $\\rho$ coefficients of the features that show higher correlation with portrait aesthetics. We can notice how face sharpness and lighting  are of crucial importance for portrait beauty, as suggested by the Lasso analysis of discriminative features, and by portrait aesthetic literature.  4 out of the top 5 positively-correlated features correspond to the landmark sharpness features. Also, the contrast in sharpness between face and background strongly correlates with portrait beauty ($\\rho=0.12$), as well as the \\textit{Overall\\_Sharpness} metric.   As hypothesized, lighting patterns are also fundamental for a good portrait. This is shown by the positive $\\rho$ of the face\/background lighting contrast feature. Moreover, our analysis shows that there is a relation between image beauty and illumination patterns ( e.g. Clusters 3 has positive $\\rho$, while Cluster 4 has negative  $\\rho$). Overall, our new lighting features show higher relation with beauty than basic brightness features ($\\rho=0.054$ for the \\textit{Average\\_V} features), confirming the need of more complex lighting features for portrait aesthetic evaluation. \nSimilarly, contrast in colors and in gray levels (\\textit{GLCM\\_Contrast} and \\textit{Contrast\\_Michelson}) also show positive correlation with aesthetic scores.\n\nMoreover, negative $\\rho$ values for \\textit{Noise}  and positive correlation with \\textit{GLCM\\_Energy} make us conclude that visually appealing portraits should have a homogeneous, smooth composition without disturbing distortions. We can also see that the amount of \\textit{Median\\_Filtering} is negatively correlated with beauty, showing that too intensive post-processing results in a decrease of the portrait appeal. Surprisingly, \\textit{Exposure\\_Quality} is negatively correlated with beauty, suggesting that playing with over\/under exposure results in more appealing pictures. Moreover, negative $\\rho$ for some Color Names indicates that beautiful portraits tend to have little regions colored with non-skin colors such as green, purple, magenta. We can also notice the good outcome of our attempt of modeling fuzzy properties, given that properties such as \\textit{Originality} and \\textit{Uniqueness} positively correlate with beauty.\n\nIt was very interesting to notice how physical\/demographic properties such as gender, eye color, glasses, age, and race show very low correlation with image beauty, suggesting that any subject, no matter his\/her traits, can be part of a stunning picture, if the photographer is able to grasp the subject's essence.\n\nBy correlating gender properties with other visual features, we could find some side curious insights about portraiture. For example, female pictures tend to be more memorable, as well as  brighter and post-processed, while male tend to be represented with darker colors, and smile less than females.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth,natwidth=1008,natheight=288 ]{fig\/Analysis.pdf}\n\\caption{Analysis of the most relevant features and components for portrait aesthetic prediction (a,b,d). Classification Performances (c).}\n\\label{fig:analysis}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\\section{Predicting Portrait Beauty}\\label{results}\nIn order to test the effectiveness of our proposed features, and verify the findings of our analysis (see Sec. \\ref{analysis}), we perform 2 different classification experiments. First, we perform a small-scale experiment on the dataset provided in \\cite{khan2012evaluating}, showing the performances of our method and comparing them with the face-specific framework proposed by \\cite{khan2012evaluating}. Then, we design a large-scale classification framework, by looking at the ability  of our features to discriminate between beautiful\/non-beautiful pictures, using the large-scale dataset we built in Sec. \\ref{datasets}. We compare the classification performances of a framework based on our different groups of features with the one of a generic aesthetic classifier, i.e. based on traditional compositional features and trained on images with diverse subjects.\n\n \\subsection{Small-Scale Experiment}\nThe work that more closely relates to ours is the portrait aesthetic framework from Khan et al. \\cite{khan2012evaluating}: they design face-specific features and computes their effectiveness on a publicly available small-scale dataset of 150 pictures. \n\\input{tables\/result_small}\nIn order to test the performances of our approach, we compute the visual features in Sec. \\ref{visual_features} on the dataset from Khan et al. \\cite{khan2012evaluating} and we prove their effectiveness by using the same experimental setup, i.e. binarization of scores based on median, 10-folds cross validation on an SVM classifier in Weka, and average accuracy as evaluation metric. For fair comparison, we first evaluate the classification performances on our portrait-specific features only (see Sec. \\ref{facefeatures}), reporting results with and without feature selection in Table \\ref{tab:small_experiments}. Our group of portrait features alone outperforms the system in \\cite{khan2012evaluating}. Moreover, when we use all the features proposed in this work for the same classification task, we reach even higher classification accuracy, observing a substantial improvement of the performances compared to our baseline (and similar works such as the one from Li et al. \\cite{li2010aesthetic}).\n\\subsection{Large-Scale Aesthetic Categorization}\nWe now test the proposed approach for aesthetic classification on a large-scale, using the dataset of Sec. \\ref{datasets}.\nTo classify the images as ``Beautiful\" and ``Non-beautiful\", we use the binaries the average AVA \nscores it by labeling as positive any image with a score\ngreater than the mean user score ($5.55$). Similar to \\cite{murray2012ava}, we learn a SVM classifier using the publicly available libSVM\npackage. For this, the dataset is randomly divided into 5 partitions, as in Sec. \\ref{analysis}, and a SVM\nclassifier is learned per partitions. We use RBF kernel where the $\\gamma$ parameter is set\nto $1\/n$ where $n$ is the number of features. The cost parameter $C$ is obtained\nusing 10-fold cross-validation. All features are standardized to be zero mean\nand unit variance. \n\nFig \\ref{fig:analysis} (c)  shows the average classification accuracy on the test set for each group of features. As we can see, our framework benefits from the combination of diverse features, since the best performance is given by all features combined with early fusion, ($64.24\\% \\pm 1.76$) . Moreover, as expected by our analysis, we confirm that the classifier based on our rich portrait features outperforms the  classifiers based on the other groups of features, suggesting that detailed information of face properties and landmarks is more discriminative for portrait classification than traditional compositional features.\n\nResults reported in \\cite{murray2012ava, interesting} proved that a classifier trained on non-specific images performs better than a portrait-specific framework.  To prove the importance of building a portrait-specific framework, we compare our results with a baseline classifier built with traditional compositional features only (as in Sec. \\ref{sec:comp}), and trained on the dataset used in \\cite{pere}, namely  a database of  images belonging to 7 different categories, including ``Portraiture\",``Flower\", etc. and annotated with the corresponding aesthetic scores from DPchallenge.com (same source as our dataset, same score range). Unlike the findings in \\cite{murray2012ava, interesting}, we confirm the hypothesis that portraits need a separate computational framework for aesthetic assessment, showing that all the classifiers based on our proposed features perform better than this baseline (with all features, the improvement is more than 16\\%).\n\nAs in ~\\cite{murray2012ava}, we also performed SVM classification by introducing a $\\delta$\nparameter to discard ambiguous images from the training set (keeping all the\nimages in the test set). The $\\delta$ parameter was ranged from $0.1$ to $1.0$,\nbut unlike~\\cite{murray2012ava} we did not experience any increase in the classification\naccuracy. However, the performance with the $\\delta = 0.5$ is similar to when\n$\\delta = 0.0$, implying that the ambiguous images do not help for the task of\nclassification and can be discarded to speed up the learning time.  \n\n\\section{Conclusions}\\label{conclusions}\nIn this paper, we presented a complete framework for large-scale portrait aesthetic assessment based on visual features. We procured a dataset of digital portraits annotated with aesthetic scores and other information regarding traits\/demographics of the subjects in the portraits. We designed a set of discriminative visual features based on portrait photography literature. We analyzed the importance of each feature for portrait beauty, showing that rich facial features play a significant role in guiding the portrait aesthetics, and that the perceived portrait beauty is largely independent of the demographic characteristics of the subject. Finally, we built a classifier that is able to successfully distinguish between beautiful and non-beautiful portraits.\n\nIn our future work, we plan to broaden our framework by extending our database to include portrait images 'in the wild', exploring portrait aesthetics with a more challenging context.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\\label{sec1}\nQuantum teleportation is a process based on classical communication\nthat transmits the quantum information from a location to another\nwith the help of shared quantum entanglement between the sender and\nreceiver. This process is a technique for transporting the state of\nan atom or photon to the remote recipient even in the absence of\nquantum communication channels connecting the sender of the quantum\nstate (called  Alice) to the recipient (called Bob) \\cite{nel}. The\noriginal protocol of this process was firstly introduced by Bennett\n\\emph{et al.} using the Einstein-Podolsky-Rosen (EPR) state as the\nquantum channel \\cite{Bennett93}. Quantum teleportation using two\nqubit systems is discussed also in \\cite{chakra10,liang13}.\n\nBecause of the strong connection between quantum entanglement and\nquantum teleportation, the usage of multiparticle entangled quantum\nstates other than two-particle entangled states for quantum\nteleportation has been the subject of various investigations\n\\cite{wang07,paul11}. In particular, quantum teleportation with\nthree-qubit GHZ state and W state is studied in\nRefs.~\\cite{karl98,wang10,gorb03,joo03,agra06,Hu1}. The\npossibility of teleportation of an unknown qubit using four-particle\nGHZ state is discussed in Ref.~\\cite{pati00}. It is also shown that\nthe state in the form $|{\\psi}\\rangle = \\frac{1}{\\sqrt{2}} \\left(\n|00q_1\\rangle + |11q_2\\rangle \\right)$ allows perfect two-party\nteleportation in which $|q_1\\rangle$ and $|q_2\\rangle$ are arbitrary\nnormalized single qubit states \\cite{jung07}.\n\nIn quantum information theory and quantum computation, fidelity is a\nmeasure to quantify the closeness of two quantum states \\cite{nel} and\nis closely related to quantum entanglement \\cite{4}, quantum phase\ntransitions \\cite{6,7,8}, and quantum chaos \\cite{5}. Fidelity can\nbe also used to quantify how much quantum information is lost due to\nnoisy channel between initial and final states. This reduction of fidelity is usually due to\nthe interaction of quantum states with environment which results in\nimperfect teleportation. Thus, the coherence of the entangled state\nmay be lost and it becomes a mixed state. Some efforts have been\nperformed in this direction to realize effective factors which cause\nthis phenomenon \\cite{Bennett93,horodecki99,banas01,jung08-1}. For\ninstance, Bennett \\emph{et al.} showed that the fidelity of\nteleportation and the range of accurately teleported states reduce in\nthe less entangled quantum channels \\cite{Bennett93}.\n\nThe existence of noise is an unavoidable property in quantum\nteleportation process which results is decoherence of states and the\nreduction of fidelity \\cite{0h02,han08,bhak08}. In particular, Oh\n\\emph{et al.} using a pair of EPR states showed that the average\nfidelity and the range of teleported states depend on the type of\nthe noise that acts on the quantum channel and confirmed Bennett\n\\emph{et al.} results \\cite{0h02}. They solved analytically and\nnumerically the master equation with Lindblad structure and found\nthe fidelity as a function of decoherence time and angles of an\nunknown teleported state.\n\nNote that analytically solving the Lindblad equation in the presence of the noise is\nnot a trivial task in general. Indeed, for multiparticle systems one\nneeds to solve many coupled differential equations that involve\ntedious computation. For example, for three-particle GHZ state\n(3GHZ) the master equation reduces to 8 diagonal coupled\ndifferential equations and 28 off-diagonal coupled differential\nequations \\cite{jung08-2}. The situation is even worse for\nfour-particle GHZ state (4GHZ) that involves 16 diagonal coupled\ndifferential equations and 120 off-diagonal coupled differential\nequations.\n\nIn this paper, we analytically solve the master equation for\n$n$-particle GHZ state ($n\\in\\{4,5,6\\}$) through various noisy\nchannels. The number of coupled differential equations for each case\nis considerably reduced by using a proper ansatz for the density\nmatrix.  The ansatz is determined from the temporal evolution of the\ninitial state of the system. We obtain the fidelity of teleportation\nand the average fidelity of teleportation that depend on the type of\nthe noisy channel and compare the results with three-particle GHZ\nstate. The goal of this paper is to find out which state is better\n(loses less quantum information) in the teleportation process with\nnoisy channels. Therefore, although various noisy channels were\nstudied in Ref.~\\cite{0h02}, we discuss noisy channels which cause\nthe quantum channels to be mixed to compare $n$GHZ states in the\nprocess of teleportation.\n\nThe organization of this paper is as follows: Section \\ref{sec2} is\ndevoted to general framework used to evaluate the two-party quantum\nteleportation circuit. In Sec.~\\ref{sec3}, we analytically solve the\nLindblad equation where the quantum channel is a four-particle GHZ\nstate, i.e., $| \\mbox{4GHZ}\\rangle$. We transmit 4GHZ state through\nisotropic and Pauli noises and compute the fidelity of\nteleportation. Moreover, we compare the robustness of 4GHZ state\nwith 3GHZ state in the noisy channels. Solving the master equation\nfor 5GHZ and 6GHZ states when Lindblad operators are in $x$ and $z$\ndirections is the subject of Secs.~\\ref{sec4} and \\ref{sec5},\nrespectively. We present our conclusions in Sec.~\\ref{sec6}.\n\n\\begin{figure}[t]\n\\input{fig2.tex}\n\\caption{\\label{fig1}A circuit for quantum teleportation through\nnoisy channels with EPR state. The two top lines belong to Alice and\nthe bottom line to Bob. $M$ denotes measurement and the dotted box\nrepresents noisy channel. The Lindblad operator is turned on inside\nthe dotted box.}\n\\end{figure}\n\n\n\\section{GHZ state, fidelity, and Lindblad equation}\\label{sec2}\nFor $n$-particle system, an $n$GHZ state is a quantum state defined\nas follows\n\\begin{equation}\n|n\\mbox{GHZ}\\rangle=\\frac{1}{\\sqrt{2}} \\left( |0\\rangle^{\\otimes n}\n+ |1\\rangle^{\\otimes n} \\right),\n\\end{equation}\nwhere $n>2$. Note that, teleportation with $|\\mbox{EPR}\\rangle$ through\nnoisy channels is depicted in Fig.~\\ref{fig1} and it is discussed in\nRef.~\\cite{0h02}. Also, teleportation of 3GHZ state through various\nnoisy channels has been previously studied in Ref.~\\cite{jung08-2}.\nHere, we are interested to investigate the teleportation process for\n$n$GHZ state through noisy channels for $n\\in\\{4,5,6\\}$. For this\npurpose, we need to solve the master equation with Lindblad form\n\\cite{lind76}\n\\begin{equation}\\label{Lindblad}\n\\frac{\\partial \\rho}{\\partial t} = -\\frac{i}{\\hbar} [H_S, \\rho] +\n\\sum_{i, \\alpha} \\left(L_{i,\\alpha} \\rho L_{i,\\alpha}^{\\dagger} -\n\\frac{1}{2} \\left\\{ L_{i,\\alpha}^{\\dagger} L_{i,\\alpha}, \\rho\n\\right\\} \\right),\n\\end{equation}\nin which $L_{i,\\alpha} = \\sqrt{\\kappa_{i,\\alpha}}\n\\sigma^{(i)}_{\\alpha}$ denote Lindblad operators that describe\ndecoherence and act on the $i$th qubit. Also,\n$\\sigma^{(i)}_{\\alpha}$ are the Pauli spin matrices of the $i$th\nqubit with $\\alpha =\\{ x,y,z\\}$, $\\kappa_{i,\\alpha}$ is the\ndecoherence rate, and $H_S$ is the Hamiltonian of the system.\n\n\\begin{figure}\n\\input{fig1.tex}\n\\caption{\\label{fig2}\n   A circuit for quantum teleportation through noisy channels with 4GHZ state.\n   The four top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\nThe unknown state to be teleportated can be written as a Bloch vector on a\nBloch sphere\n\\begin{equation}\n\\label{unknown} |\\psi_{\\text{in}}\\rangle = \\cos\n\\left(\\frac{\\theta}{2}\\right) e^{i \\phi \/ 2} |0\\rangle + \\sin\n\\left(\\frac{\\theta}{2}\\right) e^{-i \\phi \/ 2} |1\\rangle,\n\\end{equation}\nwhere $\\theta$ and $\\phi$ denote the polar and azimuthal angles,\nrespectively. Fig.~\\ref{fig2} shows a quantum teleportation circuit\nthrough noisy channels with 4GHZ state in which the input state\ninvolves five qubits as the product state of\n$|\\psi_{\\text{in}}\\rangle $ and $|\\mbox{4GHZ}\\rangle$. The four top\nlines (qubits) belong to Alice and bottom one belongs to Bob. The\ndifference of this circuit with the teleportation circuit for EPR\nstate (Fig.~\\ref{fig1}) is the presence of two more controlled-NOT\n$\\left(\\mbox{CNOT} \\right)$ gates between $\\psi_{\\text{in}}$ and\n$\\mbox{4GHZ}$ states. After measurement of the top four qubits, Bob\ngets the teleported state $|\\psi_{\\text{out}}\\rangle$. It is\nconvenient to describe the teleportation in terms of the density\noperator\n\\begin{equation}\n\\label{out1} \\rho_{\\text{out}} = \\mbox{Tr}_{1,2,3,4} \\left[\nU_{\\mbox{\\tiny tel}} \\rho_{in} \\otimes \\varepsilon\n(\\rho_{4\\mbox{\\tiny GHZ}}) U_{\\mbox{\\tiny tel}}^{\\dagger} \\right],\n\\end{equation}\nwhere $\\rho_{\\text{in}} = |\\psi_{\\text{in}}\\rangle \\langle\n\\psi_{\\text{in}}|$ is density matrix of the unknown initial state\nand $\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$ is the density matrix\nafter transmission through noisy channel which is given by the\nLindblad equation. In fact, $\\varepsilon$ is a quantum operation\nthat maps $\\rho_{4\\mbox{\\tiny GHZ}}$ to $\\varepsilon\n(\\rho_{4\\mbox{\\tiny GHZ}})$ because of noisy channel and\n$\\rho_{4\\mbox{\\tiny GHZ}}=|\\mbox{4GHZ}\\rangle\\langle\\mbox{4GHZ}|$.\nMoreover, $U_{\\mbox{\\tiny tel}}$ is the unitary operator\ncorresponding to the quantum circuit and $\\mbox{Tr}_{1,2,3,4}$ is\npartial trace over first four qubits which belong to Alice.\n\nFidelity can be used as a tool to measure how much information is\nlost or preserved through noisy quantum channels in quantum\nteleportation process. It can be written as the overlap between the\ninput state $|\\psi_{in}\\rangle$ and the density operator for the\nteleported state $|\\rho_{\\text{out}} \\rangle$,\n\\begin{equation}\n\\label{fidelity} F = \\langle \\psi_{\\text{in}} | \\rho_{\\text{out}} |\n\\psi_{\\text{in}}\\rangle,\n\\end{equation}\nthat depends on an input state and the type of noise. For the\nperfect teleportation the fidelity is equal to unity. Also, $1-F$\nindicates how much information is lost through the teleportation\nprocess. For all possible unknown input states,\nthe average fidelity is given by\n\\begin{equation}\n\\overline{}\\label{average} \\overline{F} = \\frac{1}{4 \\pi}\n\\int_{0}^{2 \\pi} \\mathrm{d} \\phi \\int_{0}^{\\pi} \\mathrm{d} \\theta\n\\sin \\theta F(\\theta, \\phi).\n\\end{equation}\nSimilarly, we find the unitary operator, fidelity and average\nfidelity for 5GHZ and 6GHZ states in the following sections.\n\n\\section{Four-qubit GHZ state with noisy channels}\\label{sec3}\nIn this section, we analytically solve the Lindblad equation,\nEq.~(\\ref{Lindblad}), for 4GHZ state through various noisy channels.\nFirst, consider $(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise channel\nwith $\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa$\nthat acts on 4GHZ state. Also, here and throughout the paper we\nassume $H_S=0$.\n\nFor this case, the Lindblad equation involves 16 diagonal and 120\noff-diagonal coupled linear differential equations which make this\nequation difficult to be solved analytically. To overcome this problem,\nwe find the time evolution of the density matrix for infinitesimal\ntime interval $\\delta t$ using the Lindblad equation as\n\\begin{equation}\\label{del}\n\\rho(\\delta t)= \\rho(0) + \\left[ \\sum_{i, \\alpha} \\left(L_{i,\\alpha}\n\\rho(0) L_{i,\\alpha}^{\\dagger} \\right) - \\frac{1}{2} \\left\\{\nL_{i,\\alpha}^{\\dagger} L_{i,\\alpha}, \\rho(0) \\right\\} \\right]\\delta\nt,\n\\end{equation}\nwhere\n\\begin{equation}\\label{ro0}\n\\rho(0)=|\\mbox{4GHZ}\\rangle\\langle\\mbox{4GHZ}|=\\frac{1}{2}\\left[|0\\rangle^{\\otimes\n4} \\langle 0 |^{\\otimes 4} + |0\\rangle^{\\otimes 4}\\langle\n1|^{\\otimes 4}+|1\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} +\n|1\\rangle ^{\\otimes 4}\\langle 1|^{\\otimes 4}\\right].\n\\end{equation}\nSubstituting $\\rho(0)$ in Eq.~(\\ref{del}) results in\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 &0 & 0 & 0 & 0 & 0 & 0 & 0 & 1-4 \\kappa \\delta t   \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 &  \\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0 &  \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t &  \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0  \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}}  & 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0 &  \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 &  \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  \\kappa \\delta t & 0 & 0   \\\\\n0 &  \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0   \\\\\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 &\n0 & 0 & 1 -4 \\kappa \\delta t \\end{smallmatrix}\\right),}\n\\end{eqnarray}\nwhere \\textcircled{\\emph{n}} denotes $n$ diagonal zeros. Now, because of the form of the density matrix at $t=\\delta t$, we\nuse the following ansatz for the density matrix for all times\n\\begin{eqnarray}\\label{mat1}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = { \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a   \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix} \\right).}\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\ngives us a set of three coupled differential equations\n\\begin{eqnarray}\\label{diff1}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 4k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-4b(t)+3c(t)\\Big),\\\\\n\\dot{c}(t)=4k\\Big(b(t)-c(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial conditions $a(0)=1\/2$ and $b(0)=c(0)=0$ (see\nEq.~(\\ref{ro0})). The solutions are readily given by\n\\begin{eqnarray}\n\\label{elements} \\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{16}\\left( 1 + 6 e^{-4 \\kappa t} +  e^{-8 \\kappa t}\\right),\\\\\nb(t) =\\frac{1}{16}\\left( 1 - e^{-8 \\kappa t}\\right),\\\\\nc(t) =\\frac{1}{16}\\left( 1 - 2 e^{-4 \\kappa t} + e^{-8 \\kappa\nt}\\right).\n\\end{array}\\right.\n\\end{eqnarray}\nIn fact, the infinitesimal temporal behavior of the density matrix\nhelped us to properly suggest the solution and consequently reduced\n136 coupled differential equations to three coupled differential\nequations which are readily solved. It is now easy to check that\n$\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$, Eq.~(\\ref{mat1}), exactly\nsatisfies the Lindblad equation, Eq.~({\\ref{Lindblad}}), and the validity\nof the ansatz is verfied.\n\nHaving $\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})$ and $U_{\\mbox{\\tiny\ntel}}$ which can be read off from Fig.~\\ref{fig2}, it is\nstraightforward to compute $\\rho_{out}$. Thus, the fidelity reads\n\\begin{eqnarray}\n\\label{xfidelity} & &F(\\theta, \\phi) = \\frac{1}{2} \\left[(1 + \\sin^2\n\\theta \\cos^2 \\phi) + e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta\n\\sin^2 \\phi) \\right],\n\\end{eqnarray}\nand the average fidelity is given by\n\\begin{eqnarray}\n\\label{xfbar} \\overline{F} = \\frac{2}{3} + \\frac{1}{3} e^{-4 \\kappa\nt}.\n\\end{eqnarray}\n\nNow consider $(L_{2,y},L_{3,y},L_{4,y},L_{5,y})$ and assume\n$\\kappa_{2,y}=\\kappa_{3,y}=\\kappa_{4,y}=\\kappa_{5,y}=\\kappa$.\nSimilar to the previous case, using the infinitesimal time evolution\nof the density matrix\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-4 \\kappa \\delta t   \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 &  -\\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & -\\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0& 0 & 0 & 0&  -\\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}}& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0& 0 &0  & \\kappa \\delta t &  -\\kappa \\delta t & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & -\\kappa \\delta t & \\kappa \\delta t & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & -\\kappa \\delta t  & 0& 0 & 0 & 0& \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}}& 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n0 & 0 &  -\\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  \\kappa \\delta t & 0 & 0 \\\\\n0 &  -\\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n1 -4 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0\n&0 & 0 & 1 -4 \\kappa \\delta t\n\\end{smallmatrix}\\right),}\n\\end{eqnarray}\nwe take the following ansatz\n\\begin{eqnarray}\n \\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & -b & 0   \\\\\n0 & 0 & b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & -b & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & -b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & -b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& -b & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &- b  & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 &- b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & -b & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right).\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.({\\ref{Lindblad}}),\ngives the previous set of coupled differential equations, Eq.~(\\ref{diff1}), and consequently the solutions agree with\nEq.~(\\ref{elements}). For this case the fidelity becomes\n\\begin{eqnarray}\n\\label{yfidelity} & &F(\\theta, \\phi) = \\frac{1}{2} \\left[1 +( \\sin^2\n\\theta \\sin^2 \\phi + \\cos^2 \\theta) e^{-4 \\kappa t} + \\sin^2 \\theta\n\\cos^2 \\phi e^{-8 \\kappa t} \\right],\n\\\\   \\nonumber\n\\end{eqnarray}\nand the average fidelity reads\n\\begin{eqnarray}\n\\label{yfbar} \\overline{F} = \\frac{1}{2} +\\frac{1}{3}  e^{-4 \\kappa\nt} +\\frac{1}{6} e^{-8 \\kappa t}.\n\\end{eqnarray}\n\n\nFor the third case consider $(L_{2,z},L_{3,z},L_{4,z},L_{5,z})$ and\nassume $\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa$.\nThe infinitesimal time evolution of the density matrix gives\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2} \\left( |0\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} +\n|1\\rangle ^{\\otimes 4}\\langle 1| ^{\\otimes 4}\\right) +\n\\frac{1-8\\kappa \\delta t}{2} \\left(|0\\rangle ^{\\otimes 4} \\langle 1|\n^{\\otimes 4}+ |1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{eqnarray}\nSo the ansatz is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = a \\left( |0\\rangle^{\\otimes\n4} \\langle 0 |^{\\otimes 4} + |1\\rangle ^{\\otimes 4}\\langle 1|\n^{\\otimes 4}\\right) + b\\left(|0\\rangle ^{\\otimes 4} \\langle 1|\n^{\\otimes 4}+ |1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{eqnarray}\nInserting this matrix in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\nresults in\n\\begin{eqnarray}\\label{diff2}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 0,\\\\\n\\dot{b}(t) =-8k\\, b(t) ,\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial condition $a(0)=b(0)=1\/2$. The solution is\n\\begin{equation}\n\\label{zmatrix} \\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 4} \\langle 0 |^{\\otimes 4} + |1\\rangle\n^{\\otimes 4}\\langle 1| ^{\\otimes 4}\\right) + \\frac{1}{2} e^{-8\n\\kappa t} \\left(|0\\rangle ^{\\otimes 4} \\langle 1| ^{\\otimes 4}+\n|1\\rangle ^{\\otimes 4}\\langle 0 |^{\\otimes 4} \\right).\n\\end{equation}\nAlso, the fidelity and its average read\n\\begin{eqnarray}\n\\label{z,f,fbar}\n\\begin{array}{l}\\displaystyle\nF(\\theta, \\phi) = 1 - \\frac{1}{2} \\left(1 - e^{-8 \\kappa t} \\right)\n\\sin^2 \\theta, \\\\\\\\ \\displaystyle\\overline{F} = \\frac{2}{3} +\n\\frac{1}{3} e^{-8 \\kappa t}.\n\\end{array}\n\\end{eqnarray}\n\nThe next noisy channel is the isotropic noisy channel. For this case, the\nmaster equation involves twelve Lindblad operators\n$(L_{2,\\alpha},L_{3,\\alpha},L_{4,\\alpha},L_{5,\\alpha})$ with\n$\\alpha\\in\\{x,y,z\\}$. At $t=\\delta t$ we have\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -8 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-16 \\kappa \\delta t \\\\\n0 & 2\\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 2\\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0& 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 1}} & 0& 0 & 0 \\\\\n0 & 0 & 0& 0 & 2\\kappa \\delta t  & 0& 0 & 0 & 0& 0 & 0 &  0 & 0& 0 \\\\\n0 & 0 & 0& 0& 0 &{\\footnotesize \\textcircled{\\tiny 2}} & 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} &0 & 0 & 0& 0& 0 \\\\\n0 & 0 & 0& 0& 0 &0 & 2\\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0  \\\\\n0 & 0 & 0& 0& 0 &0 & 0 & 2\\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 \\\\\n0 & 0 & 0& 0& 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & 0  & 0& 0 & 0 & 0&  2\\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0& 0 & 0 & 0& 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 &  0 & 0& 0 & 0& 0 & 0 & 0& 0 & 0 &  2\\kappa \\delta t & 0 & 0   \\\\\n0 &  0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 2\\kappa \\delta t & 0   \\\\\n1 -16 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0\n&0 & 0 & 1 -8 \\kappa \\delta t\n\\end{smallmatrix}\\right).}\n\\end{eqnarray}\nSo we take the ansatz\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & d   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & b & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0 & c & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & c  & 0& 0 & 0 & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & c  & 0& 0 & 0 & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & 0 & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& 0 & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & 0  & 0& 0 & c & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & 0  & 0& 0 & 0 & 0 & c & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& c & 0 & 0 & 0   \\\\\n0 & 0 &0 & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\nd & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right).\n\\end{eqnarray}\nInserting this solution in the Lindblad equation, Eq.~({\\ref{Lindblad}}),\nwe find\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 8k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = 2k\\Big(a(t)-4b(t)+3c(t)\\Big),\\\\\n\\dot{c}(t)=8k\\Big(b(t)-c(t)\\Big),\\\\\n\\dot{d}(t)=-16k\\,d(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to the initial conditions $a(0)=d(0)=1\/2$ and $b(0)=c(0)=0$.\nThe solutions are\n\\begin{eqnarray}\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{16}\\Big( 1 + 6 e^{-8 \\kappa t} + e^{-16 \\kappa t}\\Big),\\\\\nb(t) =\\frac{1}{16}\\Big( 1 - e^{-16 \\kappa t}\\Big),\\\\\nc(t) = \\frac{1}{16}\\Big(1 - 2 e^{-8 \\kappa t} +  e^{-16 \\kappa t}\\Big),\\\\\nd(t) = \\frac{1}{2} e^{-16 \\kappa t}.\n\\end{array}\\right.\n\\end{eqnarray}\nAlso the fidelity is\n\\begin{eqnarray}\n& &F(\\theta, \\phi) = \\frac{1}{2} \\left[ 1 + e^{-8 \\kappa t} \\cos^2\n\\theta + e^{-16 \\kappa t}  \\sin^2 \\theta \\right],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{dfbar} \\overline{F} = \\frac{1}{6} \\left( 3 +  e^{-8 \\kappa t}\n+ 2 e^{-16 \\kappa t} \\right).\n\\end{eqnarray}\n\n\nTo this end, we only considered the noisy channels with the same\naxis. Now, as a different-axis noisy channel, consider\n$(L_{2,x},L_{3,y},L_{4,z},L_{5,x})$ noise with\n$\\kappa_{2,x}=\\kappa_{3,y}=\\kappa_{4,z}=\\kappa_{5,x}=\\kappa$ that\nexhibits the effects of noises in different directions. After an\ninfinitesimal time interval and using the Lindblad equation, the\ndensity matrix  can be written as\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2}{\\left(\n\\begin{smallmatrix}\n1 -6 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 1-10 \\kappa \\delta t \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0& 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 \\\\\n0 & 0 & 0 & \\kappa \\delta t & 0 & 0 &0 & 0 & -\\kappa \\delta t & 0& 0 & 0 \\\\\n0 & 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & 0& \\kappa \\delta t  & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 \\\\\n0 & 0 & 0& 0 & 0& \\kappa \\delta t  & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 \\\\\n0 & 0 & 0 & 0& {\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\kappa \\delta t & 0 & 0& 0& 0 & \\kappa \\delta t & 0& 0 & 0 \\\\\n0 & 0 &{\\footnotesize \\textcircled{\\tiny 2}}& 0 & 0 & 0 & 0& 0 & 0 & {\\footnotesize \\textcircled{\\tiny 2}} & 0 & 0 \\\\\n0 & \\kappa \\delta t & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n1 -10 \\kappa \\delta t & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0\n&0 & 0 & 1 -6 \\kappa \\delta t\n\\end{smallmatrix}\\right).}\n\\end{eqnarray}\nSo, the elements of the density matrix for all time can be read off\nas\n\\begin{eqnarray}\n\\varepsilon(\\rho_{4\\mbox{\\tiny GHZ}}) =\n \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & g   \\\\\n0 & b& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & h & 0   \\\\\n0 & 0 & c & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & m & 0 & 0   \\\\\n0 & 0 & 0 & d & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& k & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b  & 0& 0 & 0 & 0& 0 & 0 & n & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & d  & 0& 0 & 0 & 0 & k & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & d  & 0& 0 & f & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& b & h & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0 &0  & 0& h & b & 0 & 0 & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0& 0&0 & f  & 0& 0 & d & 0& 0& 0 & 0& 0 & 0   \\\\\n0 & 0 & 0& 0& 0 & k  & 0& 0 & 0 & 0 & d & 0& 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & n & 0& 0 & 0 & 0& 0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & k & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& d & 0 & 0 & 0   \\\\\n0 & 0 & m & 0& 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & c & 0 & 0   \\\\\n0 & h & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0   \\\\\ng & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix}\n\\right),\n\\end{eqnarray}\nwhich leads to two sets of four and six coupled differential\nequations, namely\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 3\\kappa\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = \\kappa\\Big(a(t)-3b(t)+2d(t)\\Big),\\\\\n\\dot{c}(t) = 3\\kappa\\Big(d(t)-c(t)\\Big),\\\\\n\\dot{d}(t) = \\kappa\\Big(2b(t)-3d(t)+c(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{f}(t) = \\kappa\\Big(-5f(t)+2h(t)-m(t)\\Big),\\\\\n\\dot{g}(t) = \\kappa\\Big(-5g(t)+2h(t)-n(t)\\Big),\\\\\n\\dot{h}(t) = \\kappa\\Big(f(t)+g(t)-5h(t)-k(t)\\Big),\\\\\n\\dot{k}(t) = \\kappa\\Big(-h(t)-5k(t)+m(t)+n(t)\\Big),\\\\\n\\dot{m}(t)=\\kappa\\Big(-f(t)+2k(t)-5m(t)\\Big),\\\\\n\\dot{n}(t)=\\kappa\\Big(-g(t)+2k(t)-5n(t)\\Big),\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=g(0)=1\/2$ and\n$b(0)=c(0)=d(0)=f(0)=h(0)=k(0)=m(0)=n(0)=0$. The solutions are\nreadily found\n\\begin{eqnarray}\\left\\{\n\\begin{array}{l}\na(t) =e^{2 \\kappa t}g(t)=\\frac{1}{16}\\Big( 1 + 3 e^{-2 \\kappa t} +3 e^{-4 \\kappa t} + e^{-6 \\kappa t}\\Big),\\\\\nb(t) =e^{2 \\kappa t}h(t)=-e^{2 \\kappa t}n(t)=\\frac{1}{16}\\Big( 1 + e^{-2 \\kappa t} - e^{-4 \\kappa t} - e^{-6 \\kappa t}\\Big),\\\\\nc(t) =-e^{2 \\kappa t}m(t)=\\frac{1}{16}\\Big( 1 - 3 e^{-2 \\kappa t} +3 e^{-4 \\kappa t} - e^{-6 \\kappa t}\\Big),\\\\\nd(t) =e^{2 \\kappa t}f(t)=-e^{2 \\kappa t}k(t)=\\frac{1}{16}\\Big( 1 -\ne^{-2 \\kappa t} - e^{-4 \\kappa t} + e^{-6 \\kappa t}\\Big).\n\\end{array}\\right.\n\\end{eqnarray}\nThus, the fidelity, $F(\\theta, \\phi)$, and its average, $\\overline{F}$, are given by\n\\begin{eqnarray}\n& &F(\\theta, \\phi) = \\frac{1}{2} \\left[ 1 + e^{-2 \\kappa t} \\cos^2\n\\theta + e^{-4 \\kappa t} \\sin^2\n\\theta \\cos^2 \\phi + e^{-6 \\kappa t}  \\sin^2 \\theta \\sin^2 \\phi \\right],\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{mixfbar} \\overline{F} = \\frac{1}{6} \\left( 3 + e^{-2 \\kappa t}+ e^{-4 \\kappa t}\n+ e^{-6 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\begin{table}\n\\caption{\\label{tab1}Summary of $F(\\theta, \\phi)$ and $\\overline{F}$\nthrough various noisy channels.}\n\\begin{ruledtabular}\n\\begin{tabular}{c|ccc}\n{} & Noise & 3GHZ & 4GHZ \\\\ \\hline\n{} & Pauli-X & $\\frac{1}{2} \\bigg[(1 + \\sin^2 \\theta \\cos^2 \\phi$) & $\\frac{1}{2} \\bigg[ (1 + \\sin^2 \\theta \\cos^2 \\phi$) \\\\\n{} & {} & $+ e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta \\sin^2 \\phi) \\bigg]$ & $+ e^{-4 \\kappa t} (\\cos^2 \\theta + \\sin^2 \\theta \\sin^2 \\phi) \\bigg]$ \\\\\n$F(\\theta,\\phi)$ & Pauli-Y &$\\frac{1}{2} \\bigg[1 + \\sin^2 \\theta \\sin^2 \\phi e^{-2 \\kappa t} +\\cos^2 \\theta e^{-4 \\kappa t}$   & $\\frac{1}{2} \\bigg[1 + (\\sin^2 \\theta \\sin^2 \\phi +\\cos^2 \\theta) e^{-4 \\kappa t}$    \\\\\n{} & {} & $+ \\sin^2 \\theta \\cos^2 \\phi e^{-6 \\kappa t} \\bigg]$ & $+\\sin^2 \\theta \\cos^2 \\phi e^{-8 \\kappa t} \\bigg]$\\\\\n{} & Pauli-Z & $1 - \\frac{1}{2} (1 -e^{-6 \\kappa t}) \\sin^2 \\theta$  &   $1 - \\frac{1}{2} (1 - e^{-8\\kappa t})\\sin^2 \\theta$\\\\\n{} & isotropic & $\\frac{1}{2} (1 + \\cos^2 \\theta e^{-8 \\kappa t} +\\sin^2 \\theta e^{-12 \\kappa t} )$ & $\\frac{1}{2} (1 + \\cos^2 \\theta e^{-8 \\kappa t} + \\sin^2 \\theta e^{-16 \\kappa t} )$\\\\\\hline\n{} & Pauli-X & $\\frac{2}{3} +\\frac{1}{3} e^{-4 \\kappa t}$ & $\\frac{2}{3} + \\frac{1}{3} e^{-4\\kappa t}$\\\\\n$\\overline{F}$ & Pauli-Y &$\\frac{1}{6} (3 + e^{-2 \\kappa t} + e^{-4 \\kappa t} + e^{-6 \\kappa t})$    & $\\frac{1}{6} (3 + 2 e^{-4 \\kappa t} + e^{-8 \\kappa t})$\\\\\n{} & Pauli-Z & $\\frac{2}{3} +\\frac{1}{3} e^{-6 \\kappa t}$ & $\\frac{2}{3} + \\frac{1}{3} e^{-8\\kappa t}$\\\\\n{} & isotropic & $\\frac{1}{6} (3 + e^{-8 \\kappa t} + 2 e^{-12 \\kappa t} )$ & $\\frac{1}{6} (3 + e^{-8 \\kappa t} + 2 e^{-16 \\kappa t})$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nIn Table \\ref{tab1}, a summary of fidelity and average fidelity for\n3GHZ \\cite{jung08-2} and 4GHZ states is reported and compared. Also,\ntheir average fidelity versus time is depicted in Fig.~\\ref{fig3}\nfor various noisy channels. Comparing 3GHZ and 4GHZ states shows\nthat for $(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise both states have\nthe same fidelity. This result also agrees with Bell state\n$|\\beta_{00}\\rangle=\\frac{1}{\\sqrt{2}}(|00\\rangle+|11\\rangle)$\n\\cite{0h02}. However, for other cases 3GHZ state is more robust,\ni.e., loses less quantum information in the quantum teleportation\nprocess with respect to 4GHZ state. Note that, for the isotropic\ncase, the fidelities are approximately equal. These results and\nthose obtained in Refs.~\\cite{jung08-2,0h02} show that increasing\nthe number of qubits can enhance the rate of information lost in\nquantum teleportation process. Moreover, using a proper ansatz for\nthe density matrix, we reduced the number of coupled differential\nequations from 136 to at most four coupled equations.\nFig.~\\ref{fig7} shows average fidelity for 4GHZ state through\nvarious noises. As it can be seen from the figure,\n$(L_{2,x},L_{3,x},L_{4,x},L_{5,x})$ noise does lose less quantum\ninformation with respect to others. The next noise with small\ninformation lost is $(L_{2,x},L_{3,y},L_{4,z},L_{5,x})$  for $\\kappa\nt<0.2$. However, for $\\kappa t>0.2$,\n$(L_{2,z},L_{3,z},L_{4,z},L_{5,z})$ noise represents a better\nbehavior. Moreover, the isotropic noise and the noise in $y$\ndirection always result in low fidelity quantum teleportation. In\nthe following sections, we exactly solve the Lindblad equation for\n5GHZ and 6GHZ states through two types of noisy channels.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{fig4GHZ.eps}\n\\caption{\\label{fig7} The plot of time dependence of average\nfidelity through noisy channels for\n4GHZ state.}\n\\end{figure}\n\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{ccc}\n\\includegraphics[width=8cm]{figF1.eps}\n &\\hspace{2.cm}&\n\\includegraphics[width=8cm]{figF2.eps}\\\\\n\\includegraphics[width=8cm]{figF3.eps}\n &\\hspace{2.cm}&\n\\includegraphics[width=8cm]{figF4.eps}\n\\end{tabular}}\n\\caption{\\label{fig3} The plot of time dependence of average\nfidelity for Pauli-X (left up),\nPauli-Y (right up),\nPauli-Z (left down), and isotropic\n(right down) noisy channels.}\n\\end{figure}\n\n\\section{Five-qubit GHZ state with noisy channels}\\label{sec4}\nIn this section, we teleport 5GHZ state through noisy channels as\ndepicted in Fig.~\\ref{fig4}. For this case the solution of the\nLindblad equation is a $32\\times32$ matrix that results in a set of\n32 diagonal and 496 off-diagonal coupled differential equations.\nHowever, we show that the number of required equations can be\nconsiderably reduced by choosing appropriate ansatz for the density\nmatrix.\n\n\\begin{figure}[t]\n\\input{fig3.tex}\n\\caption{\\label{fig4}\n   A circuit for quantum teleportation through noisy channels with 5GHZ state.\n   The five top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\nFirst, consider ($L_{2,x}$,$L_{3,x}$,$L_{4,x}$,$L_{5,x}$,$L_{6,x}$)\nnoise and assume\n$\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa_{6,x}=\\kappa$.\nThe infinitesimal time evolution of the density matrix now reads\n\n\\begin{eqnarray}\n\\hspace{-1cm} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta\nt} = \\frac{1}{2}{ \\left(\n\\begin{smallmatrix}\n1- 5 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1- 5 \\kappa \\delta t  \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 &  0 & \\kappa \\delta t & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t  & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0& 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & \\kappa \\delta t  & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 &0 & 0& {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 &0 &0 & 0& {\\footnotesize \\textcircled{\\tiny 6}} & 0& 0 & {\\footnotesize \\textcircled{\\tiny 6}} & 0 & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & \\kappa \\delta t  & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0& 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} &0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \\kappa \\delta t & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0   \\\\\n0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0   \\\\\n0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0   \\\\\n1- 5 \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 &\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0\n& 1- 5 \\kappa \\delta t\n\\end{smallmatrix} \\right).}\n\\hspace{0.5cm}\n\\end{eqnarray}\nSo we take the ansatz as\n\\begin{eqnarray}\n\\hspace{-1cm} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) = \\left(\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  0 & b & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 & b & 0 & 0& 0 & 0 & 0 & 0 &0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & \\tiny\\textcircled{\\emph{c}}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{3} &0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & b  & 0 & 0 & 0 & 0 & b & 0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 &0 & 0& \\tiny\\textcircled{\\emph{c}}_{6} & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{6} &0 &  0 & 0 & 0& 0 & 0 & 0   \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0& b & b & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0& 0 & 0 & 0& 0& 0 & 0 & 0 & b & b & 0 & 0 & 0& 0& 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 &0 &0 & 0& \\tiny\\textcircled{\\emph{c}}_{6} & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{6} &0 &  0 & 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & 0 & b & 0 & 0 & 0 & 0 & b & 0& 0 & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0& 0 &0 & \\tiny\\textcircled{\\emph{c}}_{3} & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{3} &0 & 0& 0 & 0 & 0 \\\\\n0 & 0 & 0& 0 & b &0 & 0& 0 & 0 & 0 & 0 &0 & 0 & b & 0& 0 & 0 & 0   \\\\\n0 & 0 & 0 &\\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}}_{1} & 0 & 0 & 0   \\\\\n0 & 0 & b & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & b & 0 & 0   \\\\\n0 & b& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 &  0 & b & 0   \\\\\na & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & a\n\\end{smallmatrix} \\right).\n\\hspace{0.5cm}\n\\end{eqnarray}\nHere $\\textcircled{\\emph{c}}_{n}$ denotes\n$n$ diagonal $c$.\n\nNow inserting this matrix in Lindblad equation, Eq.~(\\ref{Lindblad}), four\ncoupled differential equations are obtained as follows\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 5k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-5b(t)+4c(t)\\Big),\\\\\n\\dot{c}(t)=2k\\Big(b(t)-c(t)\\Big).\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nSolving this set of equations with the initial conditions\n$a(0)=1\/2$, $b(0)=c(0)=0$, leads to the following solution\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{32}\\Big( 1 + 10 e^{-4 \\kappa t} + 5e^{-8 \\kappa t}\\Big),\\\\\nb(t) =\\frac{1}{32}\\Big( 1 + 2 e^{-4 \\kappa t}- 3e^{-8 \\kappa t}\\Big),\\\\\nc(t) = \\frac{1}{32}\\Big(1 - 2 e^{-4 \\kappa t} +  e^{-8 \\kappa\nt}\\Big).\n\\end{array}\\right.\n\\end{eqnarray}\nSubstituting $\\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})$ in\nEq.~(\\ref{out1}) and using Eqs.~(\\ref{fidelity}) and (\\ref{average})\nfidelity and its average are given by\n\\begin{eqnarray}\nF(\\theta, \\phi) &=& \\frac{1}{2} \\left[ 1 + \\sin^2 \\theta \\cos^2\n\\phi+ e^{-4 \\kappa t} (\\cos^2 \\theta +\n     \\sin^2 \\theta \\sin^2 \\phi )\\right],\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-4 \\kappa t} \\right).\n\\end{eqnarray}\n\nFor ($L_{2,z}$,$L_{3,z}$,$L_{4,z}$,$L_{5,z}$,$L_{6,z}$) noise with\n$\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa_{6,z}=\\kappa$,\nthe infinitesimal evolution matrix is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\n\\frac{1}{2} \\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} +\n|1\\rangle ^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) +\n\\frac{1-10\\kappa \\delta t}{2} \\left(|0\\rangle ^{\\otimes 5} \\langle\n1| ^{\\otimes 5}+ |1\\rangle ^{\\otimes 5}\\langle 0 |^{\\otimes 5}\n\\right).\n\\end{eqnarray}\nUsing the ansatz\n\\begin{eqnarray}\n \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) =\na \\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} + |1\\rangle\n^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) + b\\left(|0\\rangle\n^{\\otimes 5} \\langle 1| ^{\\otimes 5}+ |1\\rangle ^{\\otimes 5}\\langle\n0 |^{\\otimes 5} \\right),\n\\end{eqnarray}\nwe obtain two coupled equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) =0,\\\\\n\\dot{b}(t) = -10kb(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=1\/2$, $b(0)=0$. Therefore, the density matrix reads\n\\begin{equation}\n\\label{zmatrix5} \\varepsilon(\\rho_{5\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 5} \\langle 0 |^{\\otimes 5} + |1\\rangle\n^{\\otimes 5}\\langle 1| ^{\\otimes 5}\\right) + \\frac{1}{2} e^{-10\n\\kappa t} \\left(|0\\rangle ^{\\otimes 5} \\langle 1| ^{\\otimes 5}+\n|1\\rangle ^{\\otimes 5}\\langle 0 |^{\\otimes 5} \\right),\n\\end{equation}\nand the fidelity and its average are given by\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  1 - \\frac{1}{2} \\left( 1 - e^{-10 \\kappa t}\\right) \\sin^2 \\theta ,\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-10 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\section{Six-qubit GHZ state with noisy channels}\\label{sec5}\nA quantum circuit for  teleportation through noisy channels with\n6GHZ state is depicted in Fig.~\\ref{fig5}. In the dotted box the\nLindblad operators act on the $64\\times64$ density matrix that\ninvolves five Alice's qubits and one Bob's qubits. The Lindblad\nequation, Eq.~(\\ref{Lindblad}), leads to 64 diagonal and 2016 off-diagonal\nlinear coupled differential equations. However, similar to previous\nsections, we first study infinitesimal temporal behavior of the\ndensity matrix and use a proper ansatz to considerably reduce the\nnumber of required equations.\n\n\n\\begin{figure}[t]\n\\input{fig4.tex}\n\\caption{\\label{fig5}\n   A circuit for quantum teleportation through noisy channels with 6GHZ state.\n   The six top lines belong to Alice and the bottom line to Bob.\n   $M$ denotes measurement and the dotted box represents noisy channel.\n   The Lindblad operator is turned on inside the dotted box.}\n\\end{figure}\n\n\nFor ($L_{2,x}$,$L_{3,x}$,$L_{4,x}$,$L_{5,x}$,$L_{6,x},L_{7,x}$)\nnoise and\n$\\kappa_{2,x}=\\kappa_{3,x}=\\kappa_{4,x}=\\kappa_{5,x}=\\kappa_{6,x}=\\kappa_{7,x}=\\kappa$,\nthe Lindblad operators after an infinitesimal time transform the\ninput density matrix\n$\\rho(0)=|6\\mbox{GHZ}\\rangle\\langle6\\mbox{GHZ}|$ to\n\\begin{eqnarray}\n\\hspace{-1cm}\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta\nt}= \\frac {1}{2}\\left(\\mbox{\\footnotesize$\n\\begin{smallmatrix}\n     1 - 6 \\kappa \\delta t  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  1 - 6 \\kappa \\delta t \\\\\n    0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n    0 & 0 & \\kappa \\delta t& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 \\\\\n    0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 &{\\footnotesize \\textcircled{\\tiny 14}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 7}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 3}} & 0 & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 & 0 & 0 \\\\\n    0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\\footnotesize \\textcircled{\\tiny 1}} & 0 & 0 & 0 \\\\\n    0 & 0 & \\kappa \\delta t& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 & 0 \\\\\n    0 & \\kappa \\delta t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\kappa \\delta t & 0 \\\\\n     1 - 6 \\kappa \\delta t  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  1 - 6 \\kappa \\delta t\n\\end{smallmatrix}$}\\right).\n\\end{eqnarray}\nSo consider\nthe ansatz\n\\begin{eqnarray}\n\\hspace{-2.3cm}\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})=\n\\left(\\mbox{\\footnotesize$\n\\begin{smallmatrix}\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{d}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 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\\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\tiny\\textcircled{\\emph{c}} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c & 0 & 0 & 0 \\\\\n0 & 0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 & 0 \\\\\n0 & b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b & 0 \\\\\na & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\\\\n\\end{smallmatrix}$}\\right),\n\\end{eqnarray}\nwhere $\\small\\textcircled{\\emph{c}}$ and $\\small\\textcircled{\\emph{d}}$ denote two diagonal $c$ and $d$, respectively. Substituting this matrix\ninto the Lindblad equation leads to four coupled equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) = 6k\\Big(b(t)-a(t)\\Big),\\\\\n\\dot{b}(t) = k\\Big(a(t)-6b(t)+5c(t)\\Big),\\\\\n\\dot{c}(t)=2k\\Big(b(t)-3c(t)+2d(t)\\Big),\\\\\n\\dot{d}(t)=-6k\\Big(c(t)-d(t)\\Big),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=1\/2$ and $b(0)=c(0)=d(0)=0$. Thus, the solutions\nread\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\na(t) =\\frac{1}{64}\\left( 1 + 15 e^{-4 \\kappa t} + 15 e^{-8 \\kappa t} + e^{-12 \\kappa t}\\right),\\\\\nb(t) =\\frac{1}{64}\\left( 1 +5 e^{-4 \\kappa t} - 5 e^{-8 \\kappa t} - e^{-12 \\kappa t}\\right),\\\\\nc(t) =\\frac{1}{64}\\left( 1 -  e^{-4 \\kappa t} -  e^{-8 \\kappa t} +\ne^{-12 \\kappa\nt}\\right),\\\\\nd(t) =\\frac{1}{64}\\left( 1 - 3 e^{-4 \\kappa t} + 3 e^{-8 \\kappa t}-\ne^{-12 \\kappa t}\\right),\n\\end{array}\\right.\n\\end{eqnarray}\nand finally\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  \\frac{1}{2} \\left[ 1 + \\sin^2 \\theta \\cos^2\n\\phi+ e^{-4 \\kappa t} (\\cos^2 \\theta +\n     \\sin^2 \\theta \\sin^2 \\phi )\\right],\n   \\\\ \\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-4 \\kappa t}\n\\right).\n\\end{eqnarray}\n\n\nFor the last case, we study\n($L_{2,z}$,$L_{3,z}$,$L_{4,z}$,$L_{5,z}$,$L_{6,z},L_{7,z}$) noise\nwith\n$\\kappa_{2,z}=\\kappa_{3,z}=\\kappa_{4,z}=\\kappa_{5,z}=\\kappa_{6,z}=\\kappa_{7,z}=\\kappa$.\nFor this case, the temporal evolution matrix is\n\\begin{eqnarray}\n\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}})\\Big|_{t=\\delta t} =\\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 6} \\langle 0 |^{\\otimes 6} + |1\\rangle\n^{\\otimes 6}\\langle 1| ^{\\otimes 6}\\right) + \\frac{1-12\\kappa\\delta\nt}{2} \\left(|0\\rangle ^{\\otimes 6} \\langle 1| ^{\\otimes 6}+\n|1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{eqnarray}\nTherefore, using the ansatz\n\\begin{eqnarray}\n\\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}}) = a \\left( |0\\rangle^{\\otimes\n6} \\langle 0 |^{\\otimes 6} + |1\\rangle ^{\\otimes 6}\\langle 1|\n^{\\otimes 6}\\right) + b\\left(|0\\rangle ^{\\otimes 6} \\langle 1|\n^{\\otimes 6}+ |1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{eqnarray}\nwe obtain two simple differential equations\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\dot{a}(t) =0,\\\\\n\\dot{b}(t) = -12kb(t),\\\\\n\\end{array}\\right.\n\\end{eqnarray}\nsubject to $a(0)=b(0)=1\/2$. So the solution is given by\n\\begin{equation}\n\\label{zmatrix6} \\varepsilon(\\rho_{6\\mbox{\\tiny GHZ}}) = \\frac{1}{2}\n\\left( |0\\rangle^{\\otimes 6} \\langle 0 |^{\\otimes 6} + |1\\rangle\n^{\\otimes 6}\\langle 1| ^{\\otimes 6}\\right) + \\frac{1}{2} e^{-12\n\\kappa t} \\left(|0\\rangle ^{\\otimes 6} \\langle 1| ^{\\otimes 6}+\n|1\\rangle ^{\\otimes 6}\\langle 0 |^{\\otimes 6} \\right),\n\\end{equation}\nand the fidelity and its average read\n\\begin{eqnarray}\nF(\\theta, \\phi) &=&  1 - \\frac{1}{2} \\left( 1 - e^{-12 \\kappa t}\\right) \\sin^2 \\theta ,\\\\\n\\overline{F} &=& \\frac{1}{3} \\left( 2 + e^ {-12 \\kappa t} \\right).\n\\end{eqnarray}\n\n\\section{Conclusions}\\label{sec6}\nIn this paper, we studied quantum teleportation through noisy\nchannels for $n$GHZ states, $n\\in\\{4,5,6\\}$), so that the noisy\nchannels lead to the quantum channels to be mixed states. We\nexactly solved the Lindblad equation and obtained corresponding\ndensity matrices after the transmission process. The Lindblad\noperators are responsible for the decoherence of quantum states and\nare defined to be proportional to the Pauli matrices. Solving the\nLindblad equation for $n>2$ is not a trivial task in general. For\ninstance, we need to solve 2080 coupled differential equations to\nfind the density matrix for 6GHZ state. We overcame this problem by\nstudying the temporal evolution of the input state and using a\nproper ansatz for the density matrix. Therefore, we reduced 2080\ncoupled equations to at most four coupled equations which are\nreadily solved. We found the fidelity and the average fidelity for\nvarious cases and showed that for the Lindblad operators\ncorresponding to $x$ direction the fidelity is the same for EPR and\n$n$GHZ states where $n\\in\\{3,4,5,6\\}$. However, 3GHZ state does lose\nless quantum information for other types of noisy channel. Note\nthat, In Ref.~\\cite{jung08-2} the authors only studied the same-axis\nnoisy channels and conjectured that ``average fidelity with\nsame-axis noisy channels are in general larger than average fidelity\nwith different-axis noisy channels''. However, we showed the failure\nof this conjecture for 4GHZ state which is apparent in\nFig.~\\ref{fig7}. In the appendix we showed this conjecture also\nfails for 3GHZ state (see Fig.~\\ref{fig6}). In fact, for\ndifferent-axes noises, the analytical solutions can be obtained in\nthe same way, but the number of coupled differential equations\nusually increases with respect to the same-axes noises.\n\n\\acknowledgments\nWe would like to thank Robabeh Rahimi for fruitful discussions and\nsuggestions and for a critical reading of the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSince the seminal work of \\citet{Coo95}, \\textit{statistical shape models} became an emerging tool to capture natural shape variability within a given class of objects.\nAs a result, a large number of shape models were developed during the last decades.\nThe probably most well-known models were built for the human face using textured 3D face scans.  \nIntroduced by \\citet{Bla99}, this class of statistical shape models is commonly known as \\textit{3D Morphable Models} (3DMMs) and includes models such as the \\textit{Basel Face Model} (BFM) and \\textit{Large Scale Facial Model} (LSFM) presented by  \\citet{Pay09} and \\citet{Boo16}, respectively.\nWell-studied applications of 3DMMs include face recognition, expression transfer between individuals, face animation, and 3D face reconstruction from a single 2D photograph \\citep{Egg20}.\nEspecially in the last decade, shape analysis also gained popularity in the field of computational anatomy, where statistical shape models are successfully used to model variations of anatomical objects such as bones and organs. \nLater, these models are utilized for a variety of medical applications including (but not limited to) image segmentation, surgical simulation, therapy planning, and motion analysis \\citep{Amb19}.\nDespite the popularity of statistical shape models in the aforementioned areas, to date and to the best of our knowledge, no publicly available 3D statistical shape model of the female breast exists.\n\nWith breast cancer being the most common malignant neoplasm among women \\citep{Sun21}, successful \\textit{breast reconstruction surgery} (BRS) is crucial for patients undergoing mastectomy.\nIn order to give patients a first impression about what their breast might look like after BRS, surgical outcomes are more and more often simulated using patient-specific 3D breast scans, acquired in a standing position (see Section \\ref{sec:related_work} for an overview). \nTypically, simulations are performed using physically motivated deformable models of the breast.\nWhile these models take into account material properties and physical effects such as gravity, they may \\textit{not always} produce realistic-looking shapes as no prior knowledge in form of example shapes is included \\citep{Roo06,Su11}.\nHence, simulated outcomes might be physically plausible, but definitely lack \\textit{statistical plausibility} in the sense that generated breast shapes are somehow \\textit{likely} or similar to those typically observed within a target population.\nAs the ultimate goal of BRS is an outcome looking as \\textit{natural} as possible, we believe that simulation of surgical outcomes must not only rely on physically based deformable models but also should take into account statistical effects.\nIndeed, the phenomenon that humans tend to compare themselves with others clearly underlines the importance of the fact that simulated breast shapes should look similar to the breasts within the target population.\nAs a first step towards combining physical \\textit{and} statistical plausibility of simulated breast surgery outcomes, we propose to use 3D statistical shape models built from natural-looking female breasts.\nIn addition, by introducing such models into the breast shape domain, many of the aforementioned applications from other domains could be transferred to the breast as we will exemplary show later in this article.\n\nTo this end, this paper introduces the first publicly available 3D statistical shape model of the female breast built from 110 breast scans.\nTogether with the model, we present a fully automated, pairwise registration pipeline especially tailored for 3D breast scans and its application in the context of statistical shape modeling.\nOur method is computationally efficient and requires only four landmarks to guide the registration process.\n\n\\subsection{Challenges}\nCompared to shape modeling of most parts of the human body, building a statistical shape model for the female breast imposes some new challenges as discussed in the following.\n\n\\textbf{Data acquisition.}\nFirstly, acquiring a sufficient amount of high-quality training data is challenging.\nUsually, 3D scanning and \\textit{manual} landmark detection is an uncomfortable situation for the participants, in which their upper body is required to be naked.\nMoreover, landmarks can be identified only through palpation and by using a regular tape measure, both requiring a physical examination in a clinical environment.\nIn addition, during the \\textit{whole} examination, a specified posture needs to be held fixed ensuring a similar pose across all subjects.\nThis can be very exhausting, especially if 3D breast scans are taken in a standing position in which both arms should ideally be held away from the body in order to capture the breast as isolated as possible.\nAs a result, 3D scanning protocols used in clinical practice are often designed to be carried out relatively fast, thus lacking necessary precision for pose standardization, see Figure \\ref{fig:database_samples}.\nNote that the problem of only \\textit{quasi-similar} postures was also recently observed by \\citet{Maz21}.\nAll in all, the aforementioned factors definitely hinder the implementation of large-scale, high-quality data surveys.\n\n\\begin{figure}[b]\n\\includegraphics{.\/images\/1_database_samples}\n\\caption{Three typical 3D breast scans sampled from our database.\nAlthough a common pose was declared during data acquisition, a lot of pose variations are still present (indicated through a skeleton drawn in red). \nThese mainly emerge from the arms and shoulders.}\n\\label{fig:database_samples}\n\\end{figure}\n\n\\begin{figure}[t]\n\\includegraphics[width=88.5mm]{.\/images\/2_anatomy}\n\\caption{A brief overview of common landmarks and key anatomical structures of the female breast and thorax. \nThe illustration was adapted from \\citet{Hwa15}.}\n\\label{fig:anatomy}\n\\end{figure}\n\n\\textbf{Correspondence estimation.}\nSecondly, establishing correspondence among 3D breast scans by means of surface registration (rigid and non-rigid) is difficult due to the lack of reliable landmarks.\nIn essence, only four valid landmarks can be used for non-rigid registration.\nThese include both nipples as well as both lower breast poles (throughout this work defined as \\textit{the lowest (most caudal) point of the breast}, see Figure \\ref{fig:anatomy}). \nAnatomical landmark points outside the breast region (such as the sternal notch or processus coracoideus) cannot be used for non-rigid registration purposes as they would recover undesired shape variations during statistical shape modeling.\nThis way, the processus coracoideus disqualifies as its position depends strongly on the position of the arms and shoulders.\nOn the other hand, the xiphoid, located at the center of the thorax, \\textit{could} technically be used for registration. \nHowever, it cannot be reliably and consistently located across a wide range of differently-shaped breasts because of its dorsal tilt, effectively hindering the identification of a unique point.\nAnother fundamental problem is the complete lack of robust landmarks on the lateral part of the breast. \nAlthough \\citet{Har20} defined a lateral breast pole as the orthogonal intersection between the anterior axillary line and a line passing through the nipple, this point cannot be used for non-rigid registration as its position is also affected by the pose of the arms and shoulders.\nFinally, already the initial, rigid alignment of 3D breast scans is challenging due to the lack of reliable landmarks that do not undergo large soft tissue deformations.\n\n\\textbf{Non-separability of breast and thorax.} \nLast but most importantly, the region of interest, i.e. the breast, obviously cannot be well separated from the rest of the thorax when considering 3D breast scans.\nThis is primarily because the chest wall separating the breast from the thorax cannot be captured using 3D surface scanning devices, but also due to the reason that no commonly accepted and \\textit{exact} definition of the breast contour exists, see e.g. \\citet{Lot21}. \nA statistical shape model built from 3D breast scans will therefore \\textit{necessarily} capture also shape variations of the thorax, even after pose standardization.\nIn particular, these include morphological shape variations of the underlying chest wall and upper abdomen, but also those emerging from the arms, armpits, and shoulders due to improper pose standardization (also seen from Figure \\ref{fig:database_samples}).\nIf not reduced to a minimum, this will cause the following unwanted effect.\nSince breast and thorax shapes are tightly correlated in the subspace spanned by the model, the range of representable breast shapes is limited.\nThe reason is that a considerably large part of the subspace accounts for the unwanted shape variations of the surrounding regions.\nHence, the breast area should be decoupled from the thorax as much as possible in order to build an expressive and well-performing statistical shape model.\n\n\\subsection{Contributions}\nAs key contribution, this work presents the \\textit{Regensburg Breast Shape Model} (RBSM) -- a 3D statistical shape model of the female breast.\nIn order to weaken the strong coupling between the breast and surrounding regions, we propose to minimize the \\textit{variance} outside the breast region as much as possible.\nTo achieve this goal, a novel concept called \\textit{breast probability masks} (BPMs) is introduced. \nA BPM assigns probabilities to each point of a 3D breast scan, telling how \\textit{likely} it is that a particular point belongs to the breast region. \nLater, during pairwise registration, we use the BPMs to align the template to the target as accurately as possible \\textit{inside} the breast region and only roughly outside. \nThis way, only the most prominent and global shape variations outside the breast region will be recovered, effectively reducing the unwanted variance in these areas to a minimum. Figure \\ref{fig:overview_approach} illustrates this idea.\n\nTo summarize, the contributions of this paper are three-fold:\n\\begin{itemize}\n\\item We introduce the \\textit{Regensburg Breast Shape Model} (RBSM) -- an open-access 3D statistical shape model of the female breast built from 110 breast scans. It is available at \\url{https:\/\/www.rbsm.re-mic.de\/}.\n\\item We propose a fully automated, pairwise registration pipeline used to establish correspondence among our 3D breast scans. It uses BPMs to decouple the breast from the surrounding regions as much as possible and requires only four landmarks to guide the registration process.\n\\item We present two exemplary applications demonstrating how the RBSM can be used for surgical outcome simulation and the prediction of a missing breast from the remaining one.\n\\end{itemize}\nThe remainder of this paper is organized as follows: Section \\ref{sec:related_work} briefly reviews some related work. \nSection \\ref{sec:methodology} describes the entire model building pipeline used to construct the RBSM. \nIn particular, it formally introduces the notion of a BPM, followed by a detailed description of the proposed registration pipeline.\nSection \\ref{sec:evaluation} presents an extensive evaluation of the RBSM in terms of the common metrics compactness, generalization, and specificity. \nUsing the RBSM, two exemplary applications are showcased in Section \\ref{sec:applications}.\nFinally, Section \\ref{sec:discussion} discusses the results whereas Section \\ref{sec:conclusion} concludes this article.\n\n\n\\section{Related work}\n\\label{sec:related_work}\nIn this section, we briefly summarize some related work concerning breast surgery simulation, statistical shape models of the female breast, and popular techniques for pairwise surface registration used within the context of statistical shape modeling in general.  \nNote that we have limited our review to the 3D case. \n\n\\textbf{Breast surgery simulation.}\nMost of the existing methods for pre-operative breast surgery simulation are designed to simulate alloplastic, implant-based breast augmentation procedures, either for aesthetic reasons or after mastectomy as part of BRS.\nTypically, those methods first generate a patient-specific, geometric representation of the breast (using tetrahedral meshes, for instance). \nAfterwards, the soft tissue deformation caused by implant insertion is simulated using a geometric and biomechanical model of the implant and breast, respectively.\n\n\\begin{figure}[t]\n\\includegraphics[]{.\/images\/3_overview_approach}\n\\caption{The proposed concept of BPMs (top row) allows to minimize unwanted shape variations of the thorax by registering a template surface as precisely as possible \\textit{inside} the breast region and only roughly outside. \nThe breast region is defined by BPMs in a probabilistic manner (top left, red areas correspond to a high probability of belonging to the breast region). \nThis simple yet effective strategy decouples the breast from the surrounding regions by reducing the variance outside the breast area. \nIn the last column, the per-vertex variance over the whole dataset is visualized on the resulting mean shape. \nThe regions showing the highest variance (red) are almost coincident with the breasts in our proposed BPM-based approach.\nContrary, without BPMs (bottom row), a lot of unwanted variance is present in the surrounding regions, especially around the arms, armpits, shoulders, and upper abdomen. \nThis implies a strong coupling between breast and thorax in the final statistical shape model.}\n\\label{fig:overview_approach}\n\\end{figure}\n\nAs such, \\citet{Roo06} used the tensor-mass model introduced by \\citet{Cot00}, a combination of classical finite element and mass-spring models, for implant-based breast augmentation planning.\n\\citet{Cie12} proposed a web-based tool for breast augmentation planning which requires only 2D photographs and anthropometric measurements as input and allows the user to choose from a variety of different implants. \nTheir method automatically reconstructs a 3D breast model and subsequently applies a tissue elastic model closely resembling the finite element model.\n\\citet{Geo14} utilizes patient-specific finite element models generated from 3D breast scans. \nCombined with a novel mechanism called \\textit{displacement template}, geometric implant models are no longer required, thus breaking up the coupling between implant and enclosing breast.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics{.\/images\/4_overview_pipeline}\n\\caption{An overview of the pipeline used to build the RBSM. \nWe first establish dense correspondence by means of pairwise surface registration. \nBased on BPMs, our method starts by rigidly aligning the template to the target. \nSubsequently, a non-rigid alignment is applied in a hierarchical, multi-resolution fashion. \nIn this step, BPMs are used to precisely recover only shape variations of the breast.\nFinally, we perform classical \\textit{Generalized Procrustes Analysis} (GPA) and \\textit{Principal Component Analysis} (PCA) to build the model.}\n\\label{fig:overview_pipeline}\n\\end{figure*}\n\nBesides the simulation of breast augmentation procedures using implants, some methods exist especially addressing the simulation of BRS \\textit{without} implant insertion, for example by means of autologous fat tissue.\nBased on Pascal's principle and volume conservation, \\citet{Cos01} developed a novel approach for real-time physically based simulation of deformable objects, called \\textit{Long Elements Method} (LEM).\nAn extension of LEM, known as the \\textit{Radial Elements Method} \\citep{Bal03}, was later used by \\citet{Bal06} for cosmetic and reconstructive breast surgery simulation.\n\\citet{Wil03} employ a finite element approach incorporating the Mooney-Rivlin hyperelastic material model for the realistic simulation of soft tissue to simulate \\textit{transverse rectus abdominis myocutaneous} flap breast reconstruction.\n\nFor most of the existing methods, however, various authors highlighted the disagreement between simulated and actual outcomes \\citep{Cru15,Roo06} or only partially satisfying results for certain types of breast shapes \\citep{Vor17}.\n\n\\textbf{Statistical shape models of the female breast.}\nLiterature about 3D statistical shape models of the female breast is sparse.\nIn the early work of \\citet{Seo07}, a 3D statistical shape model was built from 28 breast scans with the goal of analyzing breast volume and surface measurements.\nHowever, they assume symmetric breasts obtained by simply mirroring the right breast.\nTo date and to the best of our knowledge, this is the only work primarily addressing the construction of a statistical shape model from 3D breast scans, which is thus closest to our work.\nBesides, \\citet{Rui18} utilized a 3D statistical shape model built from 310 breast scans for the validation of a novel weighted regularized projection method used for 3D reconstruction.\nAs their focus did not lie on the construction of a well-performing statistical shape model of the breast, they did not provide detailed information about the registration method used to establish correspondence, the training data nor a comprehensive evaluation in terms of common metrics.\n\n\\textbf{Pairwise surface registration.}\nDuring the last few decades, countless algorithms were proposed tackling the problem of (pairwise) non-rigid surface registration.\nTo date, however, none of them were used for 3D breast scan registration.\n\nOne of the most widely used methods is the \\textit{Optimal Step Non-rigid Iterative Closest Point} (NICP) framework proposed by \\citet{Amb07} and based on the work of \\citet{All03}. \nNICP is the only method already used in the female breast shape domain to reconstruct a 3D breast model from a sequence of depth images \\citep{Lac19}.\nMoreover, NICP was employed to construct the BFM and LSFM.\nA second class of non-rigid registration methods is based on splines, such as \\textit{Thin Plate Splines} (TPS) or B-splines.\nPioneered by \\citet{Boo89}, TPS were utilized by \\citet{Pau02} to construct a statistical shape model of the human ear canal.\nB-splines are extensively used in the \\textit{free-form deformations} framework and, among others, used for the creation of shape models of the human heart \\citep{Ord07}.\nA third, recently introduced framework is based on \\textit{Gaussian Process Morphable Models} (GPMMs) introduced by \\citet{Lue18}.\nGPMMs are statistical shape models themselves generalizing classical point-based models as proposed by \\citet{Coo95}.\nBy means of GPMMs, expected deformations can be modeled using analytic covariance functions and later used as a \\textit{prior} for non-rigid surface registration, thus effectively reducing the search space. \nThis approach was successfully used by \\citet{Ger17} to build an improved version of the BFM.\nLastly, pairwise non-rigid surface registration algorithms are built upon the \\textit{as-rigid-as-possible} (ARAP) real-time mesh deformation framework proposed by \\citet{Sor07}.\nThis method was successfully transferred to the registration domain, yielding to similar methods constraining deformations to be \\textit{as conformal as possible} \\citep{Yos14} or \\textit{as similar as possible} \\citep{Jia17,Yam13}. \nRecently, a variant of ARAP was utilized by \\citet{Dai20} to built a shape model of the full human head. \n\n\\section{Methodology}\n\\label{sec:methodology}\nThis section describes the entire pipeline used to build the RBSM. \nAs outlined in Figure \\ref{fig:overview_pipeline}, we start by establishing correspondence among our training data.\nBased on a novel concept called \\textit{breast probability masks} (Section \\ref{subsec:bpms}), this is achieved by means of a fully automated, pairwise registration pipeline as proposed in Section \\ref{subsec:registration}.\nFinally, we follow the typical workflow used to build a point-based statistical shape model by applying a \\textit{Generalized Procrustes Analysis} and a \\textit{Principal Component Analysis} to the registered data set (briefly summarized in Section \\ref{subsec:model_building}).\n\nIn what follows, 3D breast scans are represented using triangular surface meshes.\nA triangle mesh $\\mathcal{M}=(V,E,\\mathcal{P})$ is fully specified by a set of $n$ vertices $V\\subset\\mathbb{N}$, edges $E\\subset V\\times V$, and an embedding $\\mathcal{P}=\\{\\mathbf{p}_1,\\mathbf{p}_2\\ldots,\\mathbf{p}_n\\}\\subset\\mathbb{R}^3$. \nSometimes, however, instead of arranging points $\\mathbf{p}_i$ into a set, it is more convenient to use a matrix notation $\\mathbf{P}=(\\mathbf{p}_1,\\mathbf{p}_2,\\ldots,\\mathbf{p}_n)^\\top\\in\\mathbb{R}^{n\\times 3}$.\nHence, we will denote a triangle mesh either as $\\mathcal{M}=(V,E,\\mathcal{P})$ or equivalently as $\\mathcal{M}=(V,E,\\mathbf{P})$.\n\n\\subsection{Breast probability masks}\n\\label{subsec:bpms}\nGiven a 3D breast scan represented as triangle mesh $\\mathcal{M}=(V,E,\\mathcal{P})$, we call\n\\begin{equation}\np_\\mathcal{M}:\\mathcal{P}\\longrightarrow(0,1]\n\\end{equation}\na \\textit{breast probability mask} (BPM). Technically, a BPM is a scalar field defined over $\\mathcal{M}$ assigning each point $\\mathbf{p}_i$ of a 3D breast scan a probability $p_\\mathcal{M}(\\mathbf{p}_i)$ telling how \\textit{likely} it is that $\\mathbf{p}_i$ belongs to the breast region.\n\n\\textbf{Concrete mapping.} As a concrete mapping for $p_\\mathcal{M}$ we propose to use a normalized sum of \\textit{elliptical basis functions} (EBFs), centered at the nipples.\nWe use EBFs instead of ordinary \\textit{radial basis functions} (RBFs) because we found that they better capture the natural teardrop shape of the breast (see Figure \\ref{fig:comparison_bpms} for a comparison between RBFs and EBFs).\nTechnically, EBFs are a generalization of RBFs using the Mahalanobis distance instead of an ordinary vector norm.\nFormally, an EBF $\\phi:[0,\\infty)\\longrightarrow\\mathbb{R}$ centered at a point $\\mathbf{c}\\in\\mathbb{R}^n$ is of the form $\\phi(\\mathbf{x})=\\phi\\left(d_M\\left(\\mathbf{x},\\mathbf{c}\\right)\\right)$. \nHere, $d_M$ is the Mahalanobis distance, defined as\n\\begin{equation}\nd_M(\\mathbf{x},\\mathbf{c})\\coloneqq\\sqrt{\\left(\\mathbf{x}-\\mathbf{c}\\right)^\\top\\mathbf{S}^{-1}\\left(\\mathbf{x}-\\mathbf{c}\\right)}\\,,\n\\end{equation}\nwhere $\\mathbf{S}\\in\\mathbb{R}^{n\\times n}$ is a symmetric positive definite matrix, also called \\textit{covariance matrix}.\nTo stress that the Mahalanobis distance depends on $\\mathbf{S}$, we write $d_M(\\mathbf{x},\\mathbf{c};\\mathbf{S})$ in the following.\n\nNow, in order to define a concrete BPM using EBFs, let $\\mathbf{p}_\\text{N}^\\tau\\in\\mathcal{P}$ denote the position of the left (L) and right (R) nipple, respectively, and $\\tau\\in\\{\\text{L, R}\\}$.\nWe first construct two individual probability masks for the left and the right breast, given as\n\\begin{equation}\np_\\mathcal{M}^\\tau(\\mathbf{p}_i)=\\phi\\left(d_M\\left(\\mathbf{p}_i,\\mathbf{p}_\\text{N}^\\tau;\\mathbf{S}_\\tau\\right)\\right).\n\\end{equation}\nHereby, we define $\\phi:[0,\\infty)\\longrightarrow(0,1]$ as \n\\begin{equation}\n\\phi(x)=\\exp\\left(-x^2\\right).\n\\end{equation}\nFinally, the BPM for a whole 3D breast scan is given as the normalized sum\n\\begin{equation}\np_\\mathcal{M}(\\mathbf{p}_i)=\\frac{1}{4}\\left(p_\\mathcal{M}^\\text{L}(\\mathbf{p}_i)+\\hat{p}_\\mathcal{M}^\\text{L}(\\mathbf{p}_i)+p_\\mathcal{M}^\\text{R}(\\mathbf{p}_i)+\\hat{p}_\\mathcal{M}^\\text{R}(\\mathbf{p}_i)\\right),\n\\end{equation}\nwhere \n\\begin{equation}\n\\hat{p}_\\mathcal{M}^\\tau(\\mathbf{p}_i)=\\phi\\left(d_M\\left(\\mathbf{p}_i,\\mathbf{\\hat{p}}_\\text{N}^\\tau;\\mathbf{\\hat{S}}_\\tau\\right)\\right)\n\\end{equation}\nare shifted BPMs of the left and right breast added to better mimic the teardrop shape, and $\\mathbf{\\hat{p}}_\\text{N}^\\tau=\\mathbf{p}_\\text{N}^\\tau+\\mathbf{t}_\\tau$ with translation vectors $\\mathbf{t}_\\tau\\in\\mathbb{R}^3$.\n\n\\begin{figure}[h!]\n\\includegraphics[]{.\/images\/5_comparison_bpms}\n\\caption{From left to right: comparison between RBF, EBF, and a sum of two EBFs, illustrated as contour plots. \nWhile a simple RBF or EBF is not able to accurately mimic the typical teardrop shape of the breast, a sum of two EBFs comes close.}\n\\label{fig:comparison_bpms}\n\\end{figure}\n\n\\textbf{Parameter selection.} In order to fully define a BPM, appropriate matrices $\\mathbf{S}_\\tau,\\mathbf{\\hat{S}}_\\tau\\in\\mathbb{R}^{3\\times 3}$ and translation vectors $\\mathbf{t}_\\tau\\in\\mathbb{R}^3$ need to be chosen first.\nAs such, a total of 30 values are required to be properly determined (six for each $\\mathbf{S}_\\tau$ and $\\mathbf{\\hat{S}}_\\tau$, and three for each $\\mathbf{t}_\\tau$). \nTo simplify that task, we assume diagonal covariance matrices and utilize previously marked landmarks on the 3D breast scans.\nSpecifically, denote the landmark points shown in Figure \\ref{fig:anatomy} as $\\mathbf{p}_\\text{SN},\\mathbf{p}_\\text{XI}\\in\\mathcal{P}$ for sternal notch and xiphoid, and $\\mathbf{p}_\\text{LaBP}^\\tau,\\mathbf{p}_\\text{LBP}^\\tau\\in\\mathcal{P}$ for left and right lateral and lower breast pole, respectively.\nWe then define\n\\begin{equation}\n\\begin{split}\n\\mathbf{S}_\\tau&=\n\\!\\begin{multlined}[t][60mm]\n\\frac{1}{2}\\diag\\Bigl(d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)+d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{XI}\\right),\\\\\nd_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}^\\tau_\\text{LBP}\\right), d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)\\Bigr)\\,,\n\\end{multlined}\\\\\n\\mathbf{\\hat{S}}_\\tau&=\n\\!\\begin{multlined}[t][60mm]\n\\frac{1}{2}\\diag\\Bigl(d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)+d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{XI}\\right),\\\\\nd_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{SN}\\right),d_G\\left(\\mathbf{p}^\\tau_\\text{LaBP},\\mathbf{p}^\\tau_\\text{N}\\right)\\Bigr)\\,,\n\\end{multlined}\\\\\n\\mathbf{t}_\\tau&=\\mathbf{p}^\\tau_\\text{N}+\\frac{1}{5}\\left(0,d_G\\left(\\mathbf{p}^\\tau_\\text{N},\\mathbf{p}_\\text{SN}\\right),0\\right),\n\\end{split}\n\\end{equation}\nwhere $d_G$ denotes the Geodesic distance between two points on the surface mesh.\nNote that $\\mathbf{S}_\\tau$ and $\\mathbf{\\hat{S}}_\\tau$ differ only in the second diagonal element.\n\n\\subsection{Registration of 3D breast scans}\n\\label{subsec:registration}\nFollowing Figure \\ref{fig:overview_pipeline}, the proposed pairwise registration pipeline is mainly composed of rigid alignment (Section \\ref{subsubsec:rigid}) and non-rigid alignment (Section \\ref{subsubsec:non-rigid}). \nFollowing common practice, the latter is carried out in a hierarchical, multi-resolution fashion (Section \\ref{subsubsec:multi_res_fitting}).\n\nBoth phases make extensive use of BPMs in order to align a template surface $\\mathcal{S}=(V,E,\\mathbf{P})$ to a target $\\mathcal{T}$ as accurately as possible \\textit{inside} the breast region and only roughly outside, effectively decoupling the breast from the rest of the thorax by reducing the \\textit{variance} outside the breast region to a minimum.\nThis is justified as the covariance $\\cov(x,y)$ becomes smaller if $\\var(x)$ or $\\var(y)$ is lowered, following from the well known fact that $\\left|\\cov(x,y)\\right|\\leq\\sqrt{\\var(x)}\\sqrt{\\var(y)}$ (which holds via the Cauchy\u2013Schwarz inequality).\n\nFinally, note that the target surface $\\mathcal{T}$ can be given in any representation that allows for closest point search.\nWe use a triangular surface mesh but write $\\mathcal{T}\\subset\\mathbb{R}^3$ for the sake of notational simplicity.\n\n\\subsubsection{Rigid alignment}\n\\label{subsubsec:rigid}\nThe overall goal of the rigid alignment is to move the template as close as possible to the \\textit{rigid part} of the target, which we define as \\textit{the thorax without the breast}.\nIn particular, we expect that the thoraxes of two subjects without the breast region can be sufficiently well aligned if we assume the breast to be the only part of the thorax that deforms non-rigidly.\nBased on this assumption, the absence of suitable landmarks, and due to the fact that our initial 3D breast scans are already reasonably well aligned (see Section \\ref{subsec:data}) we propose a modified version of the \\textit{ Iterative Closest Point} (ICP) algorithm, originally introduced by \\citet{Bes92}.\n\nEssentially, compared to the standard version of the ICP algorithm, our modified version differs in the following three aspects: (i) a scaling factor is added to the rigid transformation effectively allowing for Euclidean similarity transformations \\citep{Du07,Zin05}. \nSecondly, (ii) to ensure that only the rigid parts of the 3D breast scans are used for alignment, correspondences, where both points have a high probability belonging to the breast region, are discarded.\nThis is implemented by thresholding the template and target BPMs.\nFinally, (iii) rotations are restricted to the $x$-axis corresponding to the transversal plane.\nRotations around the $y$- and $z$-axis (sagittal and coronal plane), possibly introduced due to severe overweight in conjunction with an uneven distribution of abdominal fat could destroy the initial alignment and lead to misalignment.\nIn any case, asymmetries introduced due to the thorax should \\textit{not} affect the rigid alignment of the template.\n\n\\subsubsection{Non-rigid alignment}\n\\label{subsubsec:non-rigid}\nGiven the rigidly aligned template $\\mathcal{S}=(V,E,\\mathbf{P})$, the goal of the non-rigid alignment is to gradually deform $\\mathcal{S}$ into a new surface $\\mathcal{S}'=(V,E,\\mathbf{P}')$ with identical topology such that $\\mathcal{S}'$ is as close as possible to the target $\\mathcal{T}$ \\textit{inside} the breast region.\nFollowing various authors including \\citet{Jia17} and \\citet{Yam13}, we formulate our non-rigid registration problem using the following non-linear energy functional\n\\begin{equation}\nF\\left(\\mathbf{P}'\\right)=F_D\\left(\\mathbf{P}'\\right)+\\alpha F_R\\left(\\mathbf{P}'\\right)+\\beta F_L\\left(\\mathbf{P}'\\right),\n\\label{eq:non_rigid_cost}\n\\end{equation}\nwhere $F_D$ is a distance term used to penalize the point-to-point distance between the template and target surface, $F_R$ is a regularization term constraining deformations \\textit{as similar as possible}, and $F_L$ constitutes a landmark term ensuring certain points to be matched.\n$\\alpha,\\beta\\geq 0$ are weights controlling the individual contribution of each term to the cost function.\nMinimizing $F$ finally leads to the new points $\\mathbf{P}'$ of the deformed template surface $\\mathcal{S}'$, i.e.\n\\begin{equation}\n\\mathbf{P}'=\\argmin_{\\mathbf{P'}\\in\\mathbb{R}^{n\\times 3}}F(\\mathbf{P'}).\n\\label{eq:non_rigid_opt_problem}\n\\end{equation}\nAdapting the strategy proposed by \\citet{All03}, instead of computing (\\ref{eq:non_rigid_opt_problem}) only once, we minimize $F$ several times but each time lowering the regularization weight $\\alpha$ in (\\ref{eq:non_rigid_cost}).\nAs later demonstrated by \\citet{Amb07}, this strategy is able to recover the whole range of global and local non-rigid deformations efficiently.\nFollowing various authors \\citep{Jia17,Sor07}, the optimization problem in (\\ref{eq:non_rigid_opt_problem}) is solved using an alternating minimization (AM) approach as briefly summarized in \\ref{app:solve_am}.\n\n\\textbf{Distance term.}\nThe distance term $F_D$ is used to attract the template $\\mathcal{S}$ to the target $\\mathcal{T}$.\nAssuming fixed correspondences between both surfaces, i.e. $\\left\\{(\\mathbf{p}_1,\\mathbf{q}_1),(\\mathbf{p}_2,\\mathbf{q}_2),\\ldots,(\\mathbf{p}_n,\\mathbf{q}_n)\\right\\}$ with $\\mathbf{q}_i\\in\\mathcal{T}$ being the closest point to $\\mathbf{p}_i$, the distance term can be written as\n\\begin{equation}\nF_D(\\mathbf{P}')=\\frac{1}{2}\\left\\Vert\\mathbf{C}\\left(\\mathbf{P}'-\\mathbf{Q}\\right)\\right\\Vert^2_F,\n\\end{equation}\nwhere $\\mathbf{C}\\coloneqq\\text{diag}(c_1,c_2,\\ldots,c_n)$, $c_i\\geq 0$ for all $i\\in\\left\\{1,2,\\ldots,n\\right\\}$ are weights used to quantify the reliability of a match, and $\\mathbf{Q}\\coloneqq\\left(\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_n\\right)^\\top\\in\\mathbb{R}^{n\\times 3}$.\nUsing the BPMs $p_\\mathcal{S}$ and $p_\\mathcal{T}$ of the template and target, we set\n\\begin{equation}\nc_i=\\frac{p_\\mathcal{S}(\\mathbf{p}_i)+p_\\mathcal{T}(\\mathbf{q}_i)}{2}.\n\\end{equation}\nThis way, correspondences $(\\mathbf{p}_i,\\mathbf{q}_i)$ mapping from one breast region to the other have a greater impact on the overall distance term as $c_i\\in(0,1]$ becomes large in this case.\nConversely, the influence tends to zero if $c_i\\rightarrow 0$, i.e. if both points are less likely to belong to the breast region.\nAs such, the deformation of points $\\mathbf{p}_i$ on the template with a small value for $c_i$ is mainly controlled by the regularization term, as previously described by \\citet{All03}.\n\n\\textbf{Regularization term.}\nThe regularization term $F_R$ should prevent the template surface from shearing and distortion while simultaneously ensuring structure preservation and smooth deformations.\nTo do so, we adapt the \\textit{consistent as-similar-as-possible} (CASAP) regularization technique in which deformations are constrained to be \\textit{locally} as similar as possible \\citep{Jia17,Yam13}.\nSpecifically, given a local neighborhood $E_i\\subset E$ around each point $\\mathbf{p}_i$, the template surface is only allowed to move in terms of an Euclidean similarity transformation\n\\begin{equation}\n\\mathbf{p}'_j-\\mathbf{p}'_k=s_i\\mathbf{R}_i\\left(\\mathbf{p}_j-\\mathbf{p}_k\\right)\\qquad\\forall(j,k)\\in E_i,\n\\label{eq:asap}\n\\end{equation}\nwhere $s_i>0$ is a scaling factor and $\\mathbf{R}_i\\in\\SO(3)$ a rotation matrix.\nFollowing \\citet{Cha10}, we define $E_i$ as \\textit{the set containing all (directed) edges of triangles incident to} $\\mathbf{p}_i$, also known as \\textit{spokes-and-rims}.\nFinally, our CASAP regularization term reads\n\\begin{multline}\nF_R(\\mathbf{P}')=\\frac{1}{2}\\sum_{i=1}^n w_i\\left[\\sum_{(j,k)\\in E_i}w_{jk}\\left\\Vert\\left(\\mathbf{p}'_j-\\mathbf{p}'_k\\right)-s_i\\mathbf{R}_i\\left(\\mathbf{p}_j-\\mathbf{p}_k\\right)\\right\\Vert^2_2+\\right.\\\\\n\\left.\\lambda\\sum_{l\\in N_i}w_{il}\\left\\Vert\\mathbf{R}_i-\\mathbf{R}_l\\right\\Vert^2_F\\right]\\,,\n\\label{eq:casap_regularization}\n\\end{multline}\nwhere weights $w_i>0$ are added to individually control the amount of regularization for each particular point.\nAs mentioned above, since the deformation of points $\\mathbf{p}_i$ with a small value for $c_i$ is mainly controlled by the regularization term, we define\n\\begin{equation}\nw_i=\\frac{1}{(h-1)c_i+1}\\qquad\\text{with}\\qquad\\frac{1}{h}\\leq w_i<1\n\\end{equation}\nfor all $i\\in\\{1,2,\\ldots,n\\}$ and $h\\in\\mathbb{N}^+$ (we used $h=2$ throughout this paper).\nAs seen, this strategy keeps points $\\mathbf{p}_i$ of the template relatively stiff if (i) $\\mathbf{p}_i$ has a low probability belonging to the breast region and (ii) if the corresponding point on the target is also not likely to be part of the breast region (because $w_i\\rightarrow 1$ if $c_i\\rightarrow 0$), thus effectively preventing the template from adapting too close to the target outside the breast region.\nLastly, $N_i\\subset V$ in (\\ref{eq:casap_regularization}) denotes the one-ring neighborhood of the $i$-th point and $w_{jk}\\in\\mathbb{R}$ are the popular cotangent weights, see e.g. \\citet{Bot10}.\n$\\lambda\\geq 0$ is usually set to $0.02A$, where $A\\geq 0$ is the total surface area of $\\mathcal{S}$ \\citep{Lev14}.\n\n\\textbf{Landmark term.}\nThe goal of the landmark term $F_L$ is to keep certain positions (i.e. landmarks) fix during the registration process.\nLet $I\\subset\\mathbb{N}$ be an index set containing the indices of the $m$ landmarks specified on the template surface $\\mathcal{S}$.\nDefine a matrix $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$ as\n\\begin{equation}\n\\mathbf{D}=(d_{ij}):=\n\\begin{cases}\n1, & \\text{if }j\\in I,\\\\\n0, & \\text{otherwise}\n\\end{cases}\n\\end{equation}\nfor $i=1,2,\\ldots,m$ and $j=1,2,\\ldots,n$.\nNext, denote the corresponding landmarks on the target surface by $\\{\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_m\\}\\subset\\mathcal{T}$. \nThen, the landmark term is defined as\n\\begin{equation}\nF_L(\\mathbf{P}')=\\frac{1}{2}\\left\\Vert\\mathbf{DP}'-\\mathbf{Q}_L\\right\\Vert^2_F,\n\\end{equation}\nwhere $\\mathbf{Q}_L\\coloneqq\\left(\\mathbf{q}_1,\\mathbf{q}_2,\\ldots,\\mathbf{q}_m\\right)^\\top\\in\\mathbb{R}^{m\\times 3}$.\n\n\\subsubsection{Multi-resolution fitting strategy}\n\\label{subsubsec:multi_res_fitting}\nFollowing common practice, instead of applying the previously described non-rigid alignment only once, we employ a hierarchical, multi-resolution fitting strategy composed of initial fitting, coarse fitting, and fine fitting (see also Figure \\ref{fig:overview_pipeline}).\n\n\\textbf{Initial fitting.}\nHaving a low-resolution instance of the rigidly aligned template at hand, the goal of the initial fitting is to roughly adapt the coarse template to the key features (i.e. landmarks) of the target.\nTo do so, we strictly prioritize the landmark constraints and do not use BPMs in this phase.\n\n\\textbf{Coarse fitting.}\nIn this step, the initially fitted low-resolution template is gradually deformed towards the target.\n\n\\textbf{Upsampling.}\nNext, the deformation obtained from the previous step is applied to the original, full-resolution template. \nThis is achieved using a concept called \\textit{Embedded Deformation}, introduced by \\citet{Sum07}.\nIn essence, the deformation of the coarse template obtained from the previous step is transferred to the template by linearly interpolating the transformation at each point.\n\n\\textbf{Fine fitting.}\nLastly, the upsampled template is fitted to the target again to produce the final result. \n\n\\subsection{Model building}\n\\label{subsec:model_building}\nOnce the data set is brought into correspondence, we follow the typical workflow used to build a classical point-based statistical shape model as proposed by \\citet{Coo95}.\nFor notational simplicity, instead of stacking points $\\mathcal{P}$ of a triangular mesh $\\mathcal{M}=(V,E,\\mathcal{P})$ into a matrix $\\mathbf{P}\\in\\mathbb{R}^{n\\times 3}$ as before, we use a vectorized representation, denoted as $\\mathbf{x}=\\vecz(\\mathbf{P})\\in\\mathbb{R}^{3n}$ in the following.\n\nBriefly, given a set of $k$ breast scans $\\{\\mathbf{x}_1,\\mathbf{x}_2,\\ldots,\\mathbf{x}_k\\}\\subset\\mathbb{R}^{3n}$ in correspondence, we first perform a \\textit{Generalized Procrustes Analysis} (GPA) as introduced by \\citet{Gow75}.\nGPA iteratively aligns the objects to the arithmetic mean $\\mathbf{\\bar{x}}\\in\\mathbb{R}^{3n}$ (successively estimated from the data) by using an Euclidean similarity transformation, effectively transforming the objects into the shape space.\nSecondly, a \\textit{Principal Component Analysis} (PCA) is carried out on the Procrustes-aligned shapes.\nLet $\\{\\lambda_1,\\lambda_2,\\ldots,\\lambda_q\\}\\subset\\mathbb{R}^+$ be the $q<k$ non-zero eigenvalues (also called \\textit{Principal Components} (PCs) in this context) of the empirical covariance matrix calculated from the data and sorted in a descending order.\nDenote the corresponding eigenvectors as $\\{\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_q\\}\\subset\\mathbb{R}^{3n}$.\nThen, a statistical shape model can be interpreted as a linear function $M:\\mathbb{R}^q\\longrightarrow\\mathbb{R}^{3n}$ defined as\n\\begin{equation}\nM(\\boldsymbol\\alpha)=\\mathbf{\\bar{x}}+\\boldsymbol\\alpha\\mathbf{U},\n\\end{equation}\nwhere $\\mathbf{U}\\coloneqq(\\mathbf{u}_1,\\mathbf{u}_2,\\ldots,\\mathbf{u}_q)^\\top\\in\\mathbb{R}^{q\\times 3n}$.\nTo ensure plausibility of the newly generated shapes, $\\alpha_i$ is usually restricted to $\\left|\\alpha_i\\right|\\leq 3\\sqrt{\\lambda_i}$ for all $i\\in\\{1,2,\\ldots,q\\}$.\nIf a (possibly unseen) shape $\\mathbf{x}'\\in\\mathbb{R}^{3n}$ is in correspondence with the model and properly aligned, it can be reconstructed from $M$ in a least-squares sense by using\n\\begin{equation}\n\\boldsymbol\\alpha^*=\\mathbf{U}^{-1}\\left(\\mathbf{x}' - \\mathbf{\\bar{x}}\\right)\n\\end{equation}\nas the model parameters, i.e. $\\mathbf{x}'\\approx M(\\boldsymbol\\alpha^*)$.\nThe number $q<k$ of retained PCs is chosen so that the model typically represents a fixed proportion of the total variance, e.g. 98\\%.\n\n\\section{Evaluation}\n\\label{sec:evaluation}\nBased on our 3D breast scan database which we present in Section \\ref{subsec:data}, several experiments were conducted in order to evaluate the proposed statistical shape model (Section \\ref{subsec:exp_res}).\n\n\\begin{table}[h]\n\\caption{An overview of the 3D breast scan database used to build the RBSM. It includes 110 textured 3D breast scans.}\n\\label{tab:overview_database} \n\\vspace{2mm}\n\\centering\n\\begin{tabular}{p{22.1mm} l l}\n\\hline\n& Mean ($\\pm$ SD) & Range [min -- max]\\\\\n\\hline\nAge [years] & 40.78 ($\\pm$ 14.30) & 18.00 -- 83.00\\\\\nBMI [kg\/m$^2$] & 23.66 ($\\pm$ 3.57) & 16.90 -- 38.27\\\\\n\\hline\n\\multicolumn{3}{l}{Breast volume [cc]} \\\\\n\\hspace{3mm} Left & 477.23 ($\\pm$ 242.94) & 70.60 -- 1,258.90\\\\\n\\hspace{3mm} Right & 481.45 ($\\pm$ 240.27) & 80.00 -- 1,609.30\\\\\n\\hspace{3mm} Difference & \\hspace{.95mm} 80.12 ($\\pm$ 81.34) & \\hspace{.95mm} 0.30 -- 367.40\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Data}\n\\label{subsec:data}\nOur database consists of 110 textured 3D breast scans collected at our institution (St. Josef Hospital Regensburg) using the portable Vectra H2$^{\\text{TM}}$ scanning system (Canfield Scientific, New Jersey, USA).\nThe H2 system employs photogrammetry to reconstruct a 3D surface mesh from a series of 2D images with a resolution of 3.5 mm and a maximal capture volume of $70\\times 41\\times 40$ cm$^3$ (width $\\times$ height $\\times$ depth).\nIn our case, in total three photos were taken of each subject: one from a frontal view and two from a lateral view ($\\pm 45^\\circ$).\nNote that due to the standardized distance and angles from which the photos are taken, the reconstructed 3D breast scans are already reasonably well aligned and consistently oriented.\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics{.\/images\/7_exp1_model}\n\\caption{From left to right: compactness, generalization, and specificity of statistical shape models built from 3D breast scans registered either with (i.e. the RBSM) or without using BPMs during registration.}\n\\label{fig:exp1_model}\n\\end{figure*}\n\nA previously developed standardized scanning protocol \\citep{Har20} was used to ensure a common pose and same imaging conditions for all participants.\nIn brief, all subjects were asked to stand in an upright posture and abduct both arms by an angle of $45^\\circ$ (a telescopic stick was used to support the subjects in holding their arms fixed at the specified angle).\nThis posture produces a natural-looking breast shape and allows to capture the whole breast as isolated as possible.\nMoreover, it is fast and easy to adopt for all participants regardless of age, body weight, or medical history.\n\nA compact overview of some key parameters in our database is given in Table \\ref{tab:overview_database}. \nIn addition, 62 out of 110 participants have at least one ptotic breast.\nOf these, 15 women show different ptosis grades on the left and right breast.\n72 out of 110 participants received some kind of breast surgery.\n\n\\subsection{Experiments and results}\n\\label{subsec:exp_res}\nFollowing common practice, we evaluate our statistical shape model using the well-known metrics compactness, generalization, and specificity as proposed by \\citet{Sty03}.\nRegistration results are assessed by the classical distance-based \\textit{mean squared error} (MSE) between deformed template and target.\nIn addition, the \\textit{angle distortion} between the deformed template $\\mathcal{S}'$ and the original template $\\mathcal{S}$ is determined in order to quantify the amount of shearing introduced due to non-rigid registration. \nThis is achieved by simply averaging the absolute deviation between the inner-triangle angles of $\\mathcal{S}$ and $\\mathcal{S}'$ over all triangles \\citep{Wan16}.\n\nFor all experiments, the same template $\\mathcal{S}$ was used during registration.\nIn particular, the 3D breast scan of a healthy, non-operated subject was chosen and subsequently mirrored along the sagittal plane to produce a perfectly symmetrical template. \nAfter isotropic remeshing and Laplacian smoothing, a regular surface mesh with an average edge length of 2.05 mm was obtained.\nIt consists of 30,924 vertices.\n\nThe complete registration pipeline was implemented in C++.\nStatistical shape modeling was performed with an open-source framework called Scalismo (\\url{https:\/\/scalismo.org\/}), implemented in Scala.\n\n\n\\subsubsection{On the effect of BPMs}\nThe first experiment should evaluate our novel BPM-based registration technique, its influence on the resulting statistical shape models, and, in particular, whether or not BPMs are able to weaken the strong coupling between the breast and surrounding regions.\nAs such, we compare two different models: the first one is constructed from 3D breast scans brought into correspondence using the proposed approach based on BPMs (i.e. the RBSM). \nThe second model is built from 3D breast scans registered without using BPMs.\n\n\\begin{figure}[h]\n\\includegraphics[]{.\/images\/6_exp1_registration}\n\\caption{MSE (left) and angle distortion (right), calculated from 3D breast scans registered with and without BPMs. }\n\\label{fig:exp1_registration}\n\\end{figure}\nFigure \\ref{fig:exp1_model} shows the evaluation metrics of the resulting statistical shape models. \nAs seen, although the model built without BPMs is more compact than with BPMs (i.e. the RBSM) when considering only the very first PCs, it is also the model that generalizes worse.\nSpecifically, when using all 109 available PCs a generalization error of 0.65 mm is reported.\nFor comparison, the RBSM shows a generalization error of only 0.17 mm when using the same number of PCs. \nFinally, the RBSM constantly achieves the lower specificity error of about 2.8 mm if more than 30 PCs are used.\n\nFigure \\ref{fig:exp1_registration} summarizes the registration results, quantified in terms of MSE and angle distortion.\nAs expected, registration without using BPMs clearly achieves the lower MSE of 1.05 mm as the goal here is to register the template to the target as close as possible.\nFor comparison, the MSE achieved when using BPMs is 2.98 mm.\nHowever, it is important to remember that the goal of the BPM-based registration is to align the template as precisely as possible to the target only \\textit{inside} the breast region.\nTherefore, the overall MSE between template and target might not reflect the actual registration quality well (see e.g. Figure \\ref{fig:overview_approach}).\nIn terms of angle distortion, only a small difference is noticeable between registration with and without BPMs, respectively.\nIn particular, registration without BPMs caused the angle distortion to increase by only 0.008 degrees on average.\n\n\\begin{figure}[h!]\n\\includegraphics{.\/images\/8_exp1_random_samples}\n\\caption{Comparison between five random samples from the RBSM (top row, generated from 3D breast scans registered using BPMs) and five random samples from a model constructed without using BPMs (bottom row). \nThe samples are colored according to the distance from the mean $\\mathbf{\\bar{x}}$. \nAs such, the red areas underwent a high deformation, whereas blue regions are rather stiff. For samples drawn from the RBSM, the regions showing the highest variation are almost always the breast region, indicating a solid decoupling between breast shape and thorax.}\n\\label{fig:exp1_random_samples}\n\\end{figure}\nIn order to further investigate to which extend BPM-based registration is able to weaken the strong coupling between breast region and thorax, Figure \\ref{fig:exp1_random_samples} shows some random samples from the RBSM and the model built from 3D breast scans registered without BPMs.\nThe samples are colored according to their distance to the mean shape, effectively providing a \\textit{measure of variation}.\nInterestingly, the areas of the highest variation in the samples drawn from the RBSM are almost always located at the breasts.\nThe surrounding regions are rather stiff, indicating low variance and a quite well decoupling between breast and thorax.\nHence, the RBSM is able to produce a variety of breast shapes without altering the whole thorax too much, also reflected in the principal modes of variation shown in Figure \\ref{fig:modes_rbsm}. \nContrary, the area of the highest variation in the samples generated from the model built without BPMs is oftentimes \\textit{not} the breast region.\n\n\n\n\n\\begin{figure*}[t!]\n\\includegraphics[width=\\textwidth]{.\/images\/9_modes_rbsm}\n\\caption{The first eight principal modes of variation from the RBSM, visualized by either adding (top row) or subtracting (bottom row) $3\\sqrt{\\lambda_i}\\mathbf{u}_i$ from the mean $\\mathbf{\\bar{x}}$, displayed on the left. \nTogether, they represent about 85\\% of the total variance present in the dataset.}\n\\label{fig:modes_rbsm}\n\\end{figure*}\n\n\n\n\n\\begin{figure*}[b!]\n\\centering\n\\includegraphics{.\/images\/10_exp2_model}\n\\caption{From left to right: compactness, generalization, and specificity of statistical shape models built from randomly sampled subsets of the training data.}\n\\label{fig:exp2_model}\n\\end{figure*}\n\n\\subsubsection{How much data is needed?}\nAn often arising question in the context of statistical shape modeling concerns the amount of training data needed to build a reasonably well-performing model.\nGenerally, this question very much depends on the amount of variability samples from a particular class of objects are expected to show.\nAs a rule of thumb, a good training set should always reflect the whole bandwidth of possible variations likely to occur within a target population.\nThe goal of this experiment is to test how much data is needed to build a well-performing statistical shape model of the female breast.\nTo do so, we randomly sample 30, 60, and 90 breast scans from our database and subsequently compare the resulting models with the model containing all 110 breast scans (i.e. the RBSM).\nTo avoid sampling bias, the whole procedure was repeated three times, and results were averaged.\n\n\n\n\nFigure \\ref{fig:exp2_model} shows the results in terms of compactness, generalization, and specificity.\nRegarding compactness, it can be clearly seen that the model built with 30 breast scans is the most compact one, followed by the models constructed from 60 and 90 breast scans and the RBSM.\nOn the other hand, the model built from only 30 breast scans shows the worst generalization ability (about 1.07 mm when using all 29 PCs available).\nThe model containing 90 breast scans and the RBSM achieved very similar results, both showing a generalization error of about 0.6 mm when considering the first 29 PCs.\nAs expected, whereas the RBSM is the most specific one, the model learned from only 30 breast scans clearly performs the worst. \n\n\\subsubsection{Building a statistical shape model from operated breasts}\nAs already mentioned earlier, a quite big proportion of our 3D breast scans (72 out of 110) result from subjects who already received some kind of breast surgery.\nIt is thus natural to assume that the RBSM might be biased towards operated breasts and is therefore not able to explain non-operated breasts well.\nThe goal of this experiment is thus to verify how well a model built from operated breasts is able to reconstruct 3D breast scans from participants that did not undergo breast surgery.\nIn particular, a statistical shape model is built from a random subset of 37 (out of 72) operated breasts.\nAfterwards, generalization ability is evaluated using the 38 non-operated breasts as test set.\nAgain, to avoid sampling bias, the sampling procedure was repeated three times, and results were averaged.\n37 out of the 38 non-operated breasts are the basis for a second model, calculating the generalization ability using leave-one-out cross-validation.\n\n\n\n\n\n\nThe results are shown in Figure \\ref{fig:exp3_model}.\nAs it can be seen, although our database is obviously biased towards operated breasts, a model built only from operated breasts \\textit{is} able to explain non-operated breasts quite well.\nSpecifically, starting with a generalization error of over 3.5 mm, the generalization ability increases constantly as the number of PCs increase.\nFinally, a generalization error of 0.77 mm is achieved when using 36 PCs.\nFor comparison, the model constructed from 37 non-operated breasts is able to reconstruct non-operated breasts only slightly better with an error of 0.71 mm when using the same number of PCs.\n\n\n\n\n\n\\begin{figure}[t]\n\\includegraphics{.\/images\/11_exp3_model}\n\\caption{Generalization ability of a statistical shape model built from 37 operated breasts, calculated using the 38 non-operated breasts as test set. \nFor comparison purposes, a second model was trained with the same number of non-operated breasts. \nIts generalization ability was evaluated on the 38 non-operated breasts using leave-one-out cross-validation.}\n\\label{fig:exp3_model}\n\\end{figure}\n\n\\subsubsection{Generalization ability using clinical parameters}\nOur final experiment evaluates the generalization ability of the RBSM using three common, \\textit{clinical} parameters typically obtained on the breast.\nThis way, we aim to show whether or not reconstructions obtained from the RBSM not only well explain unseen objects in general, but also preserve meaningful clinical parameters.\nTechnically, this is quite similar to the generalization ability as defined by \\citet{Sty03}.\nHowever, instead of calculating point-to-point distances between an unseen 3D breast scan and its corresponding reconstruction from the RBSM, we compare three anthropometric measurements obtained on each mesh.\nThe following three distances are measured:\n\\begin{itemize}\n\\item sternal notch to left nipple (SN-NL),\n\\item sternal notch to right nipple (SN-NR), and\n\\item left nipple to right nipple (NL-NR).\n\\end{itemize}\nAfterwards, the absolute difference between the ground-truth distance taken on the original 3D breast scan and the distance obtained from the reconstruction is calculated for each of the three anthropometric measurements.\nSimilar to ordinary generalization based on point-to-point distances, we employ leave-one-out cross-validation for all 110 breast scans to calculate our newly defined generalization ability.\n\n\n\n\n\\begin{figure}[h]\n\\includegraphics{.\/images\/12_exp4_model}\n\\caption{Generalization ability of the proposed RBSM, evaluated by comparing anthropometric measurements obtained on an original 3D breast scan and its corresponding reconstruction from the model. \nThe average generalization error of the three measurements considered is also shown.}\n\\label{fig:exp4_model}\n\\end{figure}\n\nThe results are shown in Figure \\ref{fig:exp4_model}.\nAs it can be seen, using less than 30 PCs leads to a high generalization error for all three anthropometric measurements, ranging between 15 mm and 45 mm. \nHowever, as the number of PCs increases, the generalization error drops significantly and remains low for the SN-NL and SN-NR distances.\nInterestingly, only the NL-NR distance increases again up to 20 mm.\nIn summary, all three anthropometric measurements are best preserved when using 109 PCs, showing an average generalization error of 1.82 mm.\nThe individual generalization errors when using 109 PCs are 0.66 mm (SN-NL), 3.24 mm (SN-NR), and 1.57 mm (NL-NR).\n\n\n\\section{Applications}\n\\label{sec:applications}\nThis section demonstrates two exemplary applications for the RBSM that may be used for breast surgery simulation. \n\n\\subsection{Breast shape editing}\nSo far, our statistical shape model provides a convenient way to generate new breast shapes by simply varying its parameter $\\boldsymbol\\alpha\\in\\mathbb{R}^q$ within a suitable range.\nHowever, a major drawback of this parametrization is the non-interpretability of $\\boldsymbol\\alpha$ in the sense that it does not correlate with any meaningful features of the breast.\nThis clearly hinders the generation of breast shapes with certain properties or the ability to alter an existing shape according to some clinical parameters, for instance.\nTo this end, the feature editing framework proposed by \\citet{Bla99} and extended by \\citet{All03} overcomes this drawback by linearly relating features with shape parameters.\nBriefly, being $l$ feature values $\\{f_{i1},f_{i2},\\ldots,f_{il}\\}\\subset\\mathbb{R}^l$ for each individual given and stacked into a feature vector $\\mathbf{f}_i\\coloneqq(f_{i1},f_{i2},\\ldots,f_{il},1)\\in\\mathbb{R}^{l+1}$, $i=1,2,\\ldots,k$, a matrix $\\mathbf{F}\\coloneqq\\left(\\mathbf{f}_1,\\mathbf{f}_2,\\ldots,\\mathbf{f}_k\\right)\\in\\mathbb{R}^{(l+1)\\times k}$ is defined. \nArranging the shape parameters of each individual in a matrix $\\mathbf{A}\\coloneqq\\left(\\boldsymbol\\alpha_1,\\boldsymbol\\alpha_2,\\ldots,\\boldsymbol\\alpha_k\\right)\\in\\mathbb{R}^{q\\times k}$ and following \\citet{All03}, $\\mathbf{F}$ and $\\mathbf{A}$ can be related by means of an unknown transformation matrix $\\mathbf{M}\\in\\mathbb{R}^{q\\times(l+1)}$ satisfying\n\\begin{equation}\n\\mathbf{MF}\\overset{!}{=}\\mathbf{A}\\quad\\Longrightarrow\\quad\\mathbf{M}=\\mathbf{F}^+\\mathbf{A},\n\\end{equation}\nwhere $\\mathbf{F}^+$ denotes the Moore\u2013Penrose inverse of $\\mathbf{F}$.\nUsing the matrix $\\mathbf{M}$, the shape of an individual can be edited by simply providing new feature values.\nIf $\\Delta\\mathbf{f}$ denotes the component-wise difference between a target feature vector and the actual feature vector of an individual, the new shape parameter $\\boldsymbol\\alpha'$ is obtained as $\\boldsymbol\\alpha'=\\boldsymbol\\alpha+\\mathbf{M}\\Delta\\mathbf{f}$.\nThe edited shape is given by $M(\\boldsymbol\\alpha')$.\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics{.\/images\/13_feature_editing}\n\\caption{Systematic breast shape editing for two examples (top row and bottom row) using the RBSM and three common anthropometric distances as features. \nNote that all measurements are given in centimeters.}\n\\label{fig:feature_editing}\n\\end{figure}\n\n\nFor our exemplary application, we use the clinical parameters SN-NL, SN-NR, and NL-NR as features and set $q=k-1$.\nTwo exemplary results are shown in Figure \\ref{fig:feature_editing}.\n\n\\subsection{Breast shape prediction}\nWhile our first application is designed to alter key clinical parameters of the breast, this section demonstrates how the RBSM can be used to predict a missing breast from the existing one by means of \\textit{posterior} shape models \\citep{Alb13}.\n\nGiven an unseen 3D breast scan in correspondence with the template and represented as $\\mathbf{x}\\in\\mathbb{R}^{3n}$, the area enclosing the missing breast is marked first.\nDenote the indices of marked points as $J\\subset\\mathbb{N}$.\nFollowing \\citet{Alb13}, the remaining $r\\coloneqq n-\\left|J\\right|$ points provide known observations which we use to compute a posterior mean $\\mathbf{\\bar{x}}_p\\in\\mathbb{R}^{3n}$. \nIt is the likeliest reconstruction of the missing breast.\nFormally, denote as $\\mathbf{\\bar{x}}_*\\in\\mathbb{R}^{3r}$ and $\\mathbf{U}_*\\in\\mathbb{R}^{q\\times 3r}$ the sub-vector and sub-matrix obtained by removing those entries from $\\mathbf{\\bar{x}}$ and $\\mathbf{U}$ corresponding to the selected points in $J$.\nSimilarly, $\\mathbf{x}_*\\in\\mathbb{R}^{3r}$ represents the target 3D breast scan after removing the marked points.\nThen, \n\\begin{equation}\n\\mathbf{\\bar{x}}_p=\\mathbf{\\bar{x}}+\\mathbf{U}^\\top\\left(\\mathbf{U}_*\\mathbf{U}_*^\\top+\\sigma^2\\mathbf{I}_q\\right)^{-1}\\mathbf{U}_*\\left(\\mathbf{x}_*-\\mathbf{\\bar{x}}_*\\right),\n\\end{equation}\nwhere $\\sigma^2\\geq 0$ is a small variance accounting for the deviation of $\\mathbf{x}_*$ from the model \\citep{Alb13} and $\\mathbf{I}_q\\in\\mathbb{R}^{q\\times q}$ is the identity matrix.\n\n\n\n\n\n\\begin{figure}[b]\n\\includegraphics[]{.\/images\/14_breast_prediction}\n\\caption{Predicting the right breast (marked as \\textit{missing} in red) from the left one for two examples (top row and bottom row) using the RBSM.}\n\\label{fig:breast_prediction}\n\\end{figure}\n\n\n\nTwo exemplary results are shown in Figure \\ref{fig:breast_prediction}.\nNote that, compared to simply mirroring the remaining breast, the prediction obtained from the RBSM is equipped with a statistical probability.\nHence, it may not be the most symmetrical result, but, more importantly, the likeliest and probably most natural one.\nThis is crucial especially in BRS, where the goal is to reconstruct a breast looking as natural as possible.\n\n\\section{Discussion}\n\\label{sec:discussion}\nThe following section discusses the most important insights gained from experimental evaluation and also summarizes some limitations of our method.\n\n\\subsection{Decoupling between breast and thorax}\nThe experimental evaluation showed that BPM-based registration is able to decouple the breast region from the thorax well.\nThe areas with the highest variation in random samples drawn from the RBSM are almost always coincident with the breast region whereas the thorax is kept relatively stiff.\nHence, a variety of different breast shapes can be generated independently from the surrounding areas or without altering the thorax too much, implying a quite well decoupling between breast and thorax.\nThis can be also verified from Figure \\ref{fig:overview_approach}, showing that the unwanted variance of the surrounding regions is properly reduced when using BPM-based registration.\nFurthermore, the fact that our feature editing application is able to change only the requested features without altering the shape of the thorax supports our observations.\n\nFinally, we also want to note that there might be an alternative strategy for breaking up the strong correlation between breast and surrounding regions. \nIn particular, the framework proposed by \\citet{Wil17} allows constructing statistical shape models with a locality assumption by manipulating the sample covariance matrix.\nDistant areas are decoupled by explicitly setting non-zero covariances between those points to zero.\nThis way, covariances between points in the breast region and points outside could be set to zero, effectively decoupling those regions.\nA similar but probably easier way to localize statistical shape models is by using GPMMs as described by \\citet{Lue18}.\nWe believe, however, that minimizing variances instead of covariances (between pairs of points) is a more intuitive way to weaken the coupling between breast and thorax. \nNote that there is a strong relation between both approaches since $\\left|\\cov(x,y)\\right|\\leq\\sqrt{\\var(x)}\\sqrt{\\var(y)}$.\n\n\\subsection{BPM-based registration}\nOur quantitative evaluation clearly indicates that BPM-based registration produces superior results than registration without BPMs, affecting both, registration \\textit{and} model quality.\nWhen using BPMs, only a little distortion was introduced. \nIn addition, although only four landmarks are provided, pretty well correspondences are established. \nWe observed only minor correspondence errors when randomly sampling from the RBSM. \n\nRegarding the EBF-based representation of the BPMs, we can conclude that EBFs are quite well suited to capture the natural teardrop shape of the breast.\nHowever, in some cases, we observed that the current BPMs are not always able to properly capture the \\textit{entire} breast region, especially if the breast does not follow a typical teardrop shape.\nThe reason is the restriction of the covariance matrices (occurring in the Mahalanobis distance) to be diagonal, pulling off a lot of flexibility for the BPM to capture unusual breast shapes.\nWe expect that, if the full covariance matrix would be specified, better BPMs can be constructed. \nHowever, this will clearly hinder the automatic parameter selection as 30 values need to be carefully determined for each 3D breast scan.\n\nLastly, it is important to note that our rigid alignment will completely fail to estimate the correct transformation if the initial alignment is bad.\nThis is a well-known behavior of the ICP algorithm as it does not optimize for a globally optimal transformation.\nSince our 3D breast scans are initially quite well aligned and consistently oriented due to the standardized data acquisition using photogrammetry, we did not run into this problem. \n\n\\subsection{3D breast scan database}\nAlthough our 3D breast scan database contains nearly twice as many operated than non-operated breasts (72 operated, 38 non-operated), the RBSM is likely to show only a minor bias towards operated breasts.\nThis is supported by the fact that a model built from a randomly chosen subset of 37 operated breasts is able to reconstruct non-operated breasts with an almost similar error than a model trained \\textit{and} tested on the same number of non-operated breasts. \nSpecifically, an absolute difference of 0.06 mm between both reconstructions was reported.\nIntuitively, this suggests that the operated breasts contained in our database look quite natural (or, at least, similar to non-operated ones), making it challenging to decide whether a particular patient received breast surgery or not.\nIt is, however, important to note that this experiment was conducted by considering only a subset of 37 operated breasts.\nThis is because otherwise, a model built from an equal amount of non-operated breasts could not be constructed for comparison purposes.\nOn the other hand, the RBSM contains 72 operated breasts.\nHowever, even if the absolute difference between a reconstruction obtained from a model trained on operated breasts and one that includes only non-operates breasts scales \\textit{exponentially} with the number of operated breasts considered, an absolute difference of 0.23 mm would be reported for the RBSM.\nIt is thus valid to assume that the bias introduced due to operated breasts will not hinder the RBSM from explaining non-operated breasts sufficiently well.\n\nRegarding the amount of data, our experiments confirmed what was expected: the more data, the better. \nThe RBSM (containing all 110 breast scans) clearly outperforms the other models trained only on a subset of the data.\nOur experiments do not allow a conclusion about whether or not 110 breast scans are already enough in order to represent the vast amount of possible breast shapes well.\nHowever, the difference between 90 and 110 breast scans in terms of generalization and specificity is quite small (see Figure \\ref{fig:exp2_model}) strengthening the impression of being near convergence.\n\nBesides, we believe that in order to provide better pose-standardized training data, a more advanced technical set-up is needed in which the movement of arms and shoulders is further constrained. \nThis, however, would lengthen the whole data acquisition which might then become unfeasible to be applied in a clinical environment.\n\n\\subsection{Landmarks guiding the registration process}\nUsing only four landmarks (both nipples as well as both lower breast poles) for non-rigid registration is quite uncommon.\nMost of the existing methods need a lot more points to guide the registration process reasonably well.\nFor instance, the registration of facial 3D scans for the construction of statistical shape models is usually guided by about 60 to 80 landmarks, see e.g. \\citet{Boo16} or \\citet{Ger17}.\nDue to human anatomy, collecting such a huge amount of landmarks on the female breast is challenging and a fundamental problem of 3D breast scan registration. \nExcept for the nipples (and partially the lower breast poles), there are no landmarks that can be \\textit{reliably} detected only through visual inspection and consistently across a wide range of differently-shaped breasts.\n\nBesides anatomical points, additional landmarks based on easy-detectable, non-bony structures may be used.\nThis way, the lower breast pole was simply defined as \\textit{the lowest (most caudal) point of the breast}.\nFrom a medical point of view, this is indeed a valid definition of a landmark that is also used in clinical practice.\nBased on our collected data, however, it turns out that this definition is not sufficiently precise to be used for registration purposes.\nThe actually determined lower breast poles were oftentimes \\textit{not} accurately located at the lowest point of the breast.\nAs a result, the initially detected points had to be manually refined in about 40\\% of the subjects.\n\nAnother fundamental issue is the non-existence of easy-detectable landmarks on the lateral part of the breast.\nThe absence of those points poses a serious problem in our 3D breast scan registration.\nIn particular, when not already quite well aligned initially, the template has no chance of being pulled laterally due to the missing guidance.\n\nTo conclude, detecting landmarks on 3D breast scans reliably is challenging.\nUltimately, since a physical examination is required to detect anatomical landmarks, automatic landmark detection algorithms based on 3D breast scans (only capturing the surface of the body) are almost impossible to develop.\nAdditionally, it is important to note that even in some participants clearly detectable landmark points would be present, one has to keep in mind that those landmarks always need to be consistently located across \\textit{all} subjects.\nBy means of the proposed registration pipeline, however, we demonstrated that 3D breast scan registration \\textit{is} generally possible by providing only four landmarks.\n\n\\section{Conclusion and outlook}\n\\label{sec:conclusion}\nThis paper proposed the RBSM -- the first publicly available 3D statistical shape model for the female breast, learned from 110 breast scans. \nOur extensive evaluation reveals a generalization ability of 0.17 mm and a specificity error of about 2.8 mm when using all 109 PCs available.\nTogether with the model, a fully automated, pairwise surface registration pipeline was presented which requires only four landmarks to guide the registration process.\nIn order to break up the strong coupling between breast and thorax and thus being able to capture the actual breast shape as isolated as possible, we proposed to minimize the unwanted variance of the surrounding regions by means of a novel concept called BPMs.\nDefined over a surface mesh and based on EBFs, BPMs allow for a probabilistic definition of the breast region and hence overcomes the difficulty in determining an exact delineation of the breast.\nLater, BPMs are incorporated into the registration pipeline in order to align the template as accurately as possible inside the breast region and only roughly outside.\nWith our experiments, we could show that this strategy effectively reduces the unwanted variance outside the breast region and hence, decouples the breast from the surrounding areas very well.\nAs a result, the RBSM is able to generate a variety of different breast shapes quite independently from the thorax.\n\nIn our future work, we ultimately plan to combine physically motivated deformable models and statistical shape models of the breast in order to enable more realistic and statistically plausible surgical outcome simulation for BRS. \nThe proposed RBSM is seen as a first step towards this goal.\n\nBesides the two exemplary applications shown in this paper, there is room for plenty of different applications, extensions, and improvements of the RBSM.\nIn particular, since our model is generative, it could be conveniently used for data augmentation in the machine learning domain.\nAdditionally, the RBSM may be used as a prior for surface registration algorithms, effectively reducing the search space of possible deformations.\nFollowing the ideas of \\citet{Boo16}, so-called \\textit{bespoke models} (statistical shape models trained from a dedicated subset of the training set) may be built, for example for different classes of BMIs, ptosis grades, or breast volumes.\nFurthermore, instead of one global model accounting for the whole breast, two individual models for the left and right breast could be built and subsequently combined.\nLastly, our rigid alignment needs to be improved to be able to deal with an arbitrary initial alignment.\nFor instance, more robust ICP variants such as Go-ICP \\citep{Yan16} may be used or advanced feature detectors (based on heat or wave kernel signatures \\citep{Aub11,Sun09} or multi-scale mean curvatures \\citep{Pan10}, for instance) combined with a robust outlier detection to find reliable correspondences for transformation calculation. \n\n\\section*{Declaration of Competing Interest}\nThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.\n\n\\section*{Acknowledgments}\nWe thank our colleagues from the University Center of Plastic, Aesthetic, Hand and Reconstructive Surgery at the University Hospital Regensburg and the St. Josef Hospital Regensburg for their help during data collection.\nThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\n\n\\bibliographystyle{model2-names.bst}\\biboptions{authoryear}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe momentum-dependence of single-nucleon mean-field potentials is one of the fundamental features of finite-range nucleon-nucleon interactions in nuclear matter.\nA moving nucleon experiences different potentials from its surroundings compared to static ones, either under the mean fields or those considering correlations.\nMomentum-dependent nuclear potentials in fact play very important roles in heavy-ion collisions\\,\\cite{Gale1987,Bertsch1988,Pra88,Welke,Gale90,Aichelin1990,Pan,Zhang94,Hart,Pawel,Das2003,LiBA04}.\nThe mean-field potential $U_J$ (with the subscript $J$ denoting protons (p) or neutrons (n)) of a nucleon with momentum $\\v{k}$ at zero temperature is generally a function of neutron density $\\rho_{\\rm{n}}$, the proton density $\\rho_{\\rm{p}}$, i.e., $U_J=U_J(\\rho_{\\rm{n}},\\rho_{\\rm{p}},|\\v{k}|)$.\nIt can be written equivalently in the form of $U_J(\\rho,\\delta,|\\v{k}|)$ with $\\rho=\\rho_{\\rm{n}}+\\rho_{\\rm{p}}$ the total density and $\\delta=(\\rho_{\\rm{n}}-\\rho_{\\rm{p}})\/\\rho$ the isospin asymmetry of the nucleon system under consideration.\nThe equation of state (EOS) of cold asymmetric nucleonic matter (ANM) can be conveniently denoted by the nucleon specific energy $E(\\rho,\\delta)$. The latter could be obtained by certain integration of $U_J(\\rho_{\\rm{n}},\\rho_{\\rm{p}},|\\v{k}|)$ over $\\v{k}$, i.e., $\\sim\\int^{\\v{k}}U_J(\\rho,\\delta,|\\v{k}|)\\d\\v{k}+\\rm{``kinetic\\;term''}\\to E(\\rho,\\delta)$.\nEffects of the momentum $\\v{k}$ dependence of the single-nucleon potential on the EOS are implicit as the momentum is integrated out.\nConnecting the momentum-dependence and the correlations\/fluctuations of the interactions\/potentials with the EOS as well as understanding how they influence each other are fundamentally important and interesting for resolving many related issues in both nuclear physics and astrophysics.\n\n\nThe EOS $E(\\rho,\\delta)$ of ANM can conventionally be expanded around the symmetric nuclear matter (SNM) with $\\delta=0$, leading to the approximation,\n\\begin{equation}\nE(\\rho,\\delta)\\approx E_0(\\rho)+E_{\\rm{sym}}(\\rho)\\delta^2+E_{\\rm{sym},4}(\\rho)\\delta^4+\\mathcal{O}(\\delta^6),\n\\end{equation}\nhere $E_{\\rm{sym}}(\\rho)=2^{-1}\\partial^2E(\\rho,\\delta)\/\\partial\\delta^2|_{\\delta=0}$ is the conventional nuclear symmetry energy besides the EOS $E_0(\\rho)\\equiv E(\\rho,0)$ of SNM,  $E_{\\rm{sym},4}(\\rho)\\equiv 24^{-1}\\partial^4E(\\rho,\\delta)\/\\partial\\delta^4|_{\\delta=0}$ is the fourth-order symmetry energy, etc. Owing to its fundamental importance in nuclear\nphysics\\,\\cite{LiBA98,Dan02,Bar05,Ste05,Che07a,LCK08,Tsa12,Che14}\nand astrophysics\\,\\cite{Gle00,Lat04,Lat12,Lat14,Oze16,Oer17}, \nunderstanding nuclear symmetry energy $E_{\\rm{sym}}(\\rho)$,\nespecially its density dependence (both at sub- and supra-saturation densities), is currently one of the most\nimportant goals in contemporary nuclear astrophysics\\,\\cite{Hor14,EPJA}. \n\nThanks to the significant efforts made by many people in the nuclear astrophysics community during\nthe last two decades, much knowledge on the symmetry energy around the saturation density\n$\\rho_0\\approx0.16\\,\\rm{fm}^{-3}$ has been obtained. However, the density dependence of\n$E_{\\rm{sym}}(\\rho)$ especially at supra-saturation densities is still\npoorly known. In particular, several nuclear structure observables are known as sensitive probes of symmetry energy at sub-saturation\ndensities, e.g., the correlation between the slope parameter $L(\\rho)$ of the symmetry energy and the\nneutron skin thickness of heavy nuclei\\,\\cite{Cen09,Zha13} as well\nas the correlation between $L(\\rho)$ and the electric dipole $\\alpha_{\\rm{D}}$\npolarizability in $^{208}\\rm{Pb}$\\,\\cite{Maz13,Zha14,Zha15}; the\ncorrelation between the isobaric analog states (IAS) and the\nsymmetry energy\\,\\cite{Dan14}, to name a few. While analyses of neutron star observables (mostly the radii and tidal polarizability) especially since GW170817 \nhave provided strong constraints on the symmetry energy around $1\\rho_0\\mbox{$\\sim$}2\\rho_0$\\,\\cite{Ste10,LiBA21,Huth22}, the situation at higher densities is unclear\\,\\cite{ZL21,Fr22} from studying the mass and radius data of the currently known most massive neutron star \nPSR J0740+6620 from the NICER Collaboration\\,\\cite{Miller21,Riley21}. On the other hand, intermediate-relativistic energy heavy-ion collisions (HICs) have been playing a significant role in determining the symmetry energy from sub-saturation to supra-saturation densities, see, e.g., Refs.\\,\\cite{LCK08,Tsa12,Xia09}. Unfortunately, various analyses of heavy-ion reaction observables based on transport model simulations have not reached a firm conclusion regrading the high-density behavior of nuclear symmetry energy, \nfor a recent review, see e.g., Ref.\\,\\cite{EPJA-review}. It is mostly because of the still existing relatively large model dependences in simulating heavy-ion collisions especially above the pion production threshold\\,\\cite{Xu16,Zhang18,Ono19,Colonna21,TMEP}.  The single-nucleon potential especially its momentum dependence in dense neutron-rich matter is among the most uncertain but very critical physics inputs of transport models. It is known to have significant impact on observables of heavy-ion reactions\\,\\cite{LiBA04,LiChen05,Cozma17,WangLi,Cozma21}. \n\nThe symmetry energy can conventionally be decomposed into a kinetic and a potential part in the quasi-particle picture, i.e.,\n$E_{\\rm{sym}}(\\rho)=E_{\\rm{sym}}^{\\rm{kin}}(\\rho)+E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$.\nEach individual term of this decomposition may have its own distinct and important physical ramifications on different problems. For instance, the critical\nformation density of the charged $\\Delta(1232)$ states in neutron\nstars depends on $E_{\\rm{sym}}^{\\rm{kin}}(\\rho)$ and\n$E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$ separately\\,\\cite{Cai15a}; while on\nthe other hand, the potential part $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$\nis extremely relevant to certain dynamical processes and observables in\nHICs\\,\\cite{Li15,Li15a,Hen15b,Yon17}. Thus it is not only important\nto explore the density dependence of the total symmetry energy, but also its\nindividual terms and their physical origins. Since the total symmetry energy at the saturation density $\\rho_0$, as mentioned above, is relatively well constrained\\,\\cite{LiBA13}, the decomposition of symmetry energy has to satisfy an asymptotic sum rule at $\\rho_0$. In the traditional free Fermi gas (FFG) model of nuclear matter,\nwhere the single-nucleon momentum distribution function\n$n_{\\v{k}}^J$ takes the form of a step function, the kinetic\nsymmetry energy is simply\n$E_{\\rm{sym}}^{\\rm{kin}}(\\rho)=k_{\\rm{F}}^2\/6M$ with $k_{\\rm{F}}$ being\nthe nucleon Fermi momentum in SNM, and $M$ is the average static mass of nucleons. It will be used as a reference of kinetic symmetry energy in our following discussions.\nIn this work, we use the abbreviation ``FFG'' (``HMT'') to denote the calculation\/model in which the momentum distribution $n_{\\v{k}}^J$ is a step function (the function containing the HMT), even in cases where a nuclear potential is included.\n\nIn most calculations of cold nuclear matter EOSs especially within phenomenological models and\/or input EOSs as well as initializations of transport model simulations of nuclear reactions, often a step function is used for the $n_{\\v{k}}^J$ as if the system is a FFG at zero temperature. However, this assumption has to be modified if one considers the HMT induced by the SRC in nuclear matter. Moreover, this modification is known to have significant ramifications on the EOS especially the kinetic symmetry  energy\\,\\cite{CXu10,Lee11,Lee14,Wang12}. \nIn a more realistic picture of nuclear matter, the form of the\n$n_{\\v{k}}^J$ should be different from the step function, owing to, e.g., the (poorly known) spin-isospin dependence of the three-nucleon forces and\/or the isospin dependence of SRC induced by the tensor forces and\/or the hard repulsive core. \nIndeed, It has been known for a long time that the SRC leads to a high (low)\nmomentum tail (depletion) in the single-nucleon momentum\ndistribution above (below) the nucleon Fermi surface, in cold\nnucleonic matter\\,\\cite{Bethe,Ant88,Arr12,Cio15}. Significant\nefforts have been made in the past\nyears\\,\\cite{Wei15,Wei15a,Cruz2018,Weiss2019-1,Weiss2019-2,Weiss2021,Hen14,Hen15,Col15,Egi06,kk1,kk2,kk3,kk4,kk5,\nHen2017RMP,Duer2018,Schmookler2019,Schmidt2020,Li22} to constrain the\n(isospin-dependent) parameters characterizing the SRC-modified\n$n_{\\v{k}}^J$ in neutron-rich matter using both\nexperimental data and microscopic model calculations. For example,\nit has been found via analyzing electron-nucleus scattering data\nthat the percentage of nucleons in the HMT above the Fermi surface is as high as about 28\\% in SNM but\ndecreases gradually to about only 1\\% in a pure neutron matter\n(PNM)\\,\\cite{Hen14,Hen15}, indicating a totally different momentum\nbehavior of the $n_{\\v{k}}^J$ due to the strong isospin dependence of SRC. \n\nDespite of the long history and impressive progresses especially in recent years in studying various features and impacts of SRC\/HMT, many interesting issues regarding the size, shape, density and isospin dependence of SRC\/HMT remain to be clearly resolved. \nFor example, although various many-body theories\nhave consistently predicted effects of the SRC on the HMT\nqualitatively consistent with the experimental findings, the\npredicted size of the HMT still depends on the model and interaction\nused. For instance, the self-consistent Green's function (SCGF)\ntheory adopting the AV18 interaction predicts a 11\\%$\\sim$13\\% HMT for\nSNM at saturation density\\,\\cite{Rio09,Rio14}. Moreover, the latest\nBruckner-Hartree-Fock (BHF) calculations predict a HMT ranging from\nabout 10\\% via the next-next-next-leading order 450 (N$^3$LO450)\nto over 20\\% using several microscopic interactions\\,\\cite{ZHLi},\nthe latest VMC calculations for $^{12}$C gives a 21\\%\nHMT\\,\\cite{VMC,Wir16}, etc. While based on the observation that the SRC\nbetween a neutron-proton pair is about 18$\\sim$20 times that of two\nprotons, the HMT in PNM was estimated to be about\n1\\%$\\sim$2\\%\\,\\cite{Hen15b}. Furthermore, some recent calculations\nindicate a significantly higher HMT in PNM, e.g., the SCGF\npredicted a 4\\%$\\sim$5\\% HMT in PNM\\,\\cite{Rio09,Rio14}. \n\nThe uncertain properties of SRC\/HMT in cold dense neutron-rich matter may bring large uncertainties to both the kinetic and potential parts of the symmetry energy as well as the single-nucleon potentials. \nFor example, because of the momentum-squared weighting in\ncalculating the average nucleon kinetic energy, the strong isospin\ndependence of the HMT makes the kinetic symmetry energy dramatically\ndifferent from the FFG model\nprediction\\,\\cite{Cai16c,Hen15b,CXu11,CXu13,Vid11,Lov11,Car12,Rio14,Car14,Cai15}.\nSpecifically, the kinetic symmetry energy is significantly reduced even\nto negative values in some model studies. Roughly speaking, nuclear symmetry\nenergy is the energy difference between PNM and SNM within the parabolic\napproximation of the total ANM EOS. It is also known that the SRC is dominated by the isosinglet neutron-proton tensor interaction. It thus increases significantly the average kinetic energy per nucleon\nin SNM but has little effect on that in PNM, leading to a reduction\nof the kinetic symmetry energy. Consequently, the potential part of the $E_{\\rm{sym}}(\\rho)$, i.e., $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$, should be modified correspondingly to satisfy the sum rule of symmetry energy at $\\rho_0$. Moreover, the \nSRC\/HMT may affect directly the potential symmetry energy if nuclear interactions are momentum dependent. Of course, it is also expected to affect the momentum dependence of both isoscalar and isovector parts of the single-nucleon potential and the associated effective masses of nucleons in dense neutron-rich matter.\n\nInterestingly, SRC\/HMT effects on the ANM EOS and some properties of neutron stars have been studied recently within relativistic mean field (RMF) models by several groups\\,\\cite{Cai16b,Lou22a,Lu2022,Bur22,Hong22,Lou22,Lu21,Souza20}. \nMost of these studies were carried out using nonlinear RMF models where the interaction is momentum independent. Given the results of these studies and those mentioned earlier, there still exist many interesting\/important questions to be  investigated\/answered to deepen our understanding of the SRC physics and its ramifications, e.g.,\n\\begin{enumerate}\n\\item[(a)]How does the SRC\/HMT affect the single-nucleon potential in neutron-rich matter, especially the momentum dependence of its isoscalar and isovector parts, in comparisons with existing experimental observations of nucleon optical potentials and the associated nucleon effective masses at $\\rho_0$? Answers to this question are potentially important for modeling HICs involving high-energy rare isotope beams.\n\\item[(b)]How does the SRC\/HMT affect the ANM EOS (including both the SNM EOS and the symmetry energy) qualitatively and quantitatively, if a non-relativistic potential\/interaction is adopted? This question is relevant since many such kind of interactions and energy density functionals are widely used successfully in studying nuclear structures, calculating the EOS for astrophysical purposes and\/or simulating heavy-ion reactions.\n\\item[(c)] Due to the SRC\/HMT induced sizable enhancement of SNM pressure and the reduction of total symmetry energy in nonlinear RMF models, the maximum mass of cold neutron stars was found to increase, see, e.g., Ref.\\,\\cite{Cai16b}. One may then wonder whether this conclusion would still hold or not if a non-relativistic and\/or momentum-dependent potential is used.\n\\end{enumerate}\n\nIn a Gogny-like energy density functional (EDF) that has been widely used in simulating heavy-ion reactions, see, e.g., Refs.\\,\\cite{Gale1987,Bertsch1988,Pawel,Das2003,LCK08,LiBA04,LiChen05,Cozma17,Cozma21}, the momentum-dependent part of the single-nucleon potential has the form of \n$\\int\\d\\v{k}'f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')$ while the corresponding potential energy in the ANM EOS has the form of\n$\\int\\d\\v{k}\\d\\v{k}'f_J(\\v{r},\\v{k})f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')$ where the kernel $\\Omega(\\v{k},\\v{k}')=[1+(\\v{k}-\\v{k}')^2\/\\Lambda^2)]^{-1}$ is from the Yukawa-type finite-range two-nucleon interaction, $\\Lambda$ is the momentum-scale parameter while $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ are the momentum distribution functions of the two interacting nucleons $J$ and $J'$. Analytical integrations of the two integrals involved after incorporating the HMT in the $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ have been found formidable\\,\\cite{CLC18}, while they are necessary for fixing the model parameters using empirical properties of nuclear matter and nucleon optical potentials at $\\rho_0$.\n\nIn this work, using a parametrized $\\Omega(\\v{k},\\v{k}')$ as the surrogate of its original form we derive analytical expressions for all relevant terms and their characteristic parameters in the ANM EOS and single-nucleon potential. The surrogate approach adopted here not only grasps all major physical features of the momentum-dependence of the original Gogny-like potential but also enables analytical expressions for all the relevant physical quantities. There are different physical origins of the HMT in various many-body systems under different conditions\\,\\cite{Coleman15}. We notice that while during heavy-ion collisions, a HMT is automatically generated in the $f_J(\\v{r},\\v{k})$ and $f_{J'}(\\v{r},\\v{k}')$ due to finite temperatures. Nevertheless, most of the parameters of the input single-nucleon potentials in transport models have to be determined by the EOS properties of nuclear matter at zero temperature. Thus, the SRC\/HMT in cold nuclear matter is important for determining the EOS of hot matter formed during heavy-ion reactions. \nWhile the EOS of hot dense neutron-rich matter is a necessary input for simulating neutron star mergers and understanding post-merger spectrum of high-frequency gravitational waves.\nMoreover, the HMT at zero temperature is also useful for investigating properties of cold neutron stars, e.g., the proton fraction and the core-crust transition density\/pressure in neutron stars at $\\beta$-equilibrium.\n\nWith the motivations concerning the questions outlined above, the main contributions\/findings of this work can be summarized as follows:\n\\begin{enumerate}\n\\item[(a)]Qualitatively, the pressure $P_0$ of SNM is enhanced while the $E_{\\rm{sym}}(\\rho)$ is reduced (both at sub- and supra-saturation densities) considering the SRC-induced HMT, which are consistent with the predictions from the nonlinear RMF models\\,\\cite{Cai16b}.\nQuantitatively, however, the reduction of $E_{\\rm{sym}}(\\rho)$ is much stronger than that from the RMF models, while the enhancement of $P_0$ is weaker compared with the one predicted from the RMF models.\n\\item[(b)]Due to the much stronger reduction of $E_{\\rm{sym}}(\\rho)$ as pointed out in (a), the maximum mass of neutron stars is found to decrease correspondingly, compared with calculations using the FFG momentum distribution.\nIn this sense, it is not quite clear whether the SRC-induced HMT tends to help support heavy neutron stars (around $2M_{\\odot}$).\nMore studies\/possible mechanisms for stiffening high-density EOS within the Gogny-like EDFs are still needed.\n\\item[(c)]The surrogate Gogny-like EDF is shown to produce reasonable (kinetic) energy-dependence of the single-nucleon potential compared with known nucleon optical potentials at $\\rho_0$,\nthus it improves another aspect of the ANM EOS with the HMT, compared with the nonlinear RMF models\\,\\cite{Cai16b}.\n\\item[(d)]The isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ is largely enhanced as the $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$ is enhanced (while the $E_{\\rm{sym}}^{\\rm{kin}}(\\rho)$ is reduced).\nThis is a new feature originated from the SRC\/HMT within the Gogny-like EDF compared to the nonlinear RMF models where there is no momentum-dependence. It provides a novel connection linking the SRC-induced HMT with the dynamics of HICs involving neutron-rich nuclei.\n\\item[(e)]Parametrized forms of the EOS $E_0(\\rho)$ and single-nucleon potential $U_0(\\rho,\\v{k}$) in SNM as well as the symmetry energy $E_{\\rm{sym}}(\\rho)$ and isovector potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ are provided to facilitate their future applications in both nuclear physics and astrophysics.\n\\end{enumerate}\n\nA brief report on some preliminary results of our study on some issues listed above can be found in a conference proceedings\\,\\cite{CLC18}. The rest of this paper is organized as follows: In section \\ref{sec2}, the single-nucleon momentum distribution function $n_{\\v{k}}^J$ as well as other relevant parameters in the calculations are given. The\nmomentum-dependent nucleon potential used in the present work will also be discussed in this section.\nIn section \\ref{sec4}, several analytical expressions for the characteristics of the NM EOS are given. \nSection \\ref{sec5} is devoted to numerical demonstrations of the formulas developed in section \\ref{sec4}. Section \\ref{sec6} gives the summary of the present work. The detailed\nderivations of most formulas are given in the appendix.\n\n\\setcounter{equation}{0}\n\\section{Gogny-like energy density functional encapsulating SRC\/HMT effects}\\label{sec2}\n\n\n\\subsection{Characteristics of the ANM EOS}\n\nWe first briefly recall definitions of\nseveral quantities characterizing the EOS of ANM.\nAround the saturation density $\\rho _{0}$, the $E_{0}(\\rho )$ can be\nexpanded up to the third-order in density as,\n\\begin{equation}\nE_{0}(\\rho )\\approx E_{0}(\\rho _{0})+\\frac{1}{2}K_0\\chi\n^{2}+\\frac{1}{6}J_0\\chi ^{3}, \\label{DenExp0}\n\\end{equation}%\nwhere $\\chi =(\\rho -\\rho _{0})\/3\\rho _{0} $ is a dimensionless\nvariable characterizing the deviations of the density from $\\rho _{0}$. The first term $E_{0}(\\rho _{0})$ on\nthe right-hand-side of Eq.\\,(\\ref{DenExp0}) is the binding energy per\nnucleon in SNM at $\\rho _{0}$ and,\n\\begin{align}\nK_{0} =&\\left. 9\\rho _{0}^{2}\\frac{\\text{d} ^{2}E_{0}(\\rho\n)}{\\text{d} \\rho ^{2}}\\right\\vert _{\\rho =\\rho\n_{0}},~~J_0=\\left.27\\rho_0^3\\frac{\\d^3E_0(\\rho)}{\\d\\rho^3}\\right|_{\\rho=\\rho_0},\n\\label{K0}\n\\end{align}\nare the incompressibility coefficient and the\nskewness\\,\\cite{Ste10,Cai14,Sel14} of the SNM, respectively.\n\n\nSimilarly, one can expand the $E_{\\mathrm{sym}}(\\rho )$ around the\nsaturation density as\n\\begin{equation}\nE_{\\text{sym}}(\\rho)\\approx\nE_{\\text{sym}}(\\rho_0)+L\\chi\n+\\frac{1}{2}K_{\\rm{sym}}\\chi^2+\\frac{1}{6}J_{\\rm{sym}}\\chi^3, \\label{EsymLKr}\n\\end{equation}\nwith the slope parameter $L$ defined as,\n\\begin{align}\nL\\equiv&\\left.3\\rho_0\\frac{\\text{d}E_{\\mathrm{sym}}(\\rho)}{\\text{d}\\rho}\\right|_{\\rho\n=\\rho_{0} }.\n\\end{align}\nThe curvature and the skewness\\,\\cite{ZL18,ZL19,ZL21,ZL22} of the symmetry energy are defined similarly as \n\\begin{align}\nK_{\\rm{sym}} =&\\left.9\\rho _{0}^{2}\\frac{\\text{d} ^{2}E_{\\rm{sym}}(\\rho\n)}{\\text{d} \\rho ^{2}}\\right|_{\\rho =\\rho\n_{0}},\\\\\nJ_{\\rm{sym}}=&\\left.27\\rho_0^3\\frac{\\d^3E_{\\rm{sym}}(\\rho)}{\\d\\rho^3}\\right|_{\\rho=\\rho_0}.\n\\end{align}\n\n\\subsection{Single-nucleon Momentum Distribution}\n\nWe describe the SRC-modified single-nucleon momentum\ndistribution function encapsulating a HMT used in the present work.\nMore details can be found in Ref.\\,\\cite{Cai15}. The single-nucleon\nmomentum distribution function in cold ANM takes the following\nform\\,\\cite{Cai15},\n\\begin{equation}\\label{MDGen}\nn^J_{\\v{k}}(\\rho,\\delta)=\\left\\{\\begin{array}{ll}\n\\Delta_J,~~&0<|\\v{k}|<k_{\\rm{F}}^J,\\\\\n\\displaystyle{C}_J\\left({k_{\\rm{F}}^{J}}\/{|\\v{k}|}\\right)^4,~~&k_{\\rm{F}}^J<|\\v{k}|<\\phi_Jk_{\\rm{F}}^J.\n\\end{array}\\right.\n\\end{equation}\nwhere $k_{\\rm{F}}^J=k_{\\rm{F}}(1+\\tau_3^J\\delta)^{1\/3}$ is the nucleon Fermi\nmomentum with $k_{\\rm{F}}=(3\\pi^2\\rho\/2)^{1\/3}$ and\n$\\tau_3^{\\rm{n}}=+1$, $\\tau_3^{\\rm{p}}=-1$.\nThe above form of nucleon momentum distribution function is\nconsistent with the well-known predictions of microscopic nuclear\nmany-body theories\\,\\cite{Bethe,Ant88,Arr12,Cio15} and the recent\nexperimental\nfindings\\,\\cite{Hen14,Hen15,Wei15,Wei15a,Col15}.\nVery recently, an analysis of the energy spectra of protons emitted in reactions of 47\\,MeV\/u  (several light to medium sized) projectiles on Sn and Au targets provides new evidence for the $1\/k^4$ shape of the HMTs in the intrinsic momenta spectra of the projectiles\\,\\cite{Hag21}. \n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=7.cm]{Fig1.eps}\n  \\caption{Sketch of the $n_{\\v{k}}^J$ with a HMT. Taken from Refs.\\,\\cite{Cai15,Cai16b}.}\n  \\label{mom-dis}\n\\end{figure}\n\nIn (\\ref{MDGen}), the $\\Delta_J$ measures the depletion of the Fermi\nsphere at zero momentum with respect to the FFG model. The sketch of\n$n^J_{\\v{k}}(\\rho,\\delta)$ is shown in FIG.\\,\\ref{mom-dis}. The\nisospin structure of the parameters $C_J$ and $\\phi_J$\nis found to be $Y_J=Y_0(1+Y_1\\tau_3^J\\delta)$\\,\\cite{Cai15}. The\namplitude ${C}_J$ and high-momentum cutoff coefficient $\\phi_J$\ndetermine the fraction of nucleons in the HMT via\n\\begin{equation}\\label{xPNM}\nx_J^{\\rm{HMT}}=3C_{{J}}\\left(1-\\phi_J^{-1}\\right).\n\\end{equation}\nThe normalization condition between the density $\\rho_J$\nand the distribution $n_{\\v{k}}^J$, i.e.,  $\n[{2}\/{(2\\pi)^3}]\\int_0^{\\infty}n^J_{\\v{k}}(\\rho,\\delta)\\d\\v{k}=\\rho_J={(k_{\\rm{F}}^{J})^3}\/{3\\pi^2}\n$ requires that only two of the three parameters, i.e., ${C}_J$,\n$\\phi_J$ and $\\Delta_J$, are independent. Here we choose the first\ntwo as independent and determine the $\\Delta_J$ by\n\\begin{equation}\\label{DeltaJ}\n\\Delta_J=1-3{C}_J\\left(1-\\phi_J^{-1}\\right)=1-x_J^{\\rm{HMT}}.\n\\end{equation}\n\nThe ${C}\/{|\\v{k}|^4}$ shape of the HMT both for SNM and PNM is\nstrongly supported by recent studies both theoretically and\nexperimentally. Combining the results from analyzing cross sections\nof $\\rm{d}(\\rm{e}, \\rm{e}^{\\prime}\\rm{p})$ reactions\\,\\cite{Hen15}\nand medium-energy photonuclear absorptions\\,\\cite{Wei15}, the $C_0$\nwas found to be $C_0\\approx0.161\\pm0.015$ in the HMT-exp\nmodel\\,\\cite{Cai15}. With this $C_0$ and the value of\n$x^{\\rm{HMT}}_{\\rm{SNM}}\\approx28\\%\\pm4\\%$\\,\\cite{Hen14,Hen15,Hen15b}\nobtained from systematic analyses of inclusive (e,e$'$) reactions\nand data from exclusive two-nucleon knockout reactions, the HMT\ncutoff parameter in SNM is determined to be\n$\\phi_0=(1-x_{\\rm{SNM}}^{\\rm{HMT}}\/3{C}_0)^{-1}\\approx2.38\\pm0.56$\\,\\cite{Cai15}.\nFurthermore, the value of $C_{\\rm{n}}^{\\rm{PNM}}= C_0(1+C_1)$ was\nextracted by applying the adiabatic sweep theorem\\,\\cite{Tan08} to\nthe EOS of PNM constrained by predictions of microscopic nuclear\nmany-body theories\\,\\cite{Sch05,Epe09a,Tew13,mm,Gez13,Gez10} and the\nEOS of ultra-cold atoms under unitary\ncondition\\,\\cite{Tan08,Stew10,Kuh10}. And more specifically,\n$C_{\\rm{n}}^{\\rm{PNM}}\\approx0.12$ and $C_1\\approx-0.25\\pm0.07$ were\nobtained\\,\\cite{Cai15}. Moreover, by inserting the values of\n$x_{\\rm{PNM}}^{\\rm{HMT}}\\approx1.5\\%\\pm0.5\\%$\\,\\cite{Hen14,Hen15,Hen15b}\nextracted in the same way as the $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\n$C_{\\rm{n}}^{\\rm{PNM}}$ into Eq.\\,({\\ref{xPNM}), the high momentum\ncutoff parameter for PNM was determined to be\n$\\phi_{\\rm{n}}^{\\rm{PNM}}\\equiv\n\\phi_0(1+\\phi_1)=(1-x_{\\rm{PNM}}^{\\rm{HMT}}\/3C_{\\rm{n}}^{\\rm{PNM}})^{-1}\\approx1.04\\pm0.02$\\,\\cite{Cai15},\nthus $\\phi_1\\approx-0.56\\pm0.10$\\,\\cite{Cai15} was obtained. \nSee Ref.\\,\\cite{Cai15} for more details on these discussions.\nOn the other hand, if we take $x_{\\rm{SNM}}^{\\rm{HMT}}\\approx12\\%$ and\n$x_{\\rm{PNM}}^{\\rm{HMT}}\\approx4\\%$ in the HMT-SCGF model, and using\na very similar scheme to determine the parameters in the HMT-exp\nmodel, we obtain $\\phi_0\\approx1.49, \\phi_1\\approx-0.25$,\n$C_0\\approx0.121$ and $C_1\\approx-0.01$\\,\\cite{Cai16c}.\nIn the following, the parameter sets adopting the experimental verification and the SCGF are denoted as HMT-exp and HMT-SCGF, respectively, and the corresponding calculations are made under the HMT-exp model or the HMT-SCGF model.\n\n\nBesides the effects of the SRC-induced HMT on the kinetic symmetry\nenergy, several related interesting problems are explored recently,\nincluding the enhancement of the isospin quartic term\\,\\cite{Cai15},\nthe predictions on the effective E-mass together with its isospin\nsplitting at the Fermi surface\\,\\cite{Cai16a,LiBA15,LiBA16}, the\ncorrelation between the neutron skin thickness and the corresponding\nkinetic energy density in the surface region of heavy\nnuclei\\,\\cite{Cai16c}, and the effects on the neutron star\nmass-radius relation\\,\\cite{Cai16b,Hen16}. \nRecently, the EOS of ANM considering the HMT in a general space of dimension $d$ is investigated\\,\\cite{CaiLi2022,CaiLi2022a}, see Ref.\\,\\cite{LiBA2018PPNP} for a recent review on more related issues\\,\\cite{Souza2020,YongGC2017PRC,YongGC2018PLB,\nWang2017PRC,Guo2021PRC,Yong2022,Yang2019,Bulgac2022,Bulgac2022a,Miller2019}.\n\n\n\n\\subsection{Surrogate Momentum-dependent Single-nucleon Potential in ANM}\n\nThe total EOS of ANM could be decomposed into its kinetic and potential parts as\n\\begin{equation}\\label{Gen-EOS}\nE(\\rho,\\delta)=E^{\\rm{kin}}(\\rho,\\delta)+E^{\\rm{pot}}(\\rho,\\delta)\n\\end{equation}\nwith the kinetic part given by\n\\begin{equation}\\label{Ekin}\nE^{\\rm{kin}}(\\rho,\\delta)=\\frac{1}{\\rho}\\frac{2}{(2\\pi)^3}\\sum_{J=\\rm{n,p}}\\int_0^{\\infty}\\frac{\\v{k}^2}{2M}n_{\\v{k}}^J(\\rho,\\delta)\\d\\v{k}\n\\end{equation}\nwhere $n_{\\v{k}}^J(\\rho,\\delta)$ is the single-nucleon momentum\ndistribution function, i.e., a step function in the FFG model or the one given in Eq.\\,(\\ref{MDGen}) in the HMT model. The potential part\nin the present work adopts the following form\\,\\cite{CLC18},\n\\begin{align}\\label{Epot}\nE^{\\rm{pot}}(\\rho,\\delta)=&\\sum_{J,J'}\\frac{C_{J,J'}}{\\rho\\rho_0}\\int\\d\\v{k}\\d\\v{k}'f_J(\\v{r},\\v{k})f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}')\\notag\\\\\n&+\\frac{A_\\ell(\\rho_{\\rm{p}}^2+\\rho_{\\rm{n}}^2)}{2\\rho\\rho_0}\n+\\frac{A_{\\rm{u}}\\rho_{\\rm{p}}\\rho_{\\rm{n}}}{\\rho\\rho_0}\\notag\\\\\n&+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\left(1-x\\delta^2\\right),\n\\end{align}\nhere \\begin{equation}\nA_\\ell=A_\\ell^0+{2xB}\/({1+\\sigma}),~~A_{\\rm{u}}=A_{\\rm{u}}^0-{2xB}\/({1+\\sigma}).\n\\end{equation}\nBesides the first line of (\\ref{Epot}), the remaining terms are\ntotally the same as those in the original Momentum Dependent Interaction (MDI)\\,\\cite{Gale1987,Bertsch1988,Pawel,Das2003,LCK08,LiBA04,LiChen05,Cozma17,Cozma21,Che05,XuJ10,XuJ15}. \nMoreover, $A_\\ell,\nA_{\\rm{u}},B,\\sigma$ and $x$ are five phenomenological\nparameters characterizing the density dependence of the nucleon\npotential. In particular, if the replacements\n$t_3=16B\/(\\sigma+1)\\rho_0^{\\sigma}$ and $x_3=(3x-1)\/2$ are made,\nthen the parameters $B,\\sigma$ and $x$ are reduced to the ones widely used to describe the\neffective three-nucleon forces\\,\\cite{XuJ10}. Furthermore, the\nparameter $x$ only affects the slope parameter of the symmetry energy. It does not affect the magnitude of the symmetry energy at $\\rho_0$, providing a convenient way to investigate the density dependence of the symmetry energy\\,\\cite{Das2003,Che05}.\nFor the same type of nucleons, one has $C_{J,J}\\equiv C_\\ell$, while for the\nunlike ones,  then $C_{J,\\overline{J}}\\equiv C_{\\rm{u}}$,\nhere $\\overline{\\rm{n}}=\\rm{p}$ and $\\overline{\\rm{p}}=\\rm{n}$.\n\nThe single-nucleon potential corresponding to the EOS of (\\ref{Gen-EOS}) is given by\n\\begin{align}\\label{Gen-U}\nU_J(\\rho,\\delta,|\\v{k}|)=&\\frac{A_\\ell\\rho_J}{\\rho_0}+\\frac{A_{\\rm{u}}\\rho_{\\overline{J}}}{\\rho_0}\n+B\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\left(1-x\\delta^2\\right)\\notag\\\\\n&-4x\\tau_3^J\\frac{B}{\\sigma+1}\\frac{\\rho^{\\sigma-1}}{\\rho_0^{\\sigma}}\\delta\\rho_{\\overline{J}}\n\\notag\\\\\n&+\\sum_{J'}\\frac{2C_{J,J'}}{\\rho_0}\\int\\d\\v{k}'f_{J'}(\\v{r},\\v{k}')\\Omega(\\v{k},\\v{k}').\n\\end{align}\n The momentum-dependence of the single-nucleon potential is controlled by\nthe last term, which is composed of two intrinsically different\nterms, i.e., $f_J(\\v{r},\\v{k})$ and $\\Omega(\\v{k},\\v{k}')$, both are\nfunctions of momentum, leading to a non-trivial momentum behavior of\nthe nucleon potential $U_J(\\rho,\\delta,|\\v{k}|)$ once the HMT in the $n_{\\v{k}}^J$ is\nconsidered.\n\nThe $f_J$ used in several integrations above is the phase space distribution function of nucleon $J$. It is related to the nucleon momentum distribution $n_{\\v{k}}^J(\\rho,\\delta)$ by\n\\begin{equation}\nf_J(\\v{r},\\v{k})=\\frac{2}{h^3}n_{\\v{k}}^J(\\rho,\\delta)=\\frac{1}{4\\pi^3}n_{\\v{k}}^J(\\rho,\\delta),~~\\hbar=1.\n\\end{equation}\nFor example, in the FFG model,\n$n_{\\v{k}}^J=\\Theta(k_{\\rm{F}}^J-|\\v{k}|)$ with $\\Theta$ the\nstandard step function, then\n$f_J(\\v{r},\\v{k})=(1\/4\\pi^3)\\Theta(k_{\\rm{F}}^J-|\\v{k}|)$. As mentioned briefly earlier, the $\\Omega(\\v{k},\\v{k}')$ in (\\ref{Epot}) is the function characterizing\nthe momentum dependence of the single-nucleon potential due to the finite-range of nuclear interactions.\nFor instance, for a Yukawa form of the two-nucleon interaction\n$\\exp[-\\mu|\\v{r}|]\/|\\v{r}|$ with $|\\v{r}|=|\\v{x}-\\v{x}'|$\nthe distance between two nucleons,  one obtains $\n\\Omega(\\v{k},\\v{k}')=[1+(\\v{k}-\\v{k}')^2\/\\Lambda^2)]^{-1}$, which\nis the familiar MDI interaction\\,\\cite{Gale1987,Bertsch1988,Das2003}. \nIn applying the above formalism to transport model simulations of nuclear reactions, the $f_J (\\v{r}, \\v{k})$ and $n_{\\v{k}}^J$ become time dependent and they are calculated self-consistently from solving\ndynamically the coupled Boltzmann-Uehling-Uhlenbeck (BUU) transport equations for quasi-nucleons\\,\\cite{Bertsch1988,Aichelin1990,Sto86}. \nSimilarly, in the original Gogny interaction\\,\\cite{GOG}, the finite-range two-nucleon interaction has the form of a Gaussian\nregulator $\\exp[-\\mu^2|\\v{r}|^2]$, and the corresponding $\\Omega$ takes the form of $\\exp[-\\mu^2|\\v{k}-\\v{k}'|^2\/4]$.\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=8.cm]{Fig2.eps}\n  \\caption{Sketch of the function $\\Omega(\\v{k},\\v{k}')$, where the blue line is only effectively applicable.}\n  \\label{fig_Omega}\n\\end{figure}\n\nIt is useful to emphasize here the following basic properties of $\\Omega(\\v{k},\\v{k}')$: \n\\begin{enumerate}\n\\item[(a)] It is symmetric between the momentum $\\v{k}$ and $\\v{k}'$, i.e.,\n$\\Omega(\\v{k},\\v{k}')=\\Omega(\\v{k}',\\v{k})$, reflecting the scalar\nproperty of the nucleon potential. Thus $\\Omega$ is a function of\n$\\v{k}+\\v{k}'$ and $\\v{k}\\cdot\\v{k}'$, and only the second term\ncouples the momenta of the two nucleons.\nIn general, one can show that it only depends on the magnitude of the difference $|\\v{k}-\\v{k}'|$.\nBoth the Yukawa potential and the Gogny force have such form.\n\n\\item[(b)] At zero momentum, $\\Omega(\\v{k},\\v{k}')$ is a constant (which can\nalways be normalized to unity), indicating\n$\\Omega(\\v{k},\\v{k}')\\approx1+\\rm{``corrections''}$ when\n$|\\v{k}|\\approx0,|\\v{k}'|\\approx0$. Furthermore, as the momentum\nscale increases, the momentum-dependent interaction between nucleons\ndecreases, even to become free\/negative, leading to that\n$\\Omega(|\\v{k}|\\to\\infty,|\\v{k}'|\\to\\infty)\\to0$ or approaches a\nsmall negative quantity $\\mathcal{N}$. This property is relevant to\nthe fact, e.g., that the single-nucleon potential in SNM at large energies\nsaturates. Consequently, the above ``corrections'' in\n$\\Omega(\\v{k},\\v{k}')$ beyond the leading contribution ``1'' at\nmoderate momenta should be negative.\nFor example, we have $\\Omega(\\v{k},\\v{k}')\\approx 1-|\\v{k}-\\v{k}'|^2\/\\Lambda^2$ for the Yukawa potential and \n$\\Omega(\\v{k},\\v{k}')\\approx1-\\mu^2|\\v{k}-\\v{k}'|^2\/4$ for the Gogny force.\nIn fact, both of them can be approximated as $\\Omega(\\v{k},\\v{k}')\\approx 1+c_1\\v{k}^2\/\\Lambda^2+c_2\\v{k}'^2\/\\Lambda^2+c_3\\v{k}\\cdot\\v{k}'\/\\Lambda^2$, where the coefficients $c_i$'s depend on the detailed form of the interactions.\nWe will use this perturbative expansion as a hint to construct a reasonable surrogate $\\Omega(\\v{k},\\v{k}')$, see (\\ref{MFunc}).\n\n\\item[(c)] Based on the last point, a perturbation-effective version of the\nfunction $\\Omega(\\v{k},\\v{k}')$ can be constructed generally as\n$\\Omega(\\v{k},\\v{k}')\\approx\n1+af(|\\v{k}-\\v{k}'|)+bg(|\\v{k}-\\v{k}'|)+\\cdots$, where the\nphenomenological functions $f,g,\\cdots$ are small at moderate\nmomentum scale. This effective $\\Omega(\\v{k},\\v{k}')$ can only be\nused up to a momentum scale around $\\Lambda$, see the sketch illustrating this feature in FIG.\\,\\ref{fig_Omega}, where the $\\Omega(\\v{k},\\v{k}')$ denoted by the\nblue line is perturbatively effective up to around a certain high momentum\ncutoff. Namely, the surrogate $\\Omega(\\v{k},\\v{k}')$ constructed in this manner grasps the main features of the original $\\Omega(\\v{k},\\v{k}')$ up to the high momentum cutoff.\n\\end{enumerate}\nConsidering the SRC-induced HMT in the single-nucleon momentum distribution and the kernel $\\Omega(\\v{k},\\v{k}')$ for the momentum dependent single-nucleon potential either with the Yukawa or Gogny force, their generally complicated form and nature hinder us from integrating out analytically the relevant integrations over momentum in the expressions of nucleon specific energy and single-nucleon potential. While these integrations are necessary for fixing all the EOS parameters using empirical properties of nuclear matter, symmetry energy, nucleon effective mass, and nucleon optical potentials at $\\rho_0$, etc. \n\nMaking use of the main features of $\\Omega$ outlined above, we adopt in this work the following parametrization as a surrogate of $\\Omega(\\v{k},\\v{k}')$ \n\\begin{equation}\\label{MFunc}\n\\Omega(\\v{k},\\v{k}')=1+{a}\\left[\\left(\\frac{\\v{k}\\cdot\\v{k}'}{\\Lambda^2}\\right)^2\\right]^{1\/4}\n+{b}\\left[\\left(\\frac{\\v{k}\\cdot\\v{k}'}{\\Lambda^2}\\right)^2\\right]^{1\/6},\n\\end{equation}\nwhere $a$ and $b$ are two phenomenological parameters, and\n$\\Lambda\\approx1\\,\\rm{GeV}$ is an effective momentum scale. The\nparameter $a$ should be negative according to the above discussion (see point (b)).\nAs pointed out in Ref.\\,\\cite{CLC18}, the advantages of using this surrogate is twofold: (1) the final 1\/2 and 1\/3 power of $\\v{k}\\cdot\\v{k}'\/\\Lambda^2$ is relevant for describing properly the kinetic energy dependence of nucleon optical potential, as we shall show in FIG.\\,\\ref{fig_ab_U0}; (2) it enables analytical expressions for both the EOS of ANM and $U_J(\\rho,\\delta, |\\v{k}|)$. \nMoreover,  the $\\Omega$ function is only perturbatively effective at momenta smaller than the momentum scale $\\Lambda$, indicating that the EDF thus constructed can only be used to a restricted range of momentum and\/or density.  For example,  the cutoff of the HMT about $2.4k_{\\rm{F}}\\approx631\\,\\rm{MeV}$ or the density $\\rho\\approx7\\rho_0\\mbox{$\\sim$}8\\rho_0$ corresponding to $k_{\\rm{F}}\\approx503\\,\\rm{MeV}\\mbox{$\\sim$}526\\,\\rm{MeV}$ should be safely smaller than the $\\Lambda$ parameter $\\approx1\\,\\rm{GeV}$.\nIn this sense, the parameterized $\\Omega$ (\\ref{MFunc}) is a reliable surrogate\/effective model\\,\\cite{SURR} for studying properties of neutron stars and intermediate-relativistic energy heavy-ion reactions where the maximum density reached is less than the above estimates.\n\n\\setcounter{equation}{0}\n\\section{Analytical expressions of the SNM EOS, nucleon effective mass, symmetry energy, isoscalar and isovector potentials}\\label{sec4}\nIn this section, we present all the analytical expressions of quantities relevant for describing the ANM EOS using the surrogate function (\\ref{MFunc}) for $\\Omega(\\v{k},\\v{k}')$. Detailed derivations are given in the APPENDIX \\ref{app1}. More specifically, in subsection \\ref{sb_EOS_SNM} we give the SNM EOS together with the isoscalar nucleon effective mass $M_0^{\\ast}(\\rho)$, the coefficient of incompressibility $K_0(\\rho)$ and the single-nucleon potential $U_0(\\rho)$ in SNM. In subsection \\ref{sb_Esym}, the expression for the symmetry energy is derived. For general expressions of the ANM EOS $E(\\rho,\\delta)$, see Eq.\\,(\\ref{HMT-Erd}) (together with Eqs.\\,(\\ref{HMT_Erd1}), (\\ref{HMT_Erd2}) and (\\ref{HMT_Erd3})) and Eq.\\,(\\ref{FFG-Erd}).\\\\\n\n\\subsection{EOS of SNM}\\label{sb_EOS_SNM}\n\n For the kinetic EOS of SNM, we\nhave\\,\\cite{Cai15}\n\\begin{equation}\\label{E0kin-HMT}\nE^{\\rm{kin}}_0(\\rho)=\\frac{3}{5}E_{\\rm{F}}(\\rho)\\left[\n1+{C}_0\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\\right],\n\\end{equation}\nwhere $E_{\\rm{F}}=k_{\\rm{F}}^2\/2M$ is the Fermi kinetic energy. The\nhigh momentum cutoff $\\phi_0$ is generally larger than unity,\nthus $\\Upsilon_0=1+C_0(5\\phi_0+3\/\\phi_0-8)\\geq1$, indicating\nthat the $E^{\\rm{kin}}_0(\\rho)$ in the presence of HMT is enhanced\ncompared to the prediction of using the FFG nucleon momentum distribution\\,\\cite{Cai15}. This conclusion has\nimportant consequences on the total EOS of SNM once the potential\npart is taken into account, i.e., as the kinetic part is enhanced,\nthe potential part should be correspondingly reduced, leading to a\nreduction of the single-nucleon potential. Furthermore, the\ndensity-dependent terms in the EOS is the same as those in the MDI\nmodel\\,\\cite{Das2003,LCK08,Che05,XuJ10,XuJ15}. The total EOS of SNM in\nthe HMT model is thus,\n\\begin{widetext}\n\\begin{align}\\label{HMT-E0-mm}\nE_0(\\rho)=&\\frac{3k_{\\rm{F}}^2}{10M}\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\n+\\frac{1}{4}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\\frac{1}{2}C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[1+\\frac{24a}{49}\\left[\\Delta_0+7C_0\\left(1-\\frac{1}{\\sqrt{\\phi_0}}\\right)\\right]^2\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)+\n\\frac{243b}{400}\\left[\\Delta_0+5C_0\\left(1-\\frac{1}{\\phi_0^{2\/3}}\\right)\\right]^2\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\\right],\n\\end{align}\n\\end{widetext}\nsee also (\\ref{HMT-E0}), where\n$A_{\\rm{tot}}=A_\\ell+A_{\\rm{u}}=A_\\ell^0+A_{\\rm{u}}^0$ and\n$C_{\\rm{tot}}=C_\\ell+C_{\\rm{u}}$. The two factors $\\Pi_1$ and $\\Pi_2$ characterizing the HMT\nin Eq.\\,(\\ref{HMT-E0-mm}) are given by\n\\begin{align}\n\\Pi_1=&\\Delta_0+7C_0\\left(1-\\frac{1}{\\sqrt{\\phi_0}}\\right),\\label{def_Pi1}\\\\\n\\Pi_2=&\\Delta_0+5C_0\\left(1-\\frac{1}{\\phi_0^{2\/3}}\\right).\n\\end{align}\n\nIn the FFG model (as a reminder, the ``FFG'' here means that the $n_{\\v{k}}^J$ takes the step function while the potential part is also included in calculating the EOS and related quantities), the high momentum cutoff $\\phi_0$ becomes $\\phi_0=1$ and $\\phi_1=0$, we have $\n\\Delta_0=1-3C_0(1-\\phi_0^{-1})\\to1$,  and the two factors $\\Pi_1$ and $\\Pi_2$ become $\\Pi_1\\to1,\\Pi_2\\to1$. In the same limit, the Eq.\\,(\\ref{HMT-E0-mm}) is then reduced to\n\\begin{align}\\label{FFG-E0-mm}\nE_0(\\rho)=&\\frac{3k_{\\rm{F}}^2}{10M}+\\frac{1}{4}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{B}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\\frac{1}{2}C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[1+\\frac{24a}{49}\\frac{k_{\\rm{F}}}{\\Lambda}\n+\\frac{243b}{400}\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\\right],\n\\end{align}\nwhich is derived (independently) in the FFG model, see derivations leading to Eq.\\,(\\ref{FFG-E0}).\n\nThe density structure of both the $E_0(\\rho)$ in the\nFFG and the HMT model is the same, i.e.,\n\\begin{equation}\\label{str}\nE_0(\\rho)=\\nu_1\\rho^{2\/3} +\\nu_2\\rho+\\nu_3\\rho^{\\sigma}+\n\\nu_4\\rho^{4\/3}+\\nu_5\\rho^{11\/9},\n\\end{equation}\nwith different coefficient $\\nu_j$'s in the FFG\/HMT model.\nThus, $E_0(\\rho)$ is a generalized polynomial of density $\\rho$ (or equivalently a polynomial of Fermi momentum $k_{\\rm{F}}$),\nindicating that it is only effectively applicable to low densities.\nConsequently, the density structure of the pressure of SNM and the incompressibility can be worked out as\n\\begin{align}\nP_0(\\rho)\/\\rho=&\\frac{2}{3}\\nu_1\\rho^{2\/3} +\\nu_2\\rho\n+\\sigma\\nu_3\\rho^{\\sigma}\\notag\\\\\n&+\\frac{4}{3}\\nu_4\\rho^{4\/3}\n+\\frac{11}{9}\\nu_5\\rho^{11\/9},\\\\\nK_0(\\rho)=&-2\\nu_1\\rho^{2\/3}+9\\sigma(\\sigma-1)\\nu_3\\rho^{\\sigma}\\notag\\\\\n&+4\\nu_4\\rho^{4\/3}+\\frac{22}{9}\\nu_5\\rho^{11\/9}.\n\\end{align}\nSpecifically, the expressions for the pressure $P_0(\\rho)$ and incompressibility $K_0(\\rho)$ in the FFG model are\ngiven in Eqs.\\,(\\ref{FFG-p0}) and (\\ref{FFG-K0}), while those in the\nHMT model are given in Eqs.\\,(\\ref{HMT-p0}) and (\\ref{HMT-K0}),\nrespectively.\n\nBesides the SNM EOS $E_0(\\rho)$ in the HMT model, the single-nucleon potential as a function of density $\\rho$ and momentum $\\v{k}$ in SNM could also be given,\n\\begin{align}\\label{HMT-U0-mm}\n&U_0(\\rho,|\\v{k}|)=\\frac{1}{2}A_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)+B\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\n+C_{\\rm{tot}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\notag\\\\\n&\\times\\left[1+\\frac{4a}{7}\\Pi_1\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}\n+\\frac{27b}{40}\\Pi_2\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}\\right],\n\\end{align}\nsee Eq.\\,(\\ref{HMT-U0}) for a detailed derivation. Again the above expression can be reduced to the FFG one, see Eq.\\,(\\ref{FFG-U0}). The\nmomentum structure of the $U_0$ could be written explicitly as (due to the function $\\Omega$),\n\\begin{equation}\nU_0(\\rho,|\\v{k}|)=\\zeta_0(\\rho)+\\zeta_1(\\rho)|\\v{k}|^{1\/2}+\\zeta_2(\\rho)|\\v{k}|^{1\/3},\n\\end{equation}\nwith\n$\\zeta_0(\\rho)=(A_{\\rm{tot}}\/2+C_{\\rm{tot}})(\\rho\/\\rho_0)+B(\\rho\/\\rho_0)^{\\sigma}$. The latter is unchanged in the presence of HMT, while the form of $\\zeta_1(\\rho)$ and $\\zeta_2(\\rho)$ should change correspondingly, compare (\\ref{FFG-U0}) and (\\ref{HMT-U0}).\nThe concavity of the $U_0$ at the saturation density\\,\\cite{Ham90} indicates that $C_{\\rm{tot}}$ is negative according to $\n\\zeta_1(\\rho)=4C_{\\rm{tot}}(\\rho\/\\rho_0)a\\Pi_1k_{\\rm{F}}^{1\/2}\/7\\Lambda>0$,\nwhere $a$ is a negative parameter as expected by general discussions (given in the last section) and\n$\\Pi_1$ is positive (see the expression (\\ref{def_Pi1})). The momentum-dependence of the $U_0$ is characterized by introducing the nucleon effective k-mass\\,\\cite{LiBA2018PPNP,LiBA15,LiBA16}, i.e.,\n\\begin{equation}\nM_0^{\\ast}(\\rho,|\\v{k}|)\/M=\\left[1+\\frac{M}{|\\v{k}|}\\frac{\\partial\nU_0}{\\partial|\\v{k}|}\\right]^{-1}.\n\\end{equation}\nSpecifically, we obtain for the current EDF the following expression \n\\begin{align}\n\\frac{M_0^{\\ast}(\\rho,|\\v{k}|)}{M}=\\Bigg[&1+C_{\\rm{tot}}M\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\Bigg[\\frac{2a\\Pi_1}{7}\\left(\\frac{k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}|\\v{k}|^{-3\/2}\\notag\\\\\n&+\\frac{9b\\Pi_2}{40}\\left(\\frac{k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}|\\v{k}|^{-5\/3}\\Bigg]\\Bigg]^{-1},\n\\end{align}\nwhich is further evaluated at the Fermi surface $|\\v{k}|=k_{\\rm{F}}$ to give the density dependence of the $M_0^{\\ast}(\\rho)\/M$ as,\n\\begin{align}\\label{FFG-M0-mm}\n\\frac{M_0^{\\ast}(\\rho)}{M}\n=\\Bigg[&1+C_{\\rm{tot}}M\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\Bigg[\\frac{2a\\Pi_1}{7}\\frac{1}{k_{\\rm{F}}\\Lambda}\\notag\\\\\n&+\\frac{9b\\Pi_2}{40}\\left(\\frac{1}{\\Lambda^2k_{\\rm{F}}^4}\\right)^{1\/3}\\Bigg]\\Bigg]^{-1}.\n\\end{align}\nIn the high density limit, we have\n\\begin{equation}\\label{def_11}\n\\lim_{\\rm{large }\\rho}\\frac{M_0^{\\ast}(\\rho)}{M}\n\\approx\\left[1+\\frac{2M\\Pi_1aC_{\\rm{tot}}}{7k_{\\rm{F}}\\Lambda}\\left(\\frac{\\rho}{\\rho_0}\\right)+\\mathcal{O}\\left(k_{\\rm{F}}^{-1\/3}\\right)\\right]^{-1},\n\\end{equation}\nwhich approaches zero in the limit $\\rho\\to\\infty$. In the FFG\nmodel, the density dependence of the effective mass is given in Eq.\\,(\\ref{FFG-M0}), i.e., by taking $\\Pi_1=1$ in (\\ref{def_11}).\n\n\n\\subsection{Symmetry Energy}\\label{sb_Esym}\n\nWe now discuss the symmetry energy, its slope parameter, and the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$. As derived in\ndetailed in APPENDIX \\ref{app1}, the symmetry energy for the general HMT model is given by,\n\\begin{widetext}\n\\begin{align}\\label{HMT-Esym-mm}\nE_{\\rm{sym}}(\\rho)=&\\frac{k_{\\rm{F}}^2}{6M}\\left[1+C_0(1+3C_1)\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\n+3C_0\\phi_1\\left(1+\\frac{5}{3}C_1\\right)\\left(5\\phi_0-\\frac{3}{\\phi_0}\\right)+\\frac{27C_0\\phi_1^2}{5\\phi_0}\\right]\n\\notag\\\\\n&+\\frac{1}{4}C_\\ell\\Delta_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{10}+Y_{11}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{12}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\n+\\frac{1}{4}C_{\\rm{u}}\\Delta_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[\nZ_{10}+Z_{11}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{12}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{3}{2}C_\\ell\\Delta_0C_0\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{20}+Y_{21}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{22}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\n+\\frac{3}{2}C_{\\rm{u}}\\Delta_0C_0\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nZ_{20}+Z_{21}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{22}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{9}{4}C_\\ell C_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nY_{30}+Y_{31}a\\frac{k_{\\rm{F}}}{\\Lambda}+Y_{32}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]+\\frac{9}{4} C_{\\rm{u}} C_0^2\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\nZ_{30}+Z_{31}a\\frac{k_{\\rm{F}}}{\\Lambda}+Z_{32}b\\left(\\frac{k_{\\rm{F}}}{\\Lambda}\\right)^{2\/3}\n\\right]\\notag\\\\\n&+\\frac{1}{4}A_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)-\\frac{Bx}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma},\\end{align}\n\\end{widetext}\nwith the expressions for $Y_{ij}$ and $Z_{ij}$ given in Eqs.\\,(\\ref{def_Y1}), (\\ref{def_Y20}), (\\ref{def_Y21}), (\\ref{def_Y22}),\n(\\ref{def_Y30}), (\\ref{def_Y31}), (\\ref{def_Y32}),\n (\\ref{def_Z1}), (\\ref{def_Z20}), (\\ref{def_Z21}), \n(\\ref{def_Z22}), (\\ref{def_Z30}), (\\ref{def_Z31}), and (\\ref{def_Z32}), respectively.\nThe second line of Eq.\\,(\\ref{HMT-Esym-mm}) represents the\ncontribution from the depletion in the $n_{\\v{k}}^J$ characterized by $\\Delta_0^2$, while the fourth line has the HMT contribution indicated by the\n$C_0^2$. The third line of Eq.\\,(\\ref{HMT-Esym-mm}) is the mixing of\nthe depletion and the HMT characterized by $\\Delta_0C_0$. In the FFG\nmodel, only the first two lines of Eq.\\,(\\ref{HMT-Esym-mm}) survive.\nMoreover, $A_{\\rm{d}}$ and $C_{\\rm{d}}$ are the\ndifference between the like and unlike terms,\n\\begin{equation}\\label{ad}\nA_{\\rm{d}}=A_{\\rm{d}}^0+\\frac{4xB}{1+\\sigma},~~C_{\\rm{d}}=C_\\ell-C_{\\rm{u}},\n\\end{equation}\nwith $\nA_{\\rm{d}}^0=A_\\ell^0-A_{\\rm{u}}^0$. The density structure of\nthe symmetry energy is the same as that of the SNM EOS, see Eq.\\,(\\ref{str}), thus it is straightforward to obtain the slope\nparameter $L(\\rho)$, see Eq.\\,(\\ref{HMT-L}).\n\nSimilarly, the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ is defined through the following Lane potential \\,\\cite{Lan62} \n\\begin{equation}\nU_J(\\rho,\\delta,|\\v{k}|)\\approx\nU_0(\\rho,|\\v{k}|)+U_{\\rm{sym}}(\\rho,|\\v{k}|)\\tau_3^J\\delta+\\mathcal{O}(\\delta^2).\n\\end{equation}\nSuch a decomposition of single-nucleon potential in ANM has been verified by various phenomenological model analyses of nucleon-nucleus scattering data and predictions of microscopic nuclear many-body theories, see, e.g., Refs. \\cite{mu04,Li04,Ron06,Dal1,zuo05,Beh11,LiX13,LiX15}. In the HMT model, it is given by\n\\begin{align}\\label{HMT-Usym-mm}\n&U_{\\rm{sym}}(\\rho,|\\v{k}|)=\\frac{1}{2}A_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)-\\frac{2Bx}{\\sigma+1}\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma}\\notag\\\\\n&+\nC_{\\rm{d}}\\left(\\frac{\\rho}{\\rho_0}\\right)\\left[\\gamma_0+\\gamma_1a\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/2}\n+\\gamma_2b\\left(\\frac{|\\v{k}|k_{\\rm{F}}}{\\Lambda^2}\\right)^{1\/3}\\right].\n\\end{align}\nThe expressions for $\\gamma_0,~\\gamma_1$ and $\\gamma_2$ are given in\nEqs.\\,(\\ref{def_gamma0}), (\\ref{def_gamma1}) and (\\ref{def_gamma2}), respectively.\nNotice that in the FFG model, besides $\n\\Delta_0\\to1$, we also have\n\\begin{align}\n\\Delta_0\\Delta_1=&-{3C_0[C_1(\\phi_0-1)+\\phi_1]}\/{\\phi_0}\\to0,\\\\\n\\Delta_0\\Delta_2=&-{3C_0\\phi_1(C_1-\\phi_1)}\/{\\phi_0}\\to0,\n\\end{align}\nsee (\\ref{Def_Delta012}), thus,\n\\begin{align}\n&Y_{10}\\to2,~~Y_{11}\\to{32}\/{21},~~Y_{12}\\to{33}\/{20},\\\\\n&Z_{10}\\to-2,~~Z_{11}\\to-{8}\/{7},~~Z_{12}\\to-{27}\/{20},\n\\end{align}\ntogether with $Y_{2j}\\to0,~Z_{2j}\\to0,~Y_{3j}\\to0,~Z_{3j}\\to0$ with $j=0,1,2$, see\nthe expressions (\\ref{def_Y20}), (\\ref{def_Y21}), (\\ref{def_Y22}),\n(\\ref{def_Z20}), (\\ref{def_Z21}), (\\ref{def_Z22}), (\\ref{def_Y30}),\n(\\ref{def_Y31}), (\\ref{def_Y32}), (\\ref{def_Z30}),\n(\\ref{def_Z31}) and (\\ref{def_Z32}) and the consequent result is the\nsymmetry energy in the FFG model, see Eq.\\,(\\ref{FFG-Esym}).\nMoreover, the expressions for the slope parameter $L(\\rho)$ and the symmetry potential\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)$ in the FFG model are given in Eq.\\,(\\ref{FFG-L}) and Eq.\\,(\\ref{FFG-Usym}), respectively.\n\n\nConsidering the expression for the symmetry energy, the factor characterizing the effects of the HMT, i.e.,\n\\begin{align}\\label{def_Upsilon}\n&\\Upsilon_{\\rm{sym}}=1+C_0(1+3C_1)\\left(5\\phi_0+\\frac{3}{\\phi_0}-8\\right)\\notag\\\\\n&+3C_0\\phi_1\\left(1+\\frac{5}{3}C_1\\right)\\left(5\\phi_0-\\frac{3}{\\phi_0}\\right)\n+\\frac{27C_0\\phi_1^2}{5\\phi_0},\n\\end{align}\nis generally smaller than unity, leading to a reduction of the\nkinetic symmetry energy\\,\\cite{Cai15}. According to the single-nucleon potential decomposition of the symmetry\nenergy\\,\\cite{XuC10,XuC11,CheR12} based on the  Hugenholtz-Van Hove (HVH) theorem\\,\\cite{Hug58}, a contribution of\n$2^{-1}U_{\\rm{sym}}(\\rho,|\\v{k}|=k_{\\rm{F}})$ to the\nsymmetry energy is obtained. Thus a reduction of the kinetic symmetry energy\nwill induce an enhancement of the symmetry potential, i.e., the\nsymmetry energy encapsulating the SRC effects will intrinsically\nhave impacts on the symmetry potential.\nIn addition, by using the expression of $A_{\\rm{d}}$ in Eq.\\,({\\ref{ad}) the last two terms in Eq.\\,(\\ref{HMT-Esym-mm}) can be rewritten as\n\\begin{equation}\n\\frac{1}{4}A_{\\rm{d}}^0\\left(\\frac{\\rho}{\\rho_0}\\right)+\\frac{xB}{1+\\sigma}\\left(\\frac{\\rho}{\\rho_0}\\right)\n\\left[1-\\left(\\frac{\\rho}{\\rho_0}\\right)^{\\sigma-1}\\right],\n\\end{equation}\nindicating that the $x$ does not affect the symmetry energy at\n$\\rho_0$\\,\\cite{Che05,XuJ10,XuJ15}. On the other hand, the corresponding contribution to the slope parameter of symmetry energy\n$L(\\rho)$ is $\n3A_{\\rm{d}}^0\/4+3xB({1-\\sigma})\/({1+\\sigma})$ at $\\rho=\\rho_0$. Thus\na larger (positive) $x$ corresponds to a softer symmetry energy if\n$B(1-\\sigma)<0$, once the other phenomenological parameters are\nfixed\\,\\cite{Che05}, see TABLE \\ref{tab_para}.\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Numerical Demonstrations}\\label{sec5}\n\n\n\\subsection{Schemes for Determining the Coupling Constants}\\label{sb_parameter}\n\n\nIn this subsection, we discuss the scheme to determine the model couplings, i.e., $A_{\\rm{tot}},~B,~C_{\\rm{tot}},~A_{\\rm{d}},~C_{\\rm{d}},~\\sigma,~a,~b,~x$ and $\\Lambda$.\nAs pointed out in Ref.\\,\\cite{CLC18} that the function $\\Omega(\\v{k},\\v{k}')$ is invariant under the transformations,\n\\begin{equation}\na\\to a\/\\xi^{3\/2},~~b\\to\\xi b,~~\\Lambda\\to\\Lambda\/\\xi^{3\/2},\n\\end{equation}\nwith $\\xi>0$ being any scaling factor, i.e., we have the freedom to first fix one of them without affecting the physical results. We set $b = 2$ in the following and then determine other parameters using known empirical constraints.\n\n\n\\begin{table*}[t!]\n\\caption{Coupling constants used in the two models (right side) and\nsome empirical properties of ANM used to fix\nthem (left side), $b=2$ and $\\Lambda=1.6\\,\\rm{GeV}$ are fixed,\n$K_0\\equiv K_0(\\rho_0),~M_0^{\\ast}\\equiv M_0^{\\ast}(\\rho_0),~L\\equiv\nL(\\rho_0)$.\nSee also Ref.\\,\\cite{CLC18}.\n}\\label{tab_para} {\\normalsize\\begin{tabular}{lr||cccc}\n\\hline\\hline Quantity& Value & Coupling & FFG&HMT-SCGF&HMT-exp\\\\\n\\hline $\\rho_0$ (fm$^{-3}$) & $0.16$ & $A_\\ell^0$ $(\\rm{MeV})$ &\n$-266.2934$&$520.3611$&$146.6085$\\\\\n\\hline $E_0(\\rho_0)$\n$(\\rm{MeV})$&$-16.0$&$A_{\\rm{u}}^0$ $(\\rm{MeV})$&$-86.8331$&$805.3082$&$1216.2500$\\\\\n\\hline $M_0^{\\ast}\/M$ &$0.58$&$B$ $(\\rm{MeV})$&$517.5297$&$-256.9850$&$-64.5669$\\\\\n\\hline $K_0$ ($\\rm{MeV}$) &$230.0$&$C_\\ell$ $(\\rm{MeV})$&$-155.6406$&$-154.7508$&$-37.3249$\\\\\n\\hline $U_0(\\rho_0,0)$ ($\\rm{MeV}$)\n&$-100.0$&$C_{\\rm{u}}$ $(\\rm{MeV})$&$-285.3256$&$-351.0989$&$-679.5379$\\\\\n\\hline $E_{\\rm{sym}}(\\rho_0)$ ($\\rm{MeV}$)&$31.6$&$\\sigma$&$1.0353$&$0.9273$&$0.6694$\\\\\n\\hline $L$ ($\\rm{MeV}$)&$58.9$&$a$&$-5.4511$&$-5.0144$&$-4.1835$\\\\\n\\hline $U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$ ($\\rm{MeV}$)&$-20.0$&$x$&$0.6144$&$0.3774$&$-0.2355$\\\\\n\\hline\\hline\n\\end{tabular}}\n\\end{table*}\n\n\nIn our scheme, the values of $E_0(\\rho_0),~ \\rho_0, ~K_0\\equiv\nK_0(\\rho_0), ~M_0^{\\ast}(\\rho_0)$ and $U_0(\\rho_0,0\\,\\rm{MeV})$\nfor SNM, and those of $E_{\\rm{sym}}(\\rho_0), ~L\\equiv L(\\rho_0)$ and\n$U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$ for the symmetry energy, its slope and the\nsymmetry potential are fixed at their currently known most probable empirical values. For example, the value of the incompressibility $K_0=230\\pm20\\,\\rm{MeV}$ was\ndetermined from analyzing nuclear giant resonances\n(GMR)\\,\\cite{You99,Shl06,Pie10,Che12,Col14,Stone2014PRC,Garg2018PPNP,LiBA2021PRC,XuJ2021PRC,ZhangZ2021CPC}, and the nucleon effective mass at $\\rho_0$ is fixed at $0.58M$\\,\\cite{CLC18,LiBA2018PPNP}.\nFor the $E_\\text{sym}(\\rho_0)$ and $L $, all existing constraints extracted\nso far from both terrestrial laboratory measurements and\nastrophysical observations are found to be essentially consistent\nwith the global averages of $E_{\\text{sym}}({\\rho _{0}}) \\approx\n31.6\\pm2.66$ MeV and $L \\approx 58.9 \\pm 16$ MeV\\,\\cite{LiBA13}, see also Ref.\\,\\cite{Chen17,LiBA21}. Consequently, we can fix for the SNM five phenomenological parameters, i.e.,\n$A_{\\rm{tot}},~B,~C_{\\rm{tot}},~\\sigma$ and $a$, while for the symmetry\nenergy and the symmetry potential, the remaining three parameters,\ni.e., $A_{\\rm{d}},~C_{\\rm{d}}$ and $x$ can be fixed. Equivalently,\nthe totally eight parameters, i.e.,\n$A_\\ell^0,~A_{\\rm{u}}^0,~B,~C_\\ell,~C_{\\rm{u}},~\\sigma,~a$ and $x$ can be\ndetermined.\n\nNext, we need to select the cutoff $\\Lambda$.\nAt first glance, the $\\Lambda$ could be determined by adopting another empirical constraint, e.g., the symmetry energy at two times the saturation density $E_{\\rm{sym}}(2\\rho_0)$\\,\\cite{LiBA21}.\nBut it does not work in this manner.\nIn particular, one can demonstrate that once we fix the above eight quantities, i.e., $E_0(\\rho_0),~\\rho_0,~K_0,~M_0^{\\ast}(\\rho_0),~U_0(\\rho_0,0\\,\\rm{MeV}),~E_{\\rm{sym}}(\\rho_0),~L$ and\n$U_{\\rm{sym}}(\\rho_0,1\\,\\rm{GeV})$, these quantities at other densities or other energy scales do not further depend on the cutoff $\\Lambda$, see APPENDIX \\ref{app2} for more discussions.\nThe role of the $\\Lambda$ is different from the other model parameters.\nHowever, the $\\Lambda$ parameter may affect quantities which are not used in the fixing scheme, e.g., $E_{\\rm{sym},4}(\\rho)$.\nIn the following we select the $\\Lambda$ parameter based on the empirical fact that the fourth-order symmetry energy at $\\rho_0$ in the FFG model is smaller than about 3\\,MeV\\,\\cite{Cai12,Sei14,Gon17,PuJ17,CWZC2022}.\n\nWe construct three models of EOS of ANM using\ndifferent HMT input parameters described in section \\ref{sec2},\ni.e., the FFG model as well as the HMT-SCGF and HMT-exp models. Again, we emphasize that the ``FFG'' or the ``HMT'' is only used to label the $n_{\\v{k}}^J(\\rho,\\delta)$ and the potential EOS is included in all of these models.\nSee TABLE \\ref{tab_para} for the values of the model parameters, here the cutoff $\\Lambda$ is set to be 1.6\\,GeV to fulfill the aforementioned constraint (specifically $1.40\\,\\rm{GeV}\\lesssim\\Lambda\\lesssim1.65\\,\\rm{GeV}$).\nIt is necessary to point out that since we fixed the parameters of\nthe HMT by using experimental data and\/or microscopic-theory calculations at the saturation density, the possible density dependence of those\nparameters, i.e., $C_0,~C_1,~\\phi_0$ and $\\phi_1$ is not\nexplored here (also see discussions given in Ref.\\,\\cite{Cai15}). The density dependence of the various terms\nin the kinetic EOS is thus only due to that of the nucleon Fermi momentum $k_{\\rm{F}}$.\nFrom the table, one can clearly find that the parameter $a$ is negative, i.e., the moving nucleons feel weaker interaction compared to the static ones,\nsee the discussions given after the formula (\\ref{MFunc}).\nThe EDF constructed is abbreviated as the abMDI\\,\\cite{CLC18}.\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=2.8cm]{CC-Lambda-a.eps}\\quad\n  \\includegraphics[height=2.8cm]{CC-Lambda-Ctot.eps}\\quad\n  \\includegraphics[height=2.8cm]{CC-Lambda-RA.eps}\n  \\caption{The $\\Lambda$-dependence of the coupling constants $a$ (left) and $C_{\\rm{tot}}$ (middle) and the ratio $C_{\\rm{tot}}\/a^2$ (right) in the three models.}\n  \\label{fig_CC-Lambda}\n\\end{figure}\n\nWe show in FIG.\\,\\ref{fig_CC-Lambda} the $\\Lambda$-dependence of the coupling constants $a$ (left panel) and $C_{\\rm{tot}}$ (middle panel) as well as the ratio $C_{\\rm{tot}}^2\/a$ (right panel) in the three models. The parameters $a$ and $C_{\\rm{tot}}$ evolve according to $a=a^0(\\Lambda\/\\Lambda_0)^{1\/3}$ and $C_{\\rm{tot}}=C_{\\rm{tot}}^0(\\Lambda\/\\Lambda_0)^{2\/3}$, respectively, see (\\ref{CC-aa}), and thus $C_{\\rm{tot}}\/a^2=C_{\\rm{tot}}^0\/(a^{0})^2=\\rm{const.}$, with the constant depending on the fitting scheme, i.e., it depends on the values of $\\rho_0,~E_0(\\rho_0),~K_0,~M_0^{\\ast}(\\rho_0)$ and $U_0(\\rho_0,0)$ via the specific model.\nSee the right panel of FIG.\\,\\ref{fig_CC-Lambda}, e.g., the ratio $C_{\\rm{tot}}\/a^2$ in the HMT-exp model is about \n$-40.95\\,\\rm{MeV}$ (see TABLE \\ref{tab_para}).\nOne can also show numerically that as $\\Lambda\\to0$, both $a$ and $C_{\\rm{tot}}$ approach zero, but their ratio keeps a constant.\nThe $\\Lambda$-dependence of other coupling constants ($A_{\\ell}^0,~A_{\\rm{u}}^0,~C_{\\ell},~C_{\\rm{u}}$) could be obtained similarly, and would not be analyzed here further (see APPENDIX B for more discussions).\n\n \n \n\\subsection{SNM EOS $E_0(\\rho)$ and Single-nucleon Potential $U_0(\\rho,|\\v{k}|)$}\n\nIn FIG.\\,\\ref{fig_ab_E0}, we show the SNM EOSs in the three models.\nIt is first interesting to see that the HMT models predict a\nslightly harder SNM EOS at supra-saturation densities than the\nFFG model, while by design they all have the same values of\n$M_0^{\\ast}$, $\\rho_0$, $E_0(\\rho_0)$ and $K_0$. Physically, this is\neasy to understand because of the large contribution to the kinetic EOS by the\nhigh momentum nucleons in the HMT models\\,\\cite{Cai16b}. For\nexample, the skewness of the SNM characterizing the high density\nbehavior of the EOS in the FFG model is found to be\n$J_0(\\rm{FFG})\\approx-381\\,\\rm{MeV}$, while that in the two HMT\nmodels is about $J_0(\\rm{HMT-SCGF})\\approx-376\\,\\rm{MeV}$ and\n$J_0(\\rm{HMT-exp})\\approx-329\\,\\rm{MeV}$, respectively. The small\nchange in the skewness from the FFG model to the HMT-SCGF model can\nbe traced back to the enhancement factor\n$\\Upsilon_0=1+C_0(5\\phi_0+3\/\\phi_0-8)$, which is unity in the FFG\nmodel and becomes about 1.18\/1.83 in the HMT-SCGF\/HMT-exp model. It\nis thus not surprising that the predictions on the pressure in SNM\nis also very similar, and can all pass through the constraints from analyzing \nthe collective flow date from relativistic heavy-ion collisions\\,\\cite{Dan02},\nsee FIG.\\,\\ref{fig_ab_Flow}.\nIt is necessary to point out that since the high momentum nucleon fraction in the HMT-SCGF is relatively lower than that in the HMT-exp model, the former is much closer to the FFG model (see the black dash line and the blue dash-dotted line shown in the inset of FIG.\\,\\ref{fig_ab_E0}).\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{E0-EOS.eps}\n  \\caption{EOS of SNM in the FFG model and the HMT-SCGF as well as the HMT-exp models.}\n  \\label{fig_ab_E0}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{P0Flow.eps}\n  \\caption{Comparison between the pressure $P_0$ of SNM with the experimental constraints from analyzing nuclear collective flows in heavy ion collisions\\,\\cite{Dan02}.}\n  \\label{fig_ab_Flow}\n\\end{figure}\n\n\nIn FIG.\\,\\ref{fig_ab_U0}, we show the single-nucleon potential in SNM\n$U_0$ as a function of kinetic energy $E_{\\rm{kin}}=[|\\v{k}|^2+M^2]^{1\/2}+U_0(\\rho_0,|\\v{k}|)-M$\nin the three models. As discussed earlier, the SRC-induced HMT (slightly) enhances\nthe kinetic EOS of SNM, and in order to maintain the total EOS of SNM\nat the saturation density to be consistent with empirical constraints the\npotential part should be correspondingly reduced, which is reflected\nin FIG.\\,\\ref{fig_ab_U0}. As the difference of the fraction of high\nmomentum nucleons in SNM and in PNM becomes larger, the kinetic EOS\nof SNM is much more enhanced, leading to a much softer potential\npart, i.e., the $U_0$ in the HMT-exp model\nis smaller than that in the HMT-SCGF model. Also shown in FIG.\\,\\ref{fig_ab_U0} are the predictions on the $U_0(\\rho_0,|\\v{k}|)$\nfrom other approaches, including the global relativistic fitting of\nelectron-scattering data up to about 1\\,GeV\\,\\cite{Ham90} (blue ``+''); the\npredictions by the neutron optical model up to about\n200\\,MeV\\,\\cite{LiX15} (dashed black band), and those from the chiral effective field\ntheories\\,\\cite{Hol16} (cyan band). It is obvious that the predictions on $U_0$\nboth in the HMT-SCGF and the HMT-exp models are consistent with these approaches, at kinetic energies $\\lesssim600\\,\\rm{MeV}$.\nIn this sense, the momentum-dependence in the function (\\ref{MFunc})\ngives a reasonable description of the kinetic-energy-dependence of the single-nucleon isoscalar\npotential. As the kinetic energy $E_{\\rm{kin}}$ becomes larger, the\ndeviation between the predictions on the $U_0$ in the three models and\nthe global relativistic fitting also becomes larger, indicating the breakdown of the perturbative construction for $\\Omega(\\v{k},\\v{k}')$.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{U0Hama.eps}\n  \\caption{single-nucleon potential in SNM as a function of kinetic energy in the FFG model and\n  HMT-SCGF\/HMT-exp model. Constraints from other approaches are shown for comparison.}\n  \\label{fig_ab_U0}\n\\end{figure}\n\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{Mstar.eps}\n  \\caption{Nucleon effective k-mass as a function of density in the FFG and HMT-SCGF\/HMT-exp models.}\n  \\label{fig_ab_M0}\n\\end{figure}\n\n\n\nIn FIG.\\,\\ref{fig_ab_M0}, the density dependence of the effective k-mass is\nshown. It is straightforward to understand that the three models give very\nsimilar results. On one hand, three points of the effective mass are\nfixed, i.e., $M_0^{\\ast}(0)\/M=1$, $M_0^{\\ast}(\\rho_0)\/M=0.58$ and\n$M_0^{\\ast}(\\infty)\/M=0$ (see Eq.\\,(\\ref{def_11}) since both $a$ and $C_{\\rm{tot}}$ are negative, leading to $aC_{\\rm{tot}}>0$). Moreover, the\n$M_0^{\\ast}\/M$ monotonically decreases in the whole density range\nand is concave, e.g., for large density $\\rho$, one has from (\\ref{def_11}) that $\nM_0^{\\ast}\/M\\sim\\rho^{-2\/3}$. Meeting all of these common constraints the\n$M_0^{\\ast}(\\rho)\/M$ in the three models behaves thus very similarly (see similar discussions given in Ref.\\,\\cite{Cai16b} for the nonlinear RMF model counterpart).\n\n\n\n\n\\subsection{Reduction of the Kinetic Symmetry Energy $E_{\\textmd{sym}}^{\\textmd{kin}}(\\rho)$ and Enhancement of the Symmetry Potential $U_{\\textmd{sym}}(\\rho,|\\v{k}|)$}\n\n\nWe now discuss effects of the SRC\/HMT on the symmetry energy and isovector (symmetry) potential.\nIn FIG.\\,\\ref{fig_ab_Esym}, we show the density dependence of the nuclear\nsymmetry energy in the three models\\,\\cite{CLC18}.\nAlso shown include the constraints on the $E_{\\rm{sym}}(\\rho)$\naround the saturation density from analyses of heavy-ion collisions\\,\\cite{Tsa12} and isobaric analog state studies\\,\\cite{Dan14}. Although the symmetry energy from the three models can pass through these constraints, they have very different behaviors at super-saturation\ndensities\\,\\cite{CLC18}. Specifically, as discussed in the above sections, the\nreduction of the kinetic symmetry energy should be compensated by\nthe potential part to keep the total symmetry energy around $\\rho_0$ consistent with certain empirical constraints\\,\\cite{LiBA13,Chen17,LiBA21}.\nMore quantitatively, the factor $\\Upsilon_{\\rm{sym}}$ (defined in Eq.\\,(\\ref{def_Upsilon})) in the HMT-SCGF model is found to be about\n$\\Upsilon_{\\rm{sym}}\\approx0.71$, which is not far different from\nthat in the FFG model (which is unity), while the\n$\\Upsilon_{\\rm{sym}}$ factor in the HMT-exp model is about\n$\\Upsilon_{\\rm{sym}}\\approx-1.12$\\,\\cite{Cai15}, totally different from the FFG\nprediction. Thus it is not surprising that the effects of HMT in the\nHMT-exp model is much more apparent. \n\nIn addition, there are two critical densities $\\rho_1$ and $\\rho_2$ corresponding to the point above which the symmetry energy starts decreasing and its zero-point when the symmetry energy vanishes. They are defined as $L(\\rho_1)=0$ and $E_{\\rm{sym}}(\\rho_2)=0$, respectively. The latter case indicates the onset of the so-called isospin separation instability, namely it is energetically more favorable to split SNM into pure neutron matter and proton matter when the  symmetry energy becomes negative\\,\\cite{Kut93,Kut94,LiBA02}. Possible appearances and ramifications of such instability in heavy-ion reactions and neutron stars have been studied in several works, see., e.g., Refs.\\,\\cite{Kut93,Kut94,LiBA02,Szm06,wen,Kho96,Bas07,Ban00,Kubis1}.\nWithout considering the SRC-induced HMT, the FFG model predicts $\\rho_1^{\\rm{FFG}}\\approx4.4\\rho_0$ and $\\rho_2^{\\rm{FFG}}\\approx11.2\\rho_0$, while the HMT reduces the $\\rho_1$ to $3.5\\rho_0$ ($1.9\\rho_0$) and the $\\rho_2$ to $8.5\\rho_0$ ($4.1\\rho_0$) in the HMT-SCGF (HMT-exp) model.\nWe see that the experimental HMT constraints largely reduce the zero-point density $\\rho_2$, with an effect of about 63\\%.\nSince the $\\rho_2^{\\rm{HMT}\\mbox{-}\\rm{exp}}\\approx4.1\\rho_0$ is quite low considering neutron star issues, it is expected that the reduction of the symmetry energy at supra-saturation densities in the HMT-exp model may have sizable effects on properties of neutron stars, e.g., the mass-radius relation.\nOn the other hand, the reduction of the $E_{\\rm{sym}}(\\rho)$ at sub-saturation densities (see the inset of FIG.\\,\\ref{fig_ab_Esym}) may also induce changes on, e.g., the core-crust transition densities in neutron stars.\nThese issues are investigated in some details in subsection \\ref{sb_NS}.\n\nFurthermore, we also show in FIG.\\,\\ref{fig_ab_Esym} two constraints on the $E_{\\rm{sym}}(2\\rho_0)$. The pink solid circle at $E_{\\rm{sym}}(2\\rho_0)\\approx45\\pm3\\,\\rm{MeV}$ is from chiral perturbation theories with consistent nucleon-nucleon and three-nucleon interactions up to the fourth-order\\,\\cite{Dri20}. The grey diamond at $E_{\\rm{sym}}(2\\rho_0)\\approx51\\pm13\\,\\rm{MeV}$ is the fiducial value from surveying analyses of both relativistic heavy-ion reactions and neutron star properties since GW170817\\,\\cite{LiBA21}. It is clearly seen that the calculations with the SRC\/HMT-induced reduction of $E_{\\rm{sym}}(2\\rho_0)$ are consistent with these constraints\\,\\cite{LiBA21,Dri20,CL2021}.\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[width=6.5cm]{Esym.eps}\n  \\caption{Density dependence of the symmetry energy in the FFG model and HMT-SCGF\/HMT-exp model. Constraints from heavy-ion collisions\\,\\cite{Tsa12} and the isobaric analog state studies\\,\\cite{Dan14} are shown for comparison.\nSee also Ref.\\,\\cite{CLC18} and the text for details on the pink solid circle and the grey diamond.}\n  \\label{fig_ab_Esym}\n\\end{figure}\n\n\n\nThe reduction of the\nsymmetry energy found here is qualitatively consistent with the one from the nonlinear RMF\nmodels\\,\\cite{Cai16b}. \nHowever, they give quantitatively different results. \nIn particular, due to the specific structure of the nonlinear RMF model, the symmetry energy should never becomes negative.\nOn the other hand, the EOS in a non-relativistic EDF could be treated as an effective expansion in density. \nIn order to investigate the probable origin of this quantitative difference, it is enough for our purpose to assume that the symmetry energy takes the form $E_{\\rm{sym}}(\\rho)=E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)(\\rho\/\\rho_0)^{2\/3}+s_1(\\rho\/\\rho_0)^{\\alpha}+s_2(\\rho\/\\rho_0)^{\\beta}$, where $\\beta>\\alpha>2\/3$, and $E_{\\rm{sym}}^{\\rm{kin}}(\\rm{FFG})\\approx12.3\\,\\rm{MeV}$.\nThe coefficients $s_1$ and $s_2$ can be determined via the values of $E_{\\rm{sym}}(\\rho_0)$ and $L\\equiv L(\\rho_0)$. Specifically, the expression for $s_2$ is given by,\n\\begin{equation}\ns_2=\\frac{1}{\\beta-\\alpha}\\left(\\alpha-\\frac{2}{3}\\right)E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)+\\frac{L-3\\alpha E_{\\rm{sym}}(\\rho_0)}{\\beta-\\alpha}.\n\\end{equation}\nThe second term here is negative since $\\alpha>2\/3$ and thus $L-3\\alpha E_{\\rm{sym}}(\\rho_0)<0$ while $\\beta-\\alpha>0$. \nSimilarly, the coefficient of the first term of $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)$ is positive (due to the same consideration), thus:\n\\begin{enumerate}\n\\item[(a)]One has $s_2^{\\rm{FFG}}>s_2^{\\rm{HMT}}$ as $E_{\\rm{sym}}^{\\rm{kin}}(\\rm{FFG})>E_{\\rm{sym}}^{\\rm{kin}}(\\rm{HMT})$. \n\\item[(b)]In addition, $s_2<0$ for the HMT-exp model, as the $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)\\approx-13.8\\,\\rm{MeV}$ is negative. \n\\end{enumerate}\nThe point (a) explains why the $E_{\\rm{sym}}(\\rho)$ considering the HMT should be reduced while the point (b) indicates that in the HMT-exp model (with a negative $E_{\\rm{sym}}^{\\rm{kin}}(\\rho_0)$) the $E_{\\rm{sym}}(\\rho)$ must start decreasing above some critical density, and become negative at even higher densities. In our EDF, $\\beta=4\/3$ and one can find actually that the $s_2$'s in all three models are negative, see TABLE \\ref{tab_para1}.\nThis explains why the reduction of symmetry energy using the non-relativistic EDF should be much stronger than the one in the nonlinear RMF models.\n\n\nThe reduction of the symmetry energy at supra-saturation densities also generates corresponding reduction of its curvature coefficient $K_{\\rm{sym}}$. Quantitatively, its value changes from $-109$\\,MeV in the FFG model to about\n$-121\\,\\rm{MeV}$ ($-223$\\,MeV) in the HMT-SCGF (HMT-exp) model.\nWhile the $K_{\\rm{sym}}\\approx-223\\,\\rm{MeV}$ in the HMT-exp model is somewhat smaller than its current fiducial value of about $-107\\pm 88$ MeV (see FIG.\\,2 of Ref.\\,\\cite{LiBA21}), those from the FFG and the HMT-SCGF model are very consistent with the fiducial value. Thus, the general tendency is that the SRC\/HMT reduces the symmetry energy curvature coefficient $K_{\\rm{sym}}$. This may have some other consequences. For example, the pressure of neutron-rich nucleonic matter is given by\n\\begin{align}\nP\/3\\rho\\approx&L(\\rho)\\delta^2+3\\rho\\frac{\\d E_0(\\rho)}{\\d\\rho}\\notag\\\\\n\\approx&\nL\\delta^2\\left(1+3\\chi+\\frac{9}{2}\\chi^2\\right)\n+K_0\\chi(1+3\\chi)+\\frac{1}{2}\\chi^2J_0\\notag\\\\\n&+K_{\\rm{sym}}\\delta^2\\chi(1+3\\chi)\\left[1+\\frac{J_{\\rm{sym}}}{K_{\\rm{sym}}}\n\\frac{\\chi}{2(1+3\\chi)}\\right]\\notag\\\\\n\\approx&L\\delta^2\\left(1+3\\chi+\\frac{9}{2}\\chi^2\\right)\n+K_0\\chi(1+3\\chi)\\notag\\\\\n&+\\frac{1}{2}\\chi^2J_0+K_{\\rm{sym}}\\delta^2\\chi(1+3\\chi),\n\\end{align}\nwhere the skewness coefficient $J_{\\rm{sym}}$ is neglected in the last step.\nSince the $K_0$ and $L$ are fixed in our study,  and as discussed earlier the $J_0$ is slightly increased by the SRC\/HMT, the reduction of $K_{\\rm{sym}}$ may reduce the pressure $P$ at densities below about $3\\rho_0$ above which the $J_{\\rm{sym}}$ becomes important (the $\\chi$ is 2\/3 for $\\rho=3\\rho_0$ and thus $\\chi\/[2(1+3\\chi)]=1\/9$). Moreover,  the reduction of $K_{\\rm{sym}}$ may also modify the saturation line (on which the pressure is zero) of ANM. More quantitatively, the incompressibility coefficient of ANM along its saturation line is $K(\\delta)=K_0+K_{\\rm{sat},2}\\delta^2$\\,\\cite{LWChen2009} where the $K_{\\rm{sat},2}$ is\n \\begin{equation}\nK_{\\rm{sat,2}}=K_{\\rm{sym}}-6L-\\frac{J_0L}{K_0}.\n\\end{equation}\nIt changes from $-365\\,\\rm{MeV}$ in the FFG model to $-378\\,\\rm{MeV}$ ($-492\\,\\rm{MeV}$) in the HMT-SCGF (HMT-exp) model.  \nIt is seen that the maximum reduction is about 127\\,\\rm{MeV}.\n\n\\begin{table*}[t!]\n\\caption{Parameterizations of the EOS of SNM and the symmetry energy as\nwell as the nucleon potential. Momentum $|\\v{k}|$ is measured in MeV and\nthe density $\\rho$ in $\\rm{fm}^{-3}$,\n$E_0(\\rho),E_{\\rm{sym}}(\\rho),U_0(\\rho,|\\v{k}|)$ and\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)$ are in MeV.}\\label{tab_para1}\n{\\normalsize\n\\begin{tabular}{c||c}\n\\hline\\hline Model for $n_{\\v{k}}^J$& Parametrization of the Physical Quantity\\\\\\hline \n\\hline FFG  &$x_{\\rm{SNM}}^{\\rm{HMT}}=0\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=0\\%$\n\\\\\n\\hline $E_0(\\rho)$ &\n$E_0(\\rho)\\approx75.00\\rho^{2\/3}+1695.42\\rho^{1.04}-551.76\\rho-1378.02\\rho[1-0.81\\rho^{1\/3}+0.55\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$ &$U_0(\\rho,|\\v{k}|)\\approx3450.68\\rho^{1.04}-1103.52\\rho-2756.04\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx41.67\\rho^{2\/3}+1101.25\\rho-1041.63\\rho^{1.04}-229.34\\rho^{4\/3}+180.92\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx-2083.27\\rho^{1.04}+\\rho[2202.51-40.52|\\v{k}|^{1\/2}\\rho^{1\/6}+69.80|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\\hline\n\\hline HMT-SCGF  &$x_{\\rm{SNM}}^{\\rm{HMT}}=12\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=4\\%$\n\\\\\n\\hline $E_0(\\rho)$   & $E_0(\\rho)\\approx88.28\\rho^{2\/3}-729.40\\rho^{0.93}+2071.36\\rho-1580.78\\rho[1-0.79\\rho^{1\/3}+0.57\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$  &$U_0(\\rho,|\\v{k}|)\\approx-1405.76\\rho^{0.93}+4142.72\\rho-3161.56\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx29.53\\rho^{2\/3}-146.18\\rho+275.30\\rho^{0.93}-467.07\\rho^{4\/3}+343.07\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx550.60\\rho^{0.93}+\\rho[-292.37-57.07|\\v{k}|^{1\/2}\\rho^{1\/6}+106.42|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\\hline\n\\hline HMT-exp  &$x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%,x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$\n\\\\\n\\hline $E_0(\\rho)$  & $E_0(\\rho)\\approx137.31\\rho^{2\/3}-131.89\\rho^{0.67}+2129.47\\rho-2240.19\\rho[1-0.77\\rho^{1\/3}+0.63\\rho^{2\/9}]$\\\\\n\\hline $U_0(\\rho,|\\v{k}|)$ &$U_0(\\rho,|\\v{k}|)\\approx-220.18\\rho^{0.67}+4258.93\\rho-4480.39\\rho[1-0.04|\\v{k}|^{1\/2}\\rho^{1\/6}+0.08|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline $E_{\\rm{sym}}(\\rho)$ & $E_{\\rm{sym}}(\\rho)\\approx-46.83\\rho^{2\/3}+392.52\\rho-31.06\\rho^{0.67}-1837.48\\rho^{4\/3}+1421.05\\rho^{11\/9}$\\\\\n\\hline $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ &\n$U_{\\rm{sym}}(\\rho,|\\v{k}|)\\approx-62.11\\rho^{0.67}+\\rho[785.04-156.10|\\v{k}|^{1\/2}\\rho^{1\/6}+348.23|\\v{k}|^{1\/3}\\rho^{1\/9}]$\\\\\n\\hline\\hline\n\\end{tabular}}\n\\end{table*}\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.8cm]{Usym-1.eps}\n  \\includegraphics[height=3.8cm]{Usym-2.eps}\n  \\caption{Symmetry potential in the FFG model and the HMT-SCGF\/HMT-exp model at $\\rho=\\rho_0$ (left) and\n  at $\\rho=3\\rho_0$ (right).\nConstraints on the symmetry potential at $\\rho=\\rho_0$ from other approaches are shown for\ncomparisons.}\n  \\label{fig_ab_Usym}\n\\end{figure}\n\nIn FIG.\\,\\ref{fig_ab_Usym}, we show the momentum dependence of the\nsymmetry potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ at $\\rho_0$ (left panel) and $3\\rho_0$ (right panel), respectively. The predictions on\n$U_{\\rm{sym}}(\\rho_0,|\\v{k}|)$ from other microscopic\napproaches\\,\\cite{LiX15,Hol16,Fri05} are also shown for comparisons. The empirical band is parametrized as $U_{\\rm{sym}}(E)\\approx\n(28\\pm6)\\,\\rm{MeV}-(0.15\\pm0.05)E$\\,\\cite{Hol16}. It can be found \nthat the uncertainties on the symmetry potential are larger than\nthose of the isoscalar potential $U_0$. This is mainly due to the poorly-known isospin dependence of nuclear interactions\\,\\cite{LCK08,LiBA2018PPNP}.\nWhile the prediction by the HMT-exp on the\n$U_{\\rm{sym}}(\\rho_0,|\\v{k}|)$ has certain deviation from the microscopic model predictions, the FFG and HMT-SCGF model\npredictions are quite consistent with the optical model fitting\\,\\cite{LiX15} as well as the chiral effective theory prediction\\,\\cite{Hol16}.\nSince there exists no direct constraint on the symmetry potential at, e.g., a large density $\\rho=3\\rho_0$, and it is beyond the current application domain of microscopic nuclear many-body theories, it will be interesting to test the predictions shown in the right window of  FIG.\\,\\ref{fig_ab_Usym}, thus the SRC\/HMT effects in dense neutron-rich matter, using nuclear reactions with high-energy radioactive beams. Of course, for this purpose it is first necessary to investigate experimental observables sensitive to the high-density symmetry potential. The results presented here especially the various analytical expressions and parameterizations will facilitate the explorations of this topic using transport models for nuclear reactions with radioactive beams. \nListed in TABLE \\ref{tab_para1} are parametrizations of the SNM EOS $E_0(\\rho)$, nucleon isoscalar potential $U_0(\\rho,|\\v{k}|)$, symmetry energy $E_{\\rm{sym}}(\\rho)$ and isovector potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ corresponding to the three models (with their parameters given in TABLE \\ref{tab_para}), for future applications.\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.3cm]{Esym-xpara-1.eps}\n  \\includegraphics[height=3.3cm]{Esym-xpara-2.eps}\n  \\includegraphics[height=3.3cm]{Esym-xpara-3.eps}\n  \\caption{Symmetry energy with different effective three-nucleon force parameter $x$ in the three models indicated. }\n  \\label{fig_ab_Esym_x}\n\\end{figure}\n\nIn our calculations up to now, the effective three-nucleon force parameter $x$ is self-consistently determined by fixing the $L(\\rho_0)$, see TABLE \\ref{tab_para}.\nSince the $x$ parameter only affects the slope parameter of the symmetry energy instead of its magnitude at $\\rho_0$ itself, we can adjust $x$ to investigate how the density dependence of $E_{\\rm{sym}}(\\rho)$ changes when the SRC-induced HMT is taken into account. In FIG.\\,\\ref{fig_ab_Esym_x}, the symmetry energy obtained within the three\nmodels indicated by only changing the $x$ parameter (corresponding to changing only the $L$ coefficient) while fixing at the same time all the other seven parameters listed\nin TABLE \\ref{tab_para}, is shown. For the FFG model, the $x$-dependence of $E_{\\rm{sym}}(\\rho)$ is the same as in the conventional MDI\nmodel\\,\\cite{Das2003,Che05,LCK08,XuJ10,XuJ15}. In the presence of the SRC\/HMT, the large\nuncertainties of $E_{\\rm{sym}}(\\rho)$ at high densities can come from the poorly known physics of either the three-nucleon forces and\/or the SRC\/HMT.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.7cm]{Esym-frac-1.eps}\\quad\n  \\includegraphics[height=3.7cm]{Esym-frac-2.eps}\n  \\caption{The symmetry energy at $2\\rho_0$ and $3\\rho_0$ with different fractions of high momentum nucleons in SNM and\/or in PNM, respectively. The black circles are for the default set on $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%$ and $x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$.}\n  \\label{fig_ab_EsymFrac}\n\\end{figure}\n\nAs discussed in the INTRODUCTION, there are still uncertainties about the fraction of high momentum nucleons in either SNM and\/or PNM and thus its isospin dependence\\,\\cite{Hen14,Rio14}. Thus, it is interesting to study how the relevant quantities change when the HMT fractions in SNM and\/or PNM are modified.\nIn FIG.\\,\\ref{fig_ab_EsymFrac}, we show the symmetry energy at two\nreference densities, i.e., $\\rho_{\\rm{f}}=2\\rho_0$ and\n$\\rho_{\\rm{f}}=3\\rho_0$, by continuously changing the high\nmomentum nucleon fraction $x_{\\rm{SNM}}^{\\rm{HMT}}$ in SNM when fixing the $x_{\\rm{PNM}}^{\\rm{HMT}}$ in PNM at the\ndefault value of the HMT-exp model (left panel) (in PNM when fixing\nthat in SNM at the default value of the HMT-exp model (right panel)), respectively. \nThe values of $C_0$ and $C_{\\rm{n}}^{\\rm{PNM}}=C_0(1+C_1)$ are fixed at 0.161 and 0.12, respectively, while the values of  $\\phi_0$ and $\\phi_1$ are readjusted correspondingly according to the given $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$ values. The black circles are for the default parameter set of $x_{\\rm{SNM}}^{\\rm{HMT}}$ and $x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $x_{\\rm{SNM}}^{\\rm{HMT}}=28\\%$ and $x_{\\rm{PNM}}^{\\rm{HMT}}=1.5\\%$.\nAs the difference between the $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\n$x_{\\rm{PNM}}^{\\rm{HMT}}$, i.e., $\\delta x=x_{\\rm{SNM}}^{\\rm{HMT}}-x_{\\rm{PNM}}^{\\rm{HMT}}$, increases, the reduction of the\n$E_{\\rm{sym}}(\\rho_{\\rm{f}})$ becomes much more apparent. While the\nsymmetry energy at the saturation density is fixed (at $31.6\\,\\rm{MeV}$),\nthese effects on $E_{\\rm{sym}}(2\\rho_0)$ are relatively minor compared with those at $\\rho=3\\rho_0$.\n\n\n\\begin{figure}[h!]\n\\centering\n \n  \\includegraphics[height=3.6cm]{Ksat2-frac-1.eps}\\quad\n  \\includegraphics[height=3.6cm]{Ksat2-frac-2.eps}\n  \\caption{The same as FIG.\\,\\ref{fig_ab_EsymFrac} but for the isospin incompressibility coefficient $K_{\\rm{sat},2}=K_{\\rm{sym}}-6L-J_0L\/K_0$. }\n  \\label{fig_Ksat2_frac}\n\\end{figure}\n\nSimilarly, we show in FIG.\\,\\ref{fig_Ksat2_frac} the dependence of the isospin incompressibility coefficient $K_{\\rm{sat},2}$ on the high momentum nucleon fraction.\nAs the $L$ and $K_0$ are fixed in our calculations, the dependence of the coefficient $K_{\\rm{sat},2}$ on $x_{\\rm{SNM}}^{\\rm{HMT}}$ and\/or\n$x_{\\rm{PNM}}^{\\rm{HMT}}$ mainly reflects the dependence of the curvature coefficient $K_{\\rm{sym}}$ and the skewness parameter $J_0$ on them.\nThe $K_{\\rm{sat},2}$ shown in FIG.\\,\\ref{fig_Ksat2_frac} is in good agreement with the estimate of $K_{\\rm{sat},2}\\approx-550\\pm100\\,\\rm{MeV}$ from analyzing different types\nof experimental data currently available\\,\\cite{Col14}, as shown using the cyan bands.\nFor example, if $x_{\\rm{PNM}}^{\\rm{HMT}}\\approx1.5\\%$ is fixed, then the constraint on $K_{\\rm{sat},2}$ leads to about $x_{\\rm{SNM}}^{\\rm{HMT}}\\gtrsim24\\%$. Similarly, if $x_{\\rm{SNM}}^{\\rm{HMT}}\\approx28\\%$ is fixed, then we can roughly find the constraint that $x_{\\rm{PNM}}^{\\rm{HMT}}\\lesssim12\\%$.\n\n\n\\subsection{SRC\/HMT Effects on the Proton Fraction $x_{\\textmd{p}}$, Core-crust Transition Density $\\rho_{\\textmd{t}}$ and pressure $P_{\\textmd{t}}$ as well as the Mass-radius Relation of Cold Neutron Stars at $\\beta$-equilibrium}\\label{sb_NS}\n\nAs an example of applying the Gogny-like EDFs encapsulating the SRC\/HMT in astrophysical studies, we investigate here the density profile of proton fraction $x_{\\textmd{p}}(\\rho)=[1-\\delta(\\rho)]\/2$, the core-crust transition density $\\rho_{\\rm{t}}$ and transition pressure $P_{\\rm{t}}$ as well as the mass-radius correlation in cold neutron stars at $\\beta$-equilibrium.\nThe transition density $\\rho_{\\rm{t}}$ is the nucleon number\ndensity that separates the liquid core from the inner crust in\nneutron stars. It plays an important role in determining many\nproperties of neutron\nstars\\,\\cite{Hor01,Pro06,Duc08a,Duc08b,XuJ09}.\nOne simple and widely used approach to determine the core-crust transition density $\\rho\n_{\\rm{t}}$ is the thermodynamical method. Normally, neutron star matter has to obey the intrinsic stability\ncondition\\,\\cite{Lat04,XuJ09,NBZ19,Kub07}\n\\begin{align}\n\\mathcal{U}_{\\rm{ther}}(\\rho)=&2\\rho \\frac{\\partial E(\\rho\n,x_{\\rm{p}})}{\\partial \\rho }+\\rho ^{2}\\frac{\\partial\n^{2}E(\\rho ,x_{\\rm{p}})}{\\partial \\rho ^{2}}\\notag\\\\\n& -\\left. \\left( \\frac{\\partial ^{2}E(\\rho ,x_{\\rm{p}})}{\\partial\n\\rho\n\\partial x_{\\rm{p}}}\\rho \\right) ^{2}\\right\/ \\frac{\\partial ^{2}E(\\rho ,x_{\\rm{p}})%\n}{\\partial x_{\\rm{p}}^{2}}\\notag\\\\\n=&\\rho^2\\left(\\frac{\\partial^2E(\\rho,x_{\\rm{p}})}{\\partial x_{\\rm{p}}^2}\\right)^{-1} \n\\left[\\frac{\\partial\\mu_{\\rm{n}}}{\\partial\\rho_{\\rm{n}}}\\frac{\\partial \\mu_{\\rm{p}}}{\\partial\\rho_{\\rm{p}}}-\\left(\\frac{\\partial\\mu_{\\rm{n}}}{\\partial\\rho_{\\rm{p}}}\\right)^2\\right]>0. \\label{Vther}\n\\end{align}\nIn the above, the density profile $x_{\\rm{p}}=x_{\\rm{p}}(\\rho)$ is determined by the $\\beta$-equilibrium and charge neutrality condition of neutron star matter consisting of neutrons, protons and electrons (the core-crust transition occurs around $\\rho_0\/3\\mbox{$\\sim$}\\rho_0\/2$ where muons generally do not emerge). It is uniquely determined by the density dependence of nuclear symmetry energy\\,\\cite{Lat04}.\nOnce the above stability condition is broken, the speed of sound in the uniform outer core becomes imaginary and small density oscillations will then grow exponentially, indicating the onset of a transition from the uniform liquid core to the crust.  \nIn (\\ref{Vther}), $E(\\rho ,x_{\\rm{p}})$ is the energy per nucleon in\nthe $\\beta $-equilibrated neutron star matter.  The corresponding pressure is obtained as $P=P_{\\rm{N}}+P_{\\rm{e}}$ with\n$P_{\\rm{N}}$ and $P_{\\rm{e}}$ denoting contributions from nucleons and electrons, respectively.\nThe electron pressure $P_{\\rm{e}}$ could be obtained through the FFG model\\,\\cite{XuJ09}, i.e., \n\\begin{equation}\\label{dd_e}\n\\varepsilon_{\\rm{e}}(\\rho,\\delta)=\\eta_{\\rm{e}}\\Phi_{\\rm{e}}(t_{\\rm{e}}),\n\\end{equation}\nhere \\begin{equation}\n\\eta_{\\rm{e}}={m_{\\rm{e}}}\/{8\\pi^2\\lambda_{\\rm{e}}^3},~~\n\\lambda_{\\rm{e}}={1}\/{m_{\\rm{e}}},~~t_{\\rm{e}}=\\lambda_{\\rm{e}}\\left(3\\pi^2\\rho_{\\rm{e}}\\right)^{1\/3}\n,\\end{equation} and,\n\\begin{equation}\\label{dd_e1}\n\\Phi_{\\rm{e}}(t_{\\rm{e}})=t_{\\rm{e}}\\left(1+2t_{\\rm{e}}^2\\right)\\sqrt{1+t_{\\rm{e}}^2}-\\ln\\left(t_{\\rm{e}}+\\sqrt{1+t_{\\rm{e}}^2}\\right)\n,\\end{equation} \nwhere $m_{\\rm{e}}$ and $\\rho_{\\rm{e}}$ are the electron mass and density respectively.\nConsequently, $P_{\\rm{e}}(\\rho,\\delta)=\\rho_{\\rm{e}}\\mu_{\\rm{e}}-\\varepsilon_{\\rm{e}}(\\rho,\\delta)\n$, with $\n\\mu_{\\rm{e}}=[{k_{\\rm{e}}^2+m_{\\rm{e}}^2}]^{1\/2}\\approx k_{\\rm{e}}$ and $k_{\\rm{e}}=(3\\pi^2\\rho_{\\rm{e}})^{1\/3}$.\n\nShown in FIG.\\,\\ref{fig_ab_rhot} are our results for the core-crust transition density $\\rho_{\\rm{t}}$ (upper panel) and pressure $P_{\\rm{t}}$ (lower panel) as functions of the slope parameter $L(\\rho_{\\rm{r}})$ at a sub-saturation density $\\rho_{\\rm{r}}=0.11\\,\\rm{fm}^{-3}$.}\nIt is clear that the SRC\/HMT affects significantly the\ntransition density $\\rho_{\\rm{t}}$, e.g., a reduction as high as about\n58\\% with the HMT-exp parameter set compared to the FFG model. More quantitatively, the crust-core transition density in the FFG model is found to be about\n$\\rho_{\\rm{t}}\\approx0.086\\,\\rm{fm}^{-3}$, while that in the HMT\nmodels is about $\\rho_{\\rm{t}}\\approx0.079\\,\\rm{fm}^{-3}$\n(HMT-SCGF) and $\\rho_{\\rm{t}}\\approx0.036\\,\\rm{fm}^{-3}$ (HMT-exp), respectively.\nCorrespondingly, the transition pressure in the FFG model and the\nHMT-SCGF (HMT-exp) model is about\n$0.28\\,\\rm{MeV}\/\\rm{fm}^{3}$ and $0.26\\,\\rm{MeV}\/\\rm{fm}^{3}$\n($0.17\\,\\rm{MeV}\/\\rm{fm}^{3}$), respectively. \n\nAs discussed in the\nabove sections, the symmetry energy $E_{\\rm{sym}}(\\rho)$ at sub-saturation densities is reduced by the SRC\/HMT while its magnitude and slope at $\\rho_0$ are fixed at their currently known empirical values. Thus, the $E_{\\rm{sym}}(\\rho)$ is effectively hardened within the sub-saturation density region (see the inset of FIG.\\,\\ref{fig_ab_Esym}), leading to an enhancement of the slope\nparameter $L(\\rho)$ for $\\rho\\lesssim\\rho_0$. It is known that a larger $L(\\rho_{\\rm{r}})$ with $\\rho_{\\rm{r}}\\lesssim\\rho_0$\\,\\cite{Zha13} leads to a\nsmaller core-crust transition density as studied carefully in, e.g.,\nRefs.\\,\\cite{XuJ09}. The observed SRC\/HMT effects on the crust-core transition density and pressure are qualitatively consistent with expectations based on previous studies. \n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[height=2.7cm]{rhot.eps}\\\\\n \\hspace{0.cm}\n \\includegraphics[height=2.7cm]{Pt.eps}\n  \\caption{Correlation between the core-crust transition density (upper) and the transition pressure (lower) in $\\beta$-stable neutron star matter\n  and the slope parameter $L(\\rho_{\\rm{r}})$ at $\\rho_{\\rm{r}}=0.11\\,\\rm{fm}^{-3}$.}\n  \\label{fig_ab_rhot}\n\\end{figure}\n\nAs we discussed earlier, the fundamentally important quantity that is growing from the FFG to HMT-SCGF then HMT-exp is the strength of the underlying isospin dependence of SRC\/HMT, namely, \nthe difference $\\delta x=x_{\\rm{HMT}}^{\\rm{SNM}}-x_{\\rm{HMT}}^{\\rm{PNM}}$ is increasing. Our results shown in FIG.\\,\\ref{fig_ab_rhot} indicates an anti-correlation between the $\\delta x$  and the transition density $\\rho_{\\rm{t}}$.\nIt provides an important connection to the study of the thickness of inner crusts\\,\\cite{XuJ09} as well as the related observational phenomena\\,\\cite{Gear11,Wen12,Newton12,Newton14,Hooker15,Zhou21}, such as the crustal oscillations and glitches as well as the $r$-mode stability window of rapidly rotating neutron stars. Our findings here may also have important implications on the heat transport process in neutron stars\\,\\cite{Cac08}.\n\nWithin the minimal model of neutron stars assuming no hadron-quark phase transition as well as baryon resonance and hyperon production, the compositions of neutron stars are \ndetermined by the requirement of charge neutrality and $\\beta$-equilibrium conditions. From the nucleon specific energy $E(\\rho,\\delta)$ we can calculate numerically the proton fraction $x_{\\rm{p}}(\\rho)$ for $\\beta$-stable matter and examine its dependence on the strength of SRC\/HMT. More specifically, for neutrino free $ \\beta $-stable matter, the chemical\nequilibrium for the reactions $\n\\rm{n}\\rightarrow\\rm{p}+\\rm{e}^{-}+\\overline{\\nu}_{\\rm{e}}$ and\n$\\rm{p}+\\rm{e}^{-}\\rightarrow \\rm{n}+\\nu _{\\rm{e}}$ requires $ \\mu\n_{\\rm{e}}=\\mu_{\\rm{n}}-\\mu_{\\rm{p}}\\approx\n4E_{\\rm{sym}}(\\rho)\\delta+8E_{\\rm{sym,4}}(\\rho)\\delta^3+12E_{\\rm{sym},6}(\\rho)\\delta^5+\\cdots$, here $E_{\\rm{sym},6}(\\rho)\\equiv 120^{-1}\\partial^6E(\\rho,\\delta)\/\\partial\\delta^6|_{\\delta=0}$ is the sixth-order symmetry energy.\nFor relativistically degenerate electrons, we have $ \\mu _{\\rm{e}}=[\nm_{\\rm{e}}^{2}+(3\\pi ^{2}\\rho x_{\\rm{e}})^{2\/3}] ^{1\/2}\\approx\n( 3\\pi ^{2}\\rho x_{\\rm{e}}) ^{1\/3} $, where\n$m_{\\rm{e}}\\approx0.511\\,$MeV is the electron mass, and\n$x_{\\rm{p}}=x_{\\rm{e}}$ because of charge neutrality. Above a certain density where $\\mu _{\\rm{e}}$ exceeds the muon\nmass $m_{\\mu }\\approx105.7\\,$MeV, the reactions\n$\\rm{e}^{-}\\rightarrow \\mu ^{-}+\\nu\n_{\\rm{e}}+\\overline{\\nu}_{\\mu }$, $\\rm{p}+\\mu ^{-}\\rightarrow \\rm{n}+\\nu _{\\mu }$ and $%\n\\rm{n}\\rightarrow\\rm{p}+\\mu ^{-}+\\overline{\\nu}_{\\mu }$ are\nenergetically allowed. Then, both electrons and muons are present\nin $\\beta $-stable matter. This alters the $\\beta$-stability condition\nto $\\mu_{\\rm{e}}=\\mu _{\\mu }=[ m_{\\mu }^{2}+(3\\pi ^{2}\\rho x_{\\mu\n})^{2\/3}] ^{1\/2}$ with $x_{\\rm{p}}=x_{\\rm{e}}+x_{\\mu }$. \nGenerally, when the nucleon momentum distribution function $n_{\\v{k}}^J(\\rho,\\delta)$ has a low- (high-) momentum depletion (tail), the nucleon chemical potential is not given by \n$k_{\\rm{F}}^{\\rm{n\/p},2}\/2M+U_{\\rm{n\/p}}(\\rho,\\delta,k_{\\rm{F}}^{\\rm{n\/p}})$, i.e., it could not be obtained by simply letting the momentum $|\\v{k}|$ to be the Fermi momentum in the single-nucleon potential. \nIn fact, determining the relation between the nucleon chemical potential and the single-nucleon potential for a general $n_{\\v{k}}^J(\\rho,\\delta)$ is known as a fundamental problem in nuclear many-body theories\\,\\cite{AGD1960}.\nWe thus have to calculate numerically the chemical potential $\\mu_J$ for a nucleon $J$ from the energy density $\\varepsilon$ of neutron star matter via\n$\n  \\mu_J=\\partial\\varepsilon(\\rho,\\delta)\/\\partial\\rho_J.\n$\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=4.cm]{abMDI-xp.eps}\\quad\n \\includegraphics[width=4.05cm]{abMDI-sound.eps}\n  \\caption{The proton fraction $x_{\\rm{p}}$ (left) and the square of the speed of sound $s^2$ (right) in neutron star matter within the FFG and HMT-exp model, respectively.\n  The black circle and the cyan star correspond to the central densities of the neutron star in the two models.}\n  \\label{fig_abMDI-xp}\n\\end{figure}\n\nShown in the left panel of FIG.\\,\\ref{fig_abMDI-xp} is the proton fraction $x_{\\rm{p}}$ in neutron star matter as a function of density within the FFG and HMT-exp model, respectively.\nSince the HMT-SCGF result is similar to the one using the FFG model, we thus in the following focus on the comparison between the predictions using the FFG and the HMT-exp models.\nObviously, the most important observation is that due to the reduction of the high-density symmetry energy in the HMT-exp model the proton fraction $x_{\\rm{p}}$ in this model starts to decrease quickly above some critical density. \nIt thus prompts the role played by the symmetry energy in the core of neutron stars as $\\delta$ in the HMT-exp model is much closer to 1 than that in the FFG model.\nTherefore, it may have important ramifications on properties of neutron stars. For example, it is well known that the  $x_{\\rm{p}}$ determines the cooling mechanisms of protoneutron stars and the associated neutrino emissions\\,\\citep{LPPH}. \nMore quantitatively, the critical proton fraction $x^{\\rm{DU}}_{\\rm{p}}$ enabling the direct URCA process (DU) for fast cooling is given by\n$x^{\\rm{DU}}_{\\rm{p}}=1\/[1+(1+x_{\\rm{e}}^{1\/3})^3]$\nwith the $x_{\\rm{e}}\\equiv \\rho_{\\rm{e}}\/\\rho_{\\rm{p}}$ between 1 and 0.5 leading to $x^{\\rm{DU}}_{\\rm{p}}$ between 11.1\\% to 14.8\\%\\,\\citep{Kla06}, see the red dashed lines in the left panel of FIG.\\,\\ref{fig_abMDI-xp}.\nThe black circle and the cyan star in the left panel of FIG.\\,\\ref{fig_abMDI-xp} correspond to the central densities of neutron stars in the two models studied here. It is seen that around these central densities the direct URCA is allowed in the FFG model but forbidden in the HMT-exp model. \n\nThe speed of sound squared $s^2=\\d P\/\\d\\varepsilon$ is a measure of the stiffness of nuclear EOS. Shown in the right panel of FIG.\\,\\ref{fig_abMDI-xp} are the predicted $s^2$ by the two models considered. It is seen that the EOS in the HMT-exp model is much softer than the FFG model prediction at densities above about $2.5\\rho_0$. Since the ANM EOS $E(\\rho,\\delta)\\approx E_0(\\rho)+E_{\\rm{sym}}(\\rho)\\delta^2+\\cdots$, the reduced symmetry energy but increased $\\delta^2$ at high densities together may effectively soften the EOS. However, its net effects depend on how strongly the SNM EOS $E_0(\\rho)$ is stiffened by the SRC\/HMT. The results shown here indicate clearly that the SRC\/HMT effect on the symmetry energy is winning against that on the SNM EOS. \nThis result is expected to have some dynamical effects on properties of neutron stars. Since the symmetry energy and SNM EOS in different density regions have different effects on properties of neutron stars, the SRC\/HMT effects through these two terms of the ANM EOS have to be studied quantitatively as we shall do next. \n\nThe mass-radius correlation of neutron stars is obtained from integrating the Tolman-Oppenheimer-Volkoff (TOV) equations\\,\\cite{Misner1973}\n\\begin{align}\n\\frac{\\d P(r)}{\\d r}=&-\\frac{[\\varepsilon(r)+P(r)][M(r)+4\\pi r^3P(r)]}{r[r-2M(r)]},\\\\\n\\frac{\\d M(r)}{\\d r}=&4\\pi r^2\\varepsilon(r),\n\\end{align}\nwhere $r$ is the radial distance from the center of the star, and\n$M(r)$ is the mass enclosed within $r$ (adopting natural unit in which $c=G=1$).\nThe pressure of $\\beta$-stable and charge neutral npe$\\mu$ matter is given by $\nP(\\rho,\\delta)=P_{\\rm{N}}(\\rho,\\delta)\n+P_{\\rm{e}}(\\rho,\\delta)+P_{\\mu}(\\rho,\\delta)$,\nwhere $P_{\\rm{N}}(\\rho,\\delta) $ is the nucleon pressure. \nThe lepton pressure is further given as $\nP_\\ell(\\rho,\\delta)=\\rho_\\ell\\mu_\\ell-\\varepsilon_\\ell(\\rho,\\delta)\n$, with $\n\\mu_\\ell=[{k_\\ell^2+m_\\ell^2}]^{1\/2}$ and $k_\\ell=(3\\pi^2\\rho_\\ell)^{1\/3}$ for $\\ell=\\rm{e},\\mu$, here $\\varepsilon_{\\ell}(\\rho,\\delta)=\\eta_{\\ell}\\Phi_{\\ell}(t_{\\ell})$ (see (\\ref{dd_e1}) for the definition of $\\Phi_{\\ell}$).\nFor the whole system, we have the self-consistency relation between the total pressure $P$ and the total energy density $\\varepsilon$, i.e., \n$\nP=\\rho^2{\\d(\\varepsilon\/\\rho)}\/{\\d\\rho}$.\nThe inner crust of neutron stars with densities ranging between $\\rho_{\\text{out}}=2.46\\times\n10^{-4}\\,\\rm{fm}^{-3}$ corresponding to the neutron dripline and the\ncore-crust transition density $\\rho _{\\text{t}}$ is the region where\nsome complex and exotic structures\\,---\\,the ``nuclear pasta'' may exist.  Because of our very poor knowledge\nabout this region we adopt the polytropic EOSs expressed in\nterms of the pressure $P$ as a function of the total energy density\n$\\varepsilon$ as\n$P=c+d\\varepsilon^{4\/3}$\\,\\cite{XuJ09,Hor03}. The constants $c$ and\n$d$ are determined by the quantities at $\\rho\n_{\\text{t}}$ and $\\rho _{\\text{out}}$\\,\\cite{XuJ09}, i.e.,\n\\begin{equation}\nc=\\frac{P_{\\rm{out}}\\varepsilon_{\\rm{t}}^{4\/3}-P_{\\rm{t}}\\varepsilon_{\\rm{out}}^{4\/3}}{\\varepsilon_{\\rm{t}}^{4\/3}-\\varepsilon_{\\rm{out}}^{4\/3}},\n~~\nd=\\frac{P_{\\rm{t}}-P_{\\rm{out}}}{\\varepsilon_{\\rm{t}}^{4\/3}-\\varepsilon_{\\rm{out}}^{4\/3}}\n.\\end{equation}\nFor the outer\ncrust\\,\\cite{BPS71,Iida1997}, we use the conventional Baym-Pethick-Sutherland (BPS) EOS for the region with $6.93\\times 10^{-13}\\,\\rm{fm}^{-3}\\lesssim\\rho \\lesssim\\rho _{\\text{out}}$ and the\nFeynman-Metropolis-Teller (FMT) EOS for $4.73\\times 10^{-15}\\,\\rm{fm}^{-3}\\lesssim\\rho \\lesssim6.93\\times\n10^{-13}\\,\\rm{fm}^{-3}$, respectively.\nThe total neutron star matter EOSs in the two models are shown in FIG.\\,\\ref{fig_abMDI-beta}.\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=6.5cm]{abMDI-beta.eps}\n  \\caption{The EOS of neutron star matter in the FFG model and the HMT-exp model. See the text for details.}\n  \\label{fig_abMDI-beta}\n\\end{figure}\n\nThe resulting mass-radius correlation of neutron stars is shown in FIG.\\,\\ref{fig_abMDI-MR} for the FFG and the HMT-exp model, respectively. Firstly, we notice that both models give a radius of about 12\\,km for canonical neutron stars of mass 1.4$M_{\\odot}$ consistent with many analyses of the GW170817 observation, see, e.g., Refs.\\,\\cite{EPJA-review,Baiotti,Capano20,David,Kat20,AngLi} for reviews. \nThis is mostly by design as our EDF parameters are fixed such that all the EOS characteristics at $\\rho_0$ are consistent with existing constrains from both astrophysical observations and terrestrial experiments. \nSecondly, it is interesting to see that the two models indeed predict significantly different maximum masses and the corresponding radii. More quantitatively, the maximum mass of neutron stars in the FFG model is about $M_{\\rm{NS}}^{\\max}\\approx1.98M_{\\odot}$, with the corresponding radius being about 9.9\\,km. While the $M_{\\rm{NS}}^{\\max}$ in the HMT-exp model is about $1.72M_{\\odot}$ with a radius of about 10.3\\,km. \nThe reduction of the maximum mass $M_{\\rm{NS}}^{\\max}$ due to the SRC\/HMT is thus about 13\\%. \nThe point corresponding to the maximum mass on the $P(\\varepsilon)$ plane is also shown in FIG.\\,\\ref{fig_abMDI-beta}, by the black circle (cyan star) for the FFG (HMT-exp) model.\nThirdly, since the HMT largely affects the transition density $\\rho_{\\rm{t}}$ (see FIG.\\,\\ref{fig_ab_rhot}), the radii of low-mass neutron stars in the FFG model and the HMT-exp model show obvious differences.\n\nThe central density $\\rho_{\\rm{cen}}$ corresponding to $M_{\\rm{NS}}^{\\max}$ is about $\\rho_{\\rm{cen}}^{\\rm{FFG}}\\approx7.3\\rho_0$ ($\\rho_{\\rm{cen}}^{\\rm{HMT}\\mbox{-}\\rm{exp}}\\approx7.9\\rho_0$) for the FFG (HMT-exp) model (see the points shown in FIG.\\,\\ref{fig_abMDI-xp}).\nAlthough the EDF constructed is non-relativistic in nature, it could be effectively applied to neutron stars and fulfill the principle of causality (set by the speed of sound $s^2=\\d P\/\\d\\varepsilon=1$).\nIn particular, we find that the square of the speed of sound in the FFG model at $\\rho_{\\rm{cen}}^{\\rm{FFG}}$ is about $s^2_{\\rm{FFG}}\\approx0.84$, while that for the HMT-exp model is about $s_{\\rm{HMT}\\mbox{-}\\rm{exp}}^2\\approx0.35$, see the right panel of FIG.\\,\\ref{fig_abMDI-xp} for the density dependence of the $s^2$ in the two model. Thus, due to the softening of the symmetry energy above a critical density and the non-relativistic nature of the Gogny-like EDF, the high-density $s^2$ in the HMT-exp model is \nmuch smaller than that in the FFG model. In fact, it may not exceed 1 at even larger densities when the FFG becomes acausal (the density violating the principle of causality in the FFG model is about $9.8\\rho_0$).\n\n\\begin{figure}[h!]\n\\centering\n \n \\includegraphics[width=6.5cm]{abMDI-MR.eps}\n  \\caption{The mass-radius relation for neutron stars in the FFG and the HMT-exp models.}\n  \\label{fig_abMDI-MR}\n\\end{figure}\n\nTo this end, it is interesting to note that some earlier studies on the mass-radius relation of neutron stars adopting the nonlinear RMF models found that the SRC\/HMT generally increases the maximum mass $M_{\\rm{NS}}^{\\max}$\\,\\cite{Cai16b,Lou22a,Lu2022,Hong22,Lou22,Lu21,Souza20}. In these models the SRC\/HMT induced increase in SNM pressure dominates over the reduction of symmetry energy. \nOne main reason is that in the nonlinear RMF models the symmetry energy itself plays a minor role in determining the maximum mass $M_{\\rm{NS}}^{\\max}$ as pointed out already a long time ago\\,\\cite{Ser79}. Moreover, the softening of the $E_{\\rm{sym}}(\\rho)$ in the RMF models is weaker than that in the current (non-relativistic) Gogny-like EDF. Of course, this then raises an interesting question: how can this kind of EDFs encapsulating the SRC\/HMT effects support the currently observed most massive neutron star PSR J0740+6620 which has a mass of $2.08^{+0.07}_{-0.07}M_\\odot$\\,\\cite{Fon21}? In fact, even without considering the SRC\/HMT effects, it has been a challenging problem to find mechanisms to support heavy neutron stars (with masses around and above $2M_{\\odot}$) within non-relativistic energy density functionals, see, e.g., Refs.\\,\\cite{Rios-G,Zha16,ZhouY2019,Gon18,Gon19,Lou20}. Our results presented above indicate that this problem may become more challenging and certainly deserves further investigations.\n\n\\setcounter{equation}{0}\n\\section{Summary and Outlook}\\label{sec6}\n\nIn summary, we investigated SRC\/HMT effects on the EOS and single-nucleon potential in cold neutron-rich matter within the Gogny-like EDF. The SRC effects are incorporated into the EDF by using a single-nucleon\nmomentum distribution function $n_{\\v{k}}^J(\\rho,\\delta)$ that has an isospin-dependent high-momentum (low-momentum) tail (depletion) with its parameters determined by the available SRC experiments. We introduced a parametrization as a surrogate for the momentum-dependent kernel of the Gogny-like EDF to facilitate derivations of analytical expressions for all relevant physical quantities describing the EOS and single-nucleon potential in ANM. The surrogate is shown to be effective for problems with momentum (density) scale smaller than about 1 GeV\/c (several times the saturation density). The resulting expressions for all terms in the ANM EOS and single-nucleon potential will facilitate further explorations of nuclear interactions in dense neutron-rich matter as well as their impacts on properties of neutron stars and heavy-ion reactions induced by high-energy radioactive beams. \nIn particular, \n\\begin{enumerate}\n\\item[(a)]We have derived the analytical expressions for the EOS of SNM $E_0(\\rho)$,  the corresponding pressure $P_0(\\rho)$,  the incompressibility coefficient $K_0(\\rho)$, the nucleon effective k-mass $M_0^{\\ast}(\\rho)$, the isoscalar potential $U_0(\\rho,|\\v{k}|)$, the symmetry energy $E_{\\rm{sym}}(\\rho)$ together with its slope parameter $L(\\rho)$, and the isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$.\nNumerical constructions based on these analytical expressions are given through three models for the nucleon momentum distribution, namely the FFG model without the SRC-induced HMT in the $n_{\\v{k}}^J(\\rho,\\delta)$, the\nHMT-SCGF\/HMT-exp model with different properties of the HMT, to fulfill certain empirical constraints on the EOS of ANM.  All three models contain proper potential parts.\n\\item[(b)] While the same set of available empirical constraints on the ANM EOS at $\\rho_0$ are all maintained in the three models, the EOS $E_0(\\rho)$ of SNM at high densities is made slightly harder once the SRC\neffects are considered.  In addition, the symmetry energy in the presence of HMT becomes softer at large densities even to start decreasing above a critical density. Although qualitatively consistent with the nonlinear RMF model predictions, the reduction of the high-density symmetry energy $E_{\\rm{sym}}(\\rho)$ is much stronger while the enhancement of the SNM pressure $P_0$ is much weaker in the Gogny-like EDF with SRC\/HMT. Consequently, as a result of the competition between these two effects, the maximum mass of a cold neutron star is found to be reduced compared with the prediction of the Gogny-like EDF without considering the SRC\/HMT.\n\\item[(c)] The single-nucleon potential $U_0(\\rho,|\\v{k}|)$ in SNM at $\\rho_0$ in all three models is consistent with the isoscalar nucleon optical potential from earlier analyses of various experimental data. The nucleon isovector (symmetry) potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ in ANM is largely enhanced due to the corresponding enhancement of the potential symmetry energy $E_{\\rm{sym}}^{\\rm{pot}}(\\rho)$.\nThese are the direct consequences of the momentum-dependence of the nucleon potential in the Gogny-like EDF due to the finite ranges of nuclear interactions.  Such momentum-dependence is absent in the nonlinear RMF approach.\n\\end{enumerate}\n\nThe formulas, table of coupling constants, parameterizations and quantitative demonstrations of the relevant physical quantities given in this work provide a useful and convenient starting point to further investigate the nature of strong interactions at short distances and novel effects of SRC\/HMT on properties of dense neutron-rich matter. In particular, the generally density- and momentum-dependent single-nucleon potential in neutron-rich matter is expected to play a significantly role in understanding the dynamics and observables of heavy-ion reactions induced by high-energy radioactive beams, the cooling mechanisms of protoneutron stars as well as properties of cold neutron stars at $\\beta$-equilibrium and their mergers. For instance, with the coupling parameters given in Table   \\ref{tab_para}, one can use the single-nucleon potential of Eq. (\\ref{Gen-U}) in simulating heavy-ion reaction with the $f_J(\\v{r},\\v{k})$ self-consistently generated by the BUU transport model. After comparing the predicted observables with experimental data, one can then make better informed predictions for the EOS of hot, dense and neutron-rich matter. The latter is a necessary input for simulating mergers of two neutron stars based on principles of general relativity \\cite{Ra1,Ra2}. In particular, it is important for understanding the high-frequency spectrum of gravitational waves from post-mergers that carry critical information about the nature of possible phase transitions and the separation boundary\/gap between neutron stars and blackholes. Interestingly, it has already been found that the density dependence of nuclear symmetry energy play a significant role in answering many intriguing questions in simulating  neutron star mergers \\cite{Most21}. \n\nAs discussed earlier, our approach has some caveats and some challenging issues remain to be resolved. Fortunately, multimessenger nuclear astrophysics with high-precision X-ray and gravitational wave detectors as well as advanced rare isotope beam facilities are expected to provide us with various kinds of new and more accurate observational data. Combined analyses of these data using the same Gogny-like EDF with SRC\/HMT may help us address some of the remaining issues. For example, the strong enhancement of the symmetry potential $U_{\\rm{sym}}(\\rho,|\\v{k}|)$ due to the SRC\/HMT especially at supra-saturation densities, e.g., $\\rho\\approx2\\rho_0\\mbox{$\\sim$}3\\rho_0$,  is expected to affect significantly the dynamics and some observables of nuclear reactions with high-energy radioactive beams. While the corresponding reduction of the total symmetry energy reduces significantly the maximum mass of neutron stars that the resulting EOS can support. A joint analysis of these effects using a unified Gogny-like EDF with SRC\/HMT is expected to be fruitful. Besides the traditionally used forward predictions with the different model parameter sets, the expected new data and existing ones together will enable us to get new insights from inferences of relevant model parameters using machine leaning techniques, such as the Bayesian model selection \\cite{MAL}. These studies are on the top of our working agenda. \n\n\\section*{Acknowledgement} We would like to thank Lie-Wen Chen, Xiao-Tao He, Che Ming Ko and Xiao-Hua Li for useful discussions over the years on some of the issues studied in this work.\nThis work is supported in part by the U.S. Department of Energy, Office of Science,\nunder Award No. DE-SC0013702, the CUSTIPEN (China-\nU.S. Theory Institute for Physics with Exotic Nuclei) under\nUS Department of Energy Grant No. DE-SC0009971.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}