diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbqmq" "b/data_all_eng_slimpj/shuffled/split2/finalzzbqmq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbqmq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\nThe last years have seen a generalization of Riemannian geometry to sub-Rie\\-ma\\-nnian geometry \\cite{agrachev2012, gromov144, montgomery2006, strichartz1989, strichartz1986}.\nA sub-Riemannian manifold is a connected manifold $G$ with a distribution $D\\subset TG$ such that successive Lie brackets fields in $D$ generate all the tangent space $TG$. In addition, a positive definite scalar product $\\langle\\cdot,\\cdot\\rangle$ is defined at $D$ such that it is possible to calculate the length of admissible curves, i.e., the tangent curves on $D$. As in Riemannian geometry, the distance $\\rho$ between two points $p$ and $q$ in $G$ is defined as the infimum length of admissible curves that connect the points $p$ and $q$. By means of this distance, the sub-Riemannian manifold becomes a metric space \\cite{Bellaiche1996} and can be endowed with a Hausdorff measure. Also, there is an equivalent of the Riemannian volume, the so called Popp's volume. Introduced in \\cite{montgomery2006}, it is a smooth volume which is canonically associated with the sub-Riemannian structure and was used in \\cite{agrachev2012} to define the sub-Laplacian in sub-Riemannian geometry.\n\nOn sub-Riemannian manifolds which are nilpotent Lie groups (stratified Lie gro\\-u\\-ps) the spherical Hausdorff measure, the Popp's measure and the Haar measure are mutually constant multiples one of another \\cite{Agrachev2012a, ghezzi2014, Magnani2005, mitchell1985, barilari2013}. Then, it is natural to ask what means a submanifold of\na stratified Lie group to be of minimal measure.\n\nThe aim of this paper is, using the measure proposed by Magnani and Vittone for non-horizontal submanifolds in stratified Lie groups \\cite{Magnani2008}, to calculate the first variation of these submanifolds and determine the necessary conditions for a non-horizontal submanifold to be of minimal measure. We will also give new examples of non-horizontal minimal submanifolds in the Heisenberg group, in particular minimal surfaces in 5-dimensional Heisenberg group . In order to present the main results, we briefly introduce some concepts which will be dealt with in detail in the subsequent sections.\n\nLet $\\mathbb G$ be a stratified Lie group with a graded Lie algebra $\\mathfrak{g}=\\mathfrak{g}^1\\oplus\\cdots\\oplus\\mathfrak{g}^r$ with $r\\geq1$. If $\\langle\\cdot,\\cdot\\rangle$ is a scalar product on $\\mathfrak g^1$, we can extend it to $\\mathfrak g$ by induction (see Proposition \\ref{extend}).\nWe also consider the distribution $D\\subset T\\mathbb G$ generated by $\\mathfrak g^1$ and the scalar product in $D$ generated by $\\langle\\cdot,\\cdot\\rangle$. This way, $(\\mathbb G,D,\\langle\\cdot,\\cdot\\rangle)$ becomes a sub-Riemannian manifold, also called the Carnot group. Note that, traditionally the scalar product on $D$ is extended to $T\\mathbb G$ and the Riemannian connection on $T\\mathbb G$ is used to make calculations on $\\mathbb G$. This observation will be our starting point. We shall work with a covariant derivative $\\overline\\nabla$ defined by $\\overline\\nabla X=0$ for all $X\\in\\mathfrak g$. It has intrinsic torsion which is essentially the negative of the Lie bracket in $\\mathbb G$ and the zero curvature tensor. So, this covariant derivative permits us to establish an interesting parallel between the invariants of submanifolds in $\\mathbb R^n$ and the invariants of submanifolds in $\\mathbb G$.\n\nThe geometry of a submanifold $M$ of $\\mathbb G$ at each point depends on the relative position of\n$TM$ and $D$. The submanifolds with a ``high'' contact with $D$ at one point may have\nsingularities from the metric point of view, even though being a $C^\\infty$ submanifold. In this paper,\nwe avoid these situations by considering a non-horizontal submanifold $M$ transverse to D,\ni.e., $TM+D=T\\mathbb G$. For this submanifold the horizontal normal subspace $TM^\\perp$ play the same role as the normal space does to a submanifold in $\\mathbb R^n$. We define $TM^\\perp$ as the orthogonal subspace to $TM\\cap D$ in $D$, i.e., the horizontal distribution $D=(TM\\cap D)\\oplus TM^\\perp$. Hence, $T\\mathbb G=TM\\oplus TM^\\perp$ and we can use this decomposition\n(in general not orthogonal!) to project $\\overline\\nabla$ to a connection $\\nabla$ over $TM$.\n\nLet $e_1,\\ldots, e_n$ be a orthonormal basis of $\\mathfrak{g}$ with its dual $e^1,\\ldots,e^n$. Let $f_1,\\ldots,f_n$ be a adapted frame in the $T\\mathbb{G}$ such that $f_1,\\ldots,f_p$ are orthogonal to $TM\\cap D$ and $f_{p+1},\\ldots,f_{d_1}$ is a orthonormal basis of $TM\\cap D$ in $D$. We complete $f_{p+1},\\ldots,f_{d_1}$ to a basis $f_{p+1},\\ldots,f_n$ of $TM$ taking\n$\nf_j=e_j-\\sum_{\\alpha=1}^pA_j^\\alpha f_{\\alpha},\n$\nfor $j=d_1+1,\\ldots,n$. If we denote by $f^1,\\ldots, f^n$ its dual basis, then the sub-Riemannian volume form on $\\mathbb G$ is defined as $\\dif V=e^1\\wedge\\cdots\\wedge e^n=f^1\\wedge\\cdots\\wedge f^n$. When $M$ is a hypersurface, the $H$-perimeter measure is traditionally used \\cite{Montefalcone2007, Montefalcone2012, Danielli2007, Hladky2013}. For the non-horizontal submanifolds of codimension $p\\geq 1$, the spherical Hausdorff measure has the following representation proved in \\cite{Magnani2008}:\n\\begin{equation}\\label{M}\n\\int_M\\theta(\\tau^d_M(x))\\dif S^d_\\rho(x)=\\int_M|\\tau^d_{M}(x)|\\dif\\mbox{vol}_{h}(x) \\ ,\n\\end{equation}\nwhere $h$ is a fixed Riemannian metric, $d$ is the Hausdorff dimension of $M$, $\\theta(\\tau^d_M(x))$ is the metric factor (see Section \\ref{hspace}), $S^d_\\rho$ is the\t $d$-spherical Hausdorff measure and $\\dif\\mbox{vol}_{h}$ is the Riemannian volume form on $M$ induced by $(\\mathbb G,h)$.\n\nWe will denote by $\\mu$ the measure in non-horizontal submanifolds defined by $\\dif\\mu(x)=|\\tau^d_{M}(x)|\\dif\\mbox{vol}_{h}(x)$, which is a nat\\-u\\-ral can\\-di\\-date to de\\-fine the vol\\-ume for non-horiz\\-ontal submanifolds.\nIf the metric factor $\\theta(\\tau^d_M(x))$ is constant on $M$ (the case of the Heisenberg group $\\mathbb{H}^n$, see Section \\ref{hspace}) the measure $\\dif\\mu(x)$ is a multiple of $(Q-p)$-spherical Hausdorff measure on $M$. Writing the density $\\dif\\mu$ in the adapted frame $f^1,\\ldots,f^n$ we obtain a beautiful formula which we will use for the variational calculation, namely\n$\n\\dif \\mu=f^{p+1}\\wedge\\cdots\\wedge f^n\n$ (see Theorem \\ref{dS}).\n\nThe second result of this paper (Theorem \\ref{firstvar}) is a sufficient condition for minimality of non-horizontal submanifolds. We say that a non-horizontal submanifold is minimal if $H+\\sigma=0$ on $TM^\\perp$, where $H$ is the mean curvature (Definition \\ref{curvmedia}) and $\\sigma$ is the mean torsion (Definition \\ref{colchete}). In the case of hypersurfaces, the mean torsion is null and so the definition of minimality is the same as in \\cite{Montefalcone2007, Montefalcone2012, Danielli2007, Hladky2013, Hurtado2010, Ritore2008} and also of minimal submanifolds of Riemannian geometry.\n\nAs a direct application of Theorem \\ref{firstvar} we present the following example: if $M$ is minimal submanifold of $\\mathbb R^{2n}$, then $N=M\\times\\mathbb R$ is minimal submanifold of $\\mathbb{H}^n$.\nFurthermore, an interesting application of this theorem is discussed in section \\ref{H2}, where we find minimal non-horizontal surfaces in the $5$-dimensional Heisenberg group $\\mathbb H^2$. Observe that in the $3$-dimensional Heisenberg group $\\mathbb H^1$ the tangent horizontal curves to minimal surfaces are lines and hence this surfaces are ruled surfaces \\cite{Pansu1982}. Now, for $\\mathbb H^2$ the tangent horizontal curves to minimal surfaces can be more general. In section \\ref{H2}, we will present two cases: the curves are lines and we obtain ruled surfaces; the curves are circles and we obtain a family of circles, which we will call tubular surfaces.\n\nIn the last section, we prove for non-horizontal hypersurfaces the following: $-H_{f_1}=\\mbox{div}_{\\mathbb G}\\Big(\\frac{\\mbox{grad}_{D}\\phi}{|\\mbox{grad}_{D}\\phi|}\\Big)$, where $\\phi:\\mathbb G\\rightarrow\\mathbb R$ is smooth function, $\\mbox{div}_{\\mathbb G}$ is the divergence function on $\\mathbb G$ and $\\mbox{grad}_D$ is the horizontal gradient operator. With this formula we give a proof that the hyperboloid paraboloid is minimal.\n\n\\section{Stratified Lie groups}\\label{slg}\n\nA stratified Lie group $\\mathbb{G}$ is an $n$-dimensional connected, simply connected nilpotent Lie group whose Lie algebra $\\mathfrak{g}$ decomposes as $\\mathfrak{g}=\\mathfrak{g}^1\\oplus\\mathfrak{g}^2\\oplus\\cdots\\oplus\\mathfrak{g}^r$ and satisfies the condition $[\\mathfrak{g}^1,\\mathfrak{g}^j]=\\mathfrak{g}^{j+1}, j=1,\\ldots, r-1,\\quad [\\mathfrak{g}^j,\\mathfrak g^r]=0, j=1,\\ldots,r$. Write $d_i=\\dim\\mathfrak{g}^i$, choose a basis $e_1,\\ldots,e_n$ of $\\mathfrak{g}$ such that $e_{d_{j-1}+1},\\ldots,e_{d_j}$ is a basis of $\\mathfrak{g}^j$ and denote its dual basis by $e^1,\\ldots,e^n$. Then, we define the \\emph{degree} of $e_k$ as $\\deg k=j$ if $d_{j-1} 1$}\n\\label{tab:coeff}\n\\end{table*}\n\nFigure \\ref{fig:risk} shows the distribution of predicted risk values. The figure shows that even with all the features listed in Table \\ref{tab:variables}, there is still a considerable overlap of predicted probabilities between case and control patients. Better separation between these two classes can improve the risk prediction accuracy. The plot suggests incorporation of additional diagnoses or temporal aspects of existing diagnoses may be necessary to improve model performance.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Risk.pdf}\n\\end{center}\n\\caption{Box plot of the predicted probabilities using a forward selection model on all the features.}\n\\label{fig:risk}\n\\end{figure}\n\nFigure \\ref{fig:perf} contains the performance plots for the forward selection models trained on feature set 1 and feature set 9. Figure \\ref{fig:roc} demonstrates the noticeable improvement using all the available features. Additionally, the model trained on feature set 1, demographics and family history features, barely outperforms random chance. The tradeoff between sensitivity, specificity, and positive predictive value can be seen in Figure \\ref{fig:ppv-sens}. Feature set 9 has a higher intersection between the sensitivity and specificity curves, which is summarized in Table \\ref{tab:intersect}. In addition, the full-featured model generally achieves a better positive predictive value for all threshold values. However, the positive predictive value and sensitivity curves cross at the value $\\sim 0.40$. At this point, we can accurately diagnose 40\\% of the case patients, but only 2 out of every 5 patients predicted to have a high risk of MS will be diagnosed with MS at the next office visit, a high number of false positives.\n\n\\begin{figure*}[htb]\n\\subfigure[ROC curves compared to random assignment]{\n\\includegraphics[width=0.48\\textwidth]{ROCR.pdf}\n\\label{fig:roc}\n}\n\\subfigure[Sensitivity, specificity, and positive predictive value as function of threshold]{\n\\includegraphics[width=0.48\\textwidth]{Threshold.pdf}\n\\label{fig:ppv-sens}\n}\n\\caption{Model performance plots for feature sets 1 and 9.}\n\\label{fig:perf}\n\\end{figure*}\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{l c c c c}\n\\hline\nFeature Set & Cutoff & Sensitivity & Sensitivity & PPV \\\\\n\\hline\n1 & 0.212 & 0.528 & 0.528 & 0.218\\\\\n9 & 0.241 & 0.647 & 0.647 & 0.314 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The intersection of the sensitivity and specificity curve from Figure \\ref{fig:ppv-sens}.}\n\\label{tab:intersect}\n\\end{table}\n\\subsection{Discussion}\nThe results demonstrate reasonable predictive accuracy using all the available features. One potential hindrance lies in the current feature construction. As Figure \\ref{fig:encounter} shows, there are a limited number of encounters prior to $t_0$ for case patients. Thus, it is difficult to determine whether an unobserved diagnosis may be due to the lack of longitudinal data (the patient was diagnosed prior to the study period). Additionally, certain diagnoses, such as EBV, can only be verified through culture samples which are not performed for every patient.\n\nAnother limitation of our study is the reliance on ICD-9 and procedure codes. A patient may exhibit all the clinical symptoms for a specific disease but it is not present in the encounter data because the disorder has not been diagnosed. The ambiguity of ICD9-codes and diagnostic discrepancies between medical doctors can also impact our feature construction. Moreover, the blood test results' conversion to a categorical feature may be inaccurate as the testing protocol may have changed during the study window. Therefore, a patient's feature vector may not accurately reflect their medical history.\n\nOur study also suggests incorporating additional features. Given that some of the variables were unrecorded in the structured portion of the EMR, parsing through the clinical notes could result in information regarding lifestyle factors, diet, detailed family and medical history. In addition, temporal aspects of the medical diagnoses were not included in our feature set since the data was confined to medical encounters over a 6-year period. \n\n\\section{Conclusion}\nThis paper presented a risk prediction model from EMRs to help address the difficulty of early diagnosis in MS patients. A sparse set of features were selected to minimize model complexity while maintaining reasonable predictive performance. Our results show we are able to help identify patients at high-risk of developing MS, in spite of a limited sample of patient data. In addition, our models have the ability to generalize to other healthcare systems as we rely only on components commonly found in electronic patient data.\n\nThe work demonstrates the potential of leveraging EMRs to aid medical professionals with difficult tasks, especially with early disease diagnosis. Future work will focus on incorporating temporal components, such as time of diagnosis, into the model, decreasing the false positive rate, and integrating a larger control population.\n\n\\section{Acknowledgments}\nWe thank Afif Hentati and Demetrius \"Jim\" Maraganore for their guidance, advice and comments on this study. We acknowledge comments from and conversations with Kibaek Kim, Yubin Park, Xiang Zhong, and Sanjay Mehrotra. We are indebted to Justin Lakeman for extracting data from the NorthShore Enterprise Data Warehouse.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.}\n\nTraditional Density Functional Theory (DFT) \\cite{Dre90,Koh99,DFTLN} and its time-dependent generalization \\cite{Run84,Gro94}\nhave evolved into standard tools for the description of electronic properties in condensed-matter physics and quantum chemistry\nthrough the simple local density instead of the less tractable $N$-body wave function.\nStationary DFT is based on the Hohenberg-Kohn (HK) theorem \\cite{Hoh64}, which\nproves that, for any non-degenerate system of N Fermions\nor Bosons \\cite{Dre90} put into a local external potential, the $N$-body ground-state wave function can be written as a functional of the local ground-state density.\nA similar theorem exists for the time-dependent case \\cite{Run84,Gro94}, where a dependence on the initial state appears.\nThe Kohn-Sham (KS) scheme \\cite{Koh65} and its time-dependent generalization \\cite{Run84,Gro94} provide a straightforward method \nto compute self-consistently the density in a quantum framework,\ndefining the non-interacting system (i.e.\\ the local single-particle potential) which reproduces the exact density.\n\nTraditional DFT is particularly well suited to study the electronic properties in molecules \\cite{Kre01}.\nAs a molecule is a self-bound system, the corresponding Hamiltonian is translationally invariant \n(which ensures Galilean invariance of the wave function \\cite{foot1}),\nand one can apply the Jacobi coordinates method.\nThis permits to decouple the center-of-mass (c.m.) properties from the internal ones,\nand to treat correctly the redundant coordinate problem\n(i.e.\\ the fact that one coordinate is redundant for the description of the internal properties \\cite{Schm01a})\nand the c.m. correlations.\nBut as the nuclei are much heavier than the electrons,\nwe can apply the Jacobi coordinates method to the nuclei only,\nso that only the nuclei will carry the c.m. correlations,\nand use the clamped nuclei approximation.\nThen, one recovers the ``external'' potential of traditional DFT, of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$,\nwhich accounts for the nuclear background as seen by the electrons in the frame attached to the c.m. of the nuclei.\nThus, traditional DFT is particularly adapted to the study of the electronic properties in molecules \\cite{Kre01}.\nIt is implicitely formulated in the nuclear c.m. frame \\cite{foot2} and the energy functional \ndoes not contain any c.m. correlations.\nOf course, contrary to the whole molecule, the pure electronic system is not a self-bound system:\nthe $v_{ext}$ potential breaks translational invariance and is compulsory in order to reach bound states in the stationary case.\n\nFor other self-bound systems, as isolated atomic nuclei or He droplets,\nthe situation is intrinsically different because the masses of all the particles (Fermions or Bosons) are of the same order of magnitude.\nAs a consequence, to decouple the c.m. properties from the internal ones,\none has to apply the Jacobi coordinates method\nto \\textit{all} the particules.\nThe redundant coordinate problem (thus the c.m. correlations) will now concern all the particles and should be treated properly.\nIf a DFT exists, the c.m. correlations \nshould be taken into account in the functional.\n\nMoreover, no \"external\" potential of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$ can be justified in the corresponding self-bound Hamiltonians\n(we denote $\\mathbf{r}_i$ the $N$ \nparticules\ncoordinates related to any inertial frame as the laboratory).\nOne may be tempted to formulate a DFT using the traditional DFT conclusions in the limit $v_{ext} \\to 0$,\nbut this would lead to false and incoherent results because:\n\\begin{itemize}\n\\item in the stationary case, the Hohenberg-Kohn theorem is valid only for external potentials that lead to bound many-body states \\cite{Lie83},\nwhich is not the case anymore at the limit $v_{ext} \\to 0$ \nfor translational invariant particle-particle interactions \\cite{Mes09};\n\\item the form of $v_{ext}$ is not translationally invariant, but translational invariance is a key feature of self-bound systems \\cite{Schm01a,Pei62,Mes09};\n\\item traditional DFT concepts as formulated so far are not applicable in terms of a well-defined internal density $\\rho_{int}$, i.e.\\ the density relative \nto the system's c.m.\\ , which is of experimental interest \\cite{Kre01,Eng07,Mes09} (it is for example measured in nuclear scattering experiments).\n\\end{itemize}\n\nInstead of the traditional DFT potential $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i)$, one might be tempted to introduce an arbitrary translational invariant potential of the form $\\sum_{i=1}^N v_{int}(\\mathbf{r}_i - \\mathbf{R})$, where $\\vec{R}=\\frac{1}{N}\\sum_{j=1}^{N}\\vec{r}_j$ is the total c.m.\\ of the particles.\nThis potential is an \"internal\" potential, i.e. is seen in the c.m. frame,\nand in \\cite{Mes09} we underlined that it is the only form which satisfies all the key formal properties.\nHowever, $v_{int}$ should be zero in the purely isolated self-bound case.\nThis is why in \\cite{Mes09} we presented it as a mathematical \"auxiliary\"\nto reach our goal and showed that it can be dropped properly at the end, conserving all the conclusions.\nThrough it (and using the Jacobi coordinates), we proved, by a different way than those found in \\cite{Eng07,Bar07},\nthe stationary \"Internal DFT\" theorem:\nthe internal many-body state can be written as a functional of $\\rho_{int}$.\nThen we formulated rigorously the corresponding \"Internal\" KS scheme (in the c.m.\\ frame).\nThe main interest of this work is\nto give a first step towards a fundamental justification to the use of internal density functionals\nfor stationary mean-field like calculations of nuclei \\cite{Ben03} or He droplets \\cite{Bar06} with effective interactions,\nshowing that there exists an ultimate functional which permits to reproduce the exact internal density, which was not clear up to now.\n\nIt is to be noted that the stationary Internal DFT \/ KS formalism \ngives a more fundamental justification than the Hartree-Fock (HF) framework\nto the stationary nuclear mean-field like calculations. Indeed, HF does not contain quantum correlations,\nnor treats correctly the redundant coordinate problem, which introduces a spurious coupling between the\ninternal properties and the c.m. motion \\cite{RS80,Schm01a}.\nA way to overcome this problem in the stationary case is to perform projected HF\n(projection before variation on c.m.\\ momentum), which permits to restore \nGalilean invariance, but at the price of abandoning the independent-particle \ndescription \\cite{Schm01a,Pei62,Ben03}.\nWithin the Internal DFT \/ KS formalism,\nwe proved that the c.m.\\ correlations can be included in the energy functional \/ the KS potential \\cite{Mes09},\nso that there would be no need for a c.m.\\ projection if the ultimate functional was known.\n\nIt is a question of interest to generalize the stationary Internal DFT \/ KS formalism to the time-dependent case.\nIt would provide a first step towards a fundamental justification to the use of density functionals in nuclear\ntime-dependent calculations with an effective mean-field \\cite{tdhf_nucl,Neg82},\nand would prove that the c.m.\\ correlations can be included in the functional.\nThis last point is even more interesting that the spurious c.m.\\ motion problem remains in time-dependent HF \\cite{Irv80,Uma09},\nbut that then the projected HF method becomes unmanageable and is not used in practise \\cite{Uma09}.\n\nIn this paper, we propose to set up the time-dependent Internal DFT \/ KS formalism.\nThe paper is organized as follows:\nwe first apply the Jacobi coordinates method to the\ntime-dependent full many-body Hamiltonian to decouple the internal properties from the c.m. ones, and define some useful ``internal'' observables, including the internal density (section II);\nthen we show that the internal many-body wave function (and thus the ``internal'' mean values of all the observables) can be written as a functional of the internal density (section III);\nfinally, we develop the associated time-dependent Internal KS scheme as a practical scheme to compute the internal density (section IV).\n\n\n\n\n\n\n\\section{Time-dependent N body formulation.}\n\n\n\n\\subsection{General formulation.}\n\nIn the time-dependent domain, the introduction of an \\textit{explicitely} time-dependent internal potential of the form\n\\begin{eqnarray}\n\\label{eq:v}\n\\sum_{i=1}^N v_{int}(\\mathbf{r}_i - \\mathbf{R};t)\n\\end{eqnarray}\ntakes a true meaning.\nThis is because self-bound systems are plagued by a c.m. problem.\nFor instance, in the stationary case, the c.m. will be delocalized in the whole space for \\textit{isolated} self-bound systems \\cite{Eng07,Kre01,Mes09}.\nThis does not occur in experiments because experimentally observed self-bound systems are not \\textit{isolated} anymore\n(they interact with the piece of matter they are inserted in which localizes the c.m.).\nIn the time domain, the c.m. motion remains uncomparable to the experimental one (this will be discussed in more detail later),\nso that it would not make sense to introduce a time-dependent potential\nwhich would act on the c.m. motion.\nIt are the internal properties which are of true experimental interest (experimentalists always deduce those properties \\cite{foot4}).\nThis justifies\nthe introduction of an \\textit{explicitely} time-dependent potential \nof the form (\\ref{eq:v}),\nwhich would act on the internal properties only,\nand models the internal effect (only) of time-dependent potentials used in experiments.\nSuch a potential does not appear any more simply as a mathematical auxiliary (as for the stationary Internal DFT \/ KS) and should not necessarily be dropped at the end.\n\n\nWe thus start from a general translationally invariant $N$-body Hamiltonian\ncomposed of the usual kinetic energy term, a \ntranslationally invariant \ntwo-body potential $u$, which describes the particle-particle \ninteraction,\nand an arbitrary translationally invariant \"internal\" potential $v_{int}$ which contains an explicit time dependence\n\\begin{equation}\n\\label{eq:H}\nH\n= \\sum_{i=1}^{N} \\frac{\\vec{p}^2_i}{2m} \n + \\sum_{\\stackrel{i,j=1}{i > j}}^{N} u (\\vec{r}_i-\\vec{r}_j) \n + \\sum_{i=1}^{N} v_{\\text{int}} (\\vec{r}_i - \\vec{R} ; t)\n\\; .\n\\end{equation}\nFor the sake of simplicity we assume a 2-body interaction $u$ and $N$ identical Fermions or Bosons.\nThe generalization to 3-body etc interactions is straightforward;\nthe generalization to different types of particles is underway.\n\nWe rewrite the Hamiltonian (\\ref{eq:H}) using the ($N-1$) Jacobi coordinates $\\{\\xi_\\alpha;\\alpha=1,\\dots,N-1\\}$\nand the c.m.\\ coordinate $\\vec{R}$, defined as\n\\begin{eqnarray}\n&& \\mathbf{\\xi}_{1} = \\mathbf{r}_2-\\vec{r}_1, \n\\mathbf{\\xi}_2=\\mathbf{r}_3-\\frac{\\vec{r}_2+\\vec{r}_1}{2}, \\ldots,\n\\nonumber\\\\\n&& \\mathbf{\\xi}_{N-1} = \\frac{N}{N-1} \\, (\\vec{r}_N - \\vec{R}),\n\\nonumber\\\\\n&& \\vec{R}=\\frac{1}{N}\\sum_{j=1}^{N}\\vec{r}_j\n.\n\\label{eq:jacobi}\n\\end{eqnarray}\nThe $\\xi_\\alpha$ are relative to the c.m.\\ of the other \n$1, \\ldots, \\alpha-1$ particles and are independent from $\\vec{R}$. They are to be distinguished from the $N$ \n\"laboratory coordinates\" $\\vec{r}_i$, and the $N$ \"c.m. frame coordinates\" $(\\vec{r}_i-\\vec{R})$ relative to the total c.m. $\\vec{R}$.\nAs the $\\{\\vec{r}_i-\\vec{r}_{j\\ne i}\\}$ and the $\\{\\vec{r}_i - \\vec{R}\\}$ can be rewritten as functions of the $\\xi_\\alpha$\n(in Appendix \\ref{app:jacobi} is given the expression of the $\\{\\mathbf{r}_i-\\mathbf{R}\\}$ as a function of the $\\{\\xi_\\alpha\\}$ coordinates),\nthe interaction $u$ and the internal potential $v_{int}$\ncan be rewritten as functions of the $\\xi_\\alpha$. We denote $U$ and $V$\nthe interaction potential and the internal potential in the Jacobi coordinates representation:\n\\begin{eqnarray}\n\\sum_{\\stackrel{i,j=1}{i > j}}^{N} u (\\vec{r}_i-\\vec{r}_j)\n\\quad&\\rightarrow&\\quad\nU(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1}) \n\\nonumber\\\\\n\\sum_{i=1}^{N} v_{int} (\\vec{r}_i - \\vec{R} ; t)\n\\quad&\\rightarrow&\\quad\nV(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\n\\label{eq:V_int}\n.\n\\end{eqnarray}\nOf course we have $U[u]$ and $V[v_{int}]$.\nThe $V[v_{int}]$ potential is ($N-1$) body in the Jacobi coordinates representation and\n\\textit{cannot} be written in a simple form in this representation (see Appendix \\ref{app:jacobi}).\nMoreover, various $v_{int}$ can lead to the same $V$, which we will develop later.\n\nAfter having defined the conjugate momenta of $\\vec{R}$ and $\\xi_\\alpha$,\nwe can separate (\\ref{eq:H}) into $H = H_\\text{CM} + H_\\text{int}$, where ($M = Nm$ is the total mass)\n\\begin{equation}\nH_\\text{CM} = -\\frac{\\hbar^2 \\Delta_\\vec{R}}{2M} \n\\label{eq:H_cm}\n\\end{equation}\nis a one-body operator acting in $\\vec{R}$ space only,\nand ($\\tau_\\alpha$ is the conjugate momentum of $\\xi_\\alpha$ and $\\mu_\\alpha = m\\frac{\\alpha}{\\alpha+1}$ the corresponding reduced mass)\n\\begin{eqnarray}\nH_\\text{int}=\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha} &+& U[u](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1}) \n\\nonumber\\\\\n&+& V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\n\\label{eq:H_int}\n\\end{eqnarray}\nis a $(N-1)$ body operator in the $\\{\\xi_\\alpha\\}$ space. It contains the \ninteraction and the internal potential.\n\nIn the time-dependent case, we can choose freely the initial state $\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t_0)$.\nWe start from an initial state which can be written \n\\begin{equation}\n\\label{eq:psi_init}\n\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t_0) \n= \\Gamma(\\vec{R} ; t_0) \\; \n \\psi_{int} ({\\boldmath{\\xi}}_1, \\ldots , {\\boldmath{\\xi}}_{N-1} ; t_0)\n\\end{equation}\nin the Jacobi coordinates representation.\nThis form does not mix the c.m. motion with the internal one\n(mixing them would not make sense because the c.m. motion does anyway not correspond to the experimental one)\nand corresponds to the form of the stationary state \\cite{Schm01a,Mes09}.\nAs $H_{CM}$ and $H_{int}$ act in two separate subspaces, the $\\mathbf{R}$ and $\\{\\xi_\\alpha\\}$ spaces\n(which implies $[H_{CM},H_{int}]=0$), it is easy to show that the state $|\\psi(t))$ can be built at all time $t\\geq t_0$ as a direct product of the form\n\\begin{equation}\n\\label{eq:psi}\n\\psi(\\vec{r}_1, \\ldots , \\vec{r}_N ; t) \n= \\Gamma(\\vec{R} ; t) \\; \n \\psi_{int} ({\\boldmath{\\xi}}_1, \\ldots , {\\boldmath{\\xi}}_{N-1} ; t) ,\n\\end{equation}\nwith\n\\begin{eqnarray}\n&& H_{CM}|\\Gamma(t)) = i\\hbar\\partial_t |\\Gamma(t))\n\\label{eq:schro_int}\\\\\n&& H_{int}|\\psi_{int}(t)) = i\\hbar\\partial_t |\\psi_{int}(t))\n\\label{eq:schro}\n\\; .\n\\end{eqnarray}\nHence, the $N$-body wave function $\\psi$ can be separated into a one-body wave function \n$\\Gamma$ that depends on the position $\\mathbf{R}$ of the c.m. only, \nand an \"internal\" ($N-1$) body wave function \n$\\psi_{int}$ that depends on the remaining ($N-1$) Jacobi \ncoordinates ${\\boldmath{\\xi}}_\\alpha$.\nOf course, $\\psi_{int}$ could also be written as a function of the $N$ laboratory coordinates $\\mathbf{r}_i$,\nbut one of them would be redundant.\n$\\Gamma$ is solution of the free Schr\\\"odinger equation and describes the motion of the \\textit{isolated} system as \na whole in any chosen inertial frame of reference (as the laboratory).\nIf one starts from a normalizable initial state $|\\Gamma(t_0))$, $|\\Gamma(t))$\nis condemned to spread more and more.\nIn the stationary limit, the only solutions of Eq. (\\ref{eq:schro_int}) are plane waves, which are infinitely spread (thus not normalizable).\nThis does not correspond to experimental situations, where the system is not isolated anymore: interactions with other systems\nof the experimental apparatus localize the c.m.\nBut the formal decoupling between the c.m.\\ motion and the internal properties obtained when using the Jacobi coordinates method\npermits to let the c.m.\\ motion to the choice of experimental conditions,\nthe internal properties being comparable to the experimental ones.\n\n\n\n\n\n\n\\subsection{Some useful definitions.}\n\\label{par:def}\n\nWe define some quantities and relations that will be useful for the next considerations.\nIn \\cite{Mes09,Gir08b,Kaz86} is defined the internal one-body density\n\\begin{eqnarray}\n\\lefteqn{\\rho_{int}(\\vec{r},t)\/N}\n\\label{eq:rho_int0}\n\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\;\n \\delta(\\mathbf{R})\n |\\psi_{int}(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2\n \\delta \\big( \\vec{r} - (\\vec{r}_i-\\mathbf{R}) \\big)\\, \n\\nonumber\\\\\n& = & \\Big(\\frac{N}{N-1}\\Big)^3 \n \\int \\! d\\vec{\\xi}_1 \\cdots d\\mathbf{\\xi}_{N-2} \\; \n \\big| \\psi_{int} \\big(\\mathbf{\\xi}_1, \\ldots, \\mathbf{\\xi}_{N-2},\n \\tfrac{N\\vec{r}}{N-1} ;t \\big) \\big|^2\n\\nonumber .\n\\end{eqnarray}\nIt is is normalized to $N$.\nThe laboratory density $\\rho(\\mathbf{r},t)$ is obtained by convolution of $\\rho_{int}$ with the c.m.\\ wave \nfunction (following \\cite{Gir08b,Kaz86}):\n$\n\\rho(\\mathbf{r},t) = \\int d\\mathbf{R} |\\Gamma(\\mathbf{R},t)|^2 \\rho_{int}(\\mathbf{r} - \\mathbf{R},t) .\n$\n\nWe also introduced in \\cite{Mes09} the local part of the two-body internal density matrix\n\\begin{eqnarray}\n\\label{eq:gamint0}\n\\lefteqn{\\gamma_{int}(\\vec{r},\\vec{r'};t)}\n \\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \n \\delta(\\mathbf{R}) |\\psi_{int}(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2 \\,\n \\nonumber \\\\\n& & \\hspace{1.cm} \\times \n \\delta \\big( \\vec{r} - (\\vec{r}_i-\\mathbf{R}) \\big)\n \\delta \\big( \\vec{r'} - (\\vec{r}_{j\\ne i}-\\mathbf{R}) \\big)\n \\nonumber \\\\\n& = & \\frac{N(N-1)}{2} \\Big(\\frac{N-1}{N-2}\\Big)^3 \\Big(\\frac{N}{N-1}\\Big)^3\n \\int \\! d\\mathbf{\\xi}_1 \\cdots d\\mathbf{\\xi}_{N-3}\n \\nonumber\\\\\n& & \\hspace{1.cm} \\times \\Big| \\psi_{int} \\Big(\\mathbf{\\xi}_1, \\ldots, \\mathbf{\\xi}_{N-3}, \n \\tfrac{\\vec{r'}+(N-1)\\vec{r} }{N-2},\\tfrac{N\\vec{r'}}{N-1} ;t \\Big) \\Big|^2\n \\nonumber\n.\n\\end{eqnarray}\nIt has the required normalisation to $N(N-1)\/2$.\nFollowing similar steps than in \\cite{Gir08b,Kaz86},\nwe can show that the local part of the two-body laboratory density matrix \n$\\gamma(\\vec{r},\\vec{r'},t)$ is obtained by convolution of $\\gamma_{int}$ with the c.m.\\ wave \nfunction:\n$\n\\gamma(\\vec{r},\\vec{r'};t) = \\int d\\mathbf{R} |\\Gamma(\\mathbf{R},t)|^2 \\gamma_{int}(\\vec{r} - \\mathbf{R},\\vec{r'} - \\mathbf{R};t) .\n$\n\nThe definitions of $\\rho_{int}(\\vec{r},t)$ and $\\gamma_{int}(\\vec{r},\\vec{r'};t)$ show clearly that they are defined in the c.m. frame,\ni.e.\\ that the $\\vec{r}$, $\\vec{r'}$ coordinates are measured in the c.m. frame (see the $\\delta$ relations in (\\ref{eq:rho_int0}) and (\\ref{eq:gamint0})).\nCompared to the traditional definitions, a $\\delta(\\mathbf{R})$ appears in the definition of the internal densities calculated\nwith $\\psi_{int}$ in $\\{\\mathbf{r}_i\\}$ coordinates. As one of them is redundant, the $\\delta(\\mathbf{R})$ represents the dependence of the redundant coordinate on the others \\cite{foot3}.\n\nAnother quantity that will be very useful is the one-body internal probability current, defined in Appendix \\ref{app:int_current} ($c.c.$ denotes the complex conjugate)\n\\begin{eqnarray}\n\\lefteqn{\\mathbf{j}_{int}(\\mathbf{r},t)\/N}\n\\label{eq:j_int1}\\\\\n&=& \\frac{\\hbar}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2} \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2}, \\nu;t)\n\\nonumber\\\\\n&& \\hspace{1.5cm}\n\\times \\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2}, \\nu;t) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}} + c.c.\n\\nonumber\n\\end{eqnarray}\nwhich satisfies the ``internal'' continuity equation\n\\begin{eqnarray}\n\\partial_t \\rho_{int}(\\mathbf{r},t) + \\mathbf{\\nabla} . \\mathbf{j}_{int}(\\mathbf{r},t) =0 .\n\\label{eq:cont_rel}\n\\end{eqnarray}\nUsing (\\ref{eq:j_int1}), (\\ref{eq:H_int}) and (\\ref{eq:schro}), we obtain the relation\n\\begin{widetext}\n\\begin{eqnarray}\n\\lefteqn{i \\frac{\\partial}{\\partial t} \\mathbf{j}_{int}(\\mathbf{r},t)}\n\\label{eq:partial_j_int}\n\\\\\n&=& \\frac{N}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2} \\Big\\{ \n\\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) i\\hbar\\partial_t \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) \\nonumber\\\\\n&& \\hspace{4.6cm} +\n\\mathbf{\\nabla_{\\nu}} \\Big( i\\hbar \\partial_t\\psi_{int}(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) \\Big) \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-2},\\nu;t) + c.c.\n\\Big\\}\\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n\\nonumber\\\\\n&=& \\frac{N}{2m i} \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2} \\Big\\{ \n\\mathbf{\\nabla_{\\nu}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t) \\frac{\\hbar^2\\Delta_{\\nu}}{2\\mu_{N-1}} \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\nu;t)\n- \\psi_{int}^*(\\mathbf{\\xi}_1, ..., \\nu;t) \\mathbf{\\nabla_{\\nu}} \\frac{\\hbar^2\\Delta_{\\nu}}{2\\mu_{N-1}} \\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t)\n\\nonumber\\\\\n&& \\hspace{4.5cm} +\n\\psi^*_{int}(\\mathbf{\\xi}_1, ..., \\nu;t) \\mathbf{\\nabla_{\\nu}} \\Big( U[u](\\mathbf{\\xi}_1, ..., \\nu) + V[v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t) \\Big) \\psi_{int}(\\mathbf{\\xi}_1, ... ,\\nu;t) \n+c.c.\n\\Big\\} \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}},\n\\nonumber\n\\end{eqnarray}\n\\end{widetext}\nwhich will be a key equation for the next considerations.\n\n\n\n\n\n\n\\section{Time-dependent Internal DFT theorem.}\n\n\n\n\\subsection{Preliminaries.}\n\\label{sub:int}\n\nTo prove the time-dependent Internal DFT theorem, we adapt the considerations of \\cite{Run84,Gro94}\nto the internal Schr\\\"odinger equation (\\ref{eq:schro}).\nThe main differences lie in the definition of the corresponding internal density (\\ref{eq:rho_int0}) and probability current (\\ref{eq:j_int1}),\nand in the fact that the potential $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$\ncannot be written as the sum of one-body potentials in the Jacobi coordinates representation\n(which introduces some subtleties due to the c.m. correlations and will bring us to use the integral mean value theorem to reach our goal).\n\nIn what follows, we consider a given type of Fermions or Bosons, i.e. a given particle-particle interaction $u$.\nSolving the ``internal'' Schr\\\"odinger equation (\\ref{eq:schro}) for a fixed initial state $|\\psi_{int}(t_0))$ and for various internal potentials $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ defines two maps \\cite{Run84,Gro94}\n\\begin{eqnarray}\n&& F: V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\rightarrow |\\psi_{int}(t))\n\\nonumber\\\\\n&& G: V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\rightarrow \\rho_{int}(\\mathbf{r},t).\n\\label{eq:map}\n\\end{eqnarray}\nWe first notice that two potentials $v_{int}$ and $v'_{int}$ which lead to two potentials\n$V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ and $V[v'_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ that differ by a scalar function of time only $c(t)$, \nwill give two wave functions that differ by a phase $e^{-i\\alpha(t)\/\\hbar}$ only \\cite{Run84,Gro94}:\n\\begin{eqnarray}\n&&V[v'_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) - V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) = c(t)\n\\nonumber\\\\\n&&\\quad\\quad\\Rightarrow\\hspace{1mm}\n|\\psi'_{int}(t))=e^{-i\\alpha(t)\/\\hbar}|\\psi_{int}(t)) ,\n\\nonumber\\\\\n&& \\hspace{2.5cm} \\rm{with} \\hspace{2mm} \\Dot{\\alpha}(t)=c(t)\n\\label{eq:v'}\n.\n\\end{eqnarray}\nThen, $|\\psi_{int}(t))$ and $|\\psi'_{int}(t))$ will give the same density $\\rho_{int}(\\mathbf{r},t)=\\rho'_{int}(\\mathbf{r},t)$.\nThe consequence is that the map $G$ is not fully invertible.\n\nLet us discuss a bit about the condition (\\ref{eq:v'}).\nThe form (\\ref{eq:V_int}) for $V[v_{int}]$ implies $V[v'_{int}]-V[v_{int}]=V[v'_{int}-v_{int}]$.\nWe define\n\\begin{eqnarray}\n\\Delta v_{int}(\\mathbf{r};t)=v'_{int}(\\mathbf{r};t)-v_{int}(\\mathbf{r};t).\n\\label{eq:delta_v}\n\\end{eqnarray}\nIt is to be noted that the condition $\\Delta v_{int}(\\mathbf{r};t)\\ne c(t)\/N$\nis necessary but not sufficient to ensure the condition (\\ref{eq:v'}),\nwhich can be rewritten $V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$.\nIndeed, it is possible to have $\\Delta v_{int}(\\mathbf{r};t)\\ne c(t)\/N$ and nevertheless\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) = c(t)$,\nbecause compensations due to the c.m. correlations can happen.\n\nLet us reason on the two particules case, where only one Jacobi coordinate is sufficient to describe the internal properties.\nWe have (see Appendix \\ref{app:jacobi}):\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1;t)=\\Delta v_{int} (-\\frac{1}{2}\\xi_1;t)+\\Delta v_{int} (\\frac{1}{2}\\xi_1;t) \\big( =\\sum_{i=1}^{2} v_{\\text{int}} (\\vec{r}_i - \\vec{R} ; t) \\big)$.\nWe see that if $\\Delta v_{int}(\\mathbf{r};t)$ is an odd function of $\\mathbf{r}$ at all $t$\n(up to an additional time-dependent function),\nwe have $V[\\Delta v_{int}](\\mathbf{\\xi}_1;t)=c(t)$\n$\\Rightarrow$ $\\rho_{int}=\\rho'_{int}$.\nThis is due to the c.m. correlations, that the non-trivial form of $V$ reflects.\nIf\n$\\Delta v_{int}$ tends to move the first particule in one direction,\nthe second particule will tend to move in the opposite direction because of the c.m. correlations.\nBut if this potential counter-acts perfectly the motion of the second particule (as does an odd potential in the c.m. frame), then the particules remain stuck and the density unchanged.\n\nThe same can occur for an arbitrary number of particules.\nFor instance, as $\\sum_{i=1}^N (\\mathbf{r}_i-\\mathbf{R})=0$,\nit is obvious with (\\ref{eq:V_int}) and (\\ref{eq:delta_v}) that every $\\Delta v_{int}(\\mathbf{r};t)=\\mathbf{b}(t).\\mathbf{r}+c(t)\/N$\nwill yield $V[\\Delta v_{int}] = c(t)$ (even if this form for $\\Delta v_{int}$ leads to internal potentials which are not null at infinity).\nAgain, this is because if a potential counter acts perfectly the motion due to the c.m. correlations, the particules remain stuck and the density unchanged.\nIn what follows, we consider only internal potentials $v_{int}$ and $v'_{int}$ that lead to $V[\\Delta v_{int}]\\ne c(t)$.\n\nWe come back to Eq. (\\ref{eq:v'}) and denote $|\\psi_{int}(t))=e^{-i\\alpha(t)\/\\hbar}|\\psi^0_{int}(t))$ where we define $\\psi^0_{int}$ as the wave function obtained for the choice $c(t)=0$,\ni.e. associated to a $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)$ where no additive time-dependent function can be split.\nIf we prove that the map $G$ is invertible up to an additive time-dependent function $c(t)$, then $\\psi^0_{int}$ is fixed by $\\rho_{int}$\nthrough the relation $|\\psi^0_{int}(t))=F G^{-1} \\rho_{int}(\\mathbf{r},t)$,\nwhich implies that $|\\psi^0_{int}(t))$ can be written as a functional of the internal density $\\rho_{int}$ defined in (\\ref{eq:rho_int0}).\nConsequently, any expectation value of an operator $\\hat{O}$ which does not contain a time derivative can be written as a functional of the internal density (as the phase cancels out):\n$(\\psi_{int}(t)|\\hat{O}|\\psi_{int}(t))=(\\psi^0_{int}[\\rho_{int}](t)|\\hat{O}|\\psi^0_{int}[\\rho_{int}](t))$.\n\nWe thus have to show that \na propagation of (\\ref{eq:schro}) with two potentials $v_{int}$ and $v_{int}'$ that yield\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$\nwill produce two different internal densities $\\rho_{int}$ and $\\rho_{int}'$.\n\n\n\n\n\\subsection{The proof.}\n\nWe start from a \\textit{fixed initial state} $|\\psi_{int}(t_0))$ and\npropagate it with two with two potentials $v_{int}$ and $v_{int}'$ that give\n$V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$.\nWe deduce from Eq. (\\ref{eq:partial_j_int})\n\\begin{widetext}\n\\begin{eqnarray}\ni \\frac{\\partial}{\\partial t} \\Big( \\mathbf{j}_{int}(\\mathbf{r},t) - \\mathbf{j}'_{int}(\\mathbf{r},t) \\Big) \\Big|_{t=t_0} =\n\\frac{N}{m i} \\Big(\\frac{N}{N-1}\\Big)^3\n\\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2} |\\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t_0)|^2\n\\mathbf{\\nabla_{\\nu}} V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}} .\n\\end{eqnarray}\nUsing the ``internal'' continuity relation (\\ref{eq:cont_rel}) we obtain\n\\begin{eqnarray}\n\\lefteqn{\\frac{\\partial^2}{\\partial t^2} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho'_{int}(\\mathbf{r},t) \\Big) \\Big|_{t=t_0} =}\n\\label{eq:partial_j_int0}\\\\\n&& \\frac{N}{m} \\Big(\\frac{N}{N-1}\\Big)^3 \\mathbf{\\nabla_{\\mathbf{r}}} .\\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2}\n|\\psi_{int}(\\mathbf{\\xi}_1, ..., \\frac{N}{N-1}\\mathbf{r};t_0)|^2\n\\mathbf{\\nabla}_\\nu V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n.\n\\nonumber\n\\end{eqnarray}\n\nWe now make the only hypothesis which is used in this derivation. Following \\cite{Run84,Gro94} we restrict the set of\npotentials $v_{int}$ to those that can be expanded into Taylor series with respect to the time at the initial time $t_0$\n(which is a reasonable hypothesis for physical potentials).\nAs we supposed that $V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)$, we have ($k$ is a positive integer)\n\\begin{eqnarray}\nV[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\ne c(t)\n\\quad\\Rightarrow\\quad\n\\exists k :\nw_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0) \\ne constant ,\n\\label{eq:Vint}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nw_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0)\n= \\frac{\\partial^k}{\\partial t^k} V[\\Delta v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t) \\Big|_{t=t_0}\n\\label{eq:wk}\n.\n\\end{eqnarray}\nIt is to be noted that the condition $\\frac{\\partial^k}{\\partial t^k} \\Delta v_{int}(\\mathbf{r};t)\\Big|_{t=t_0} \\ne constant$\n$\\Rightarrow$ $\\mathbf{\\nabla_r} \\frac{\\partial^k}{\\partial t^k} \\Delta v_{int}(\\mathbf{r};t)\\Big|_{t=t_0}\\ne \\overrightarrow{0}$\nis necessary to ensure the condition (\\ref{eq:Vint}) (see (\\ref{eq:V_int}) and (\\ref{eq:wk})), but not sufficient.\n\nIn what follows, we consider $k$ as the \\textit{smallest} positive integer such that (\\ref{eq:Vint})\nis verified. Then, if we apply $k$ time derivatives to the Eq. (\\ref{eq:partial_j_int0}), we straightforwardly obtain\n\\begin{eqnarray}\n\\frac{\\partial^{k+2}}{\\partial t^{k+2}} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho_{int}'(\\mathbf{r},t) \\Big) \\Big|_{t=t_0} =\n\\frac{N}{m} \\Big(\\frac{N}{N-1}\\Big)^3 \\mathbf{\\nabla_{\\mathbf{r}}} . \\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2} |\\psi_{int}(\\mathbf{\\xi}_1, ..., \\nu;t_0)|^2 \\mathbf{\\nabla_\\nu} w_k(\\mathbf{\\xi}_1, ..., \\nu;t_0) \\Big) \\Big|_{\\nu=\\frac{N}{N-1}\\mathbf{r}}\n.\n\\end{eqnarray}\nAs, for every physical potential, $\\mathbf{\\nabla}_{\\mathbf{\\xi}_{N-1}} w_k(\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t_0)$ is a real and continuous function in the whole position space, and as $|\\psi_{int}(\\mathbf{\\xi}_1, ...,\\mathbf{\\xi}_{N-1};t_0)|^2$ is a real and positive function in the whole position space, we can apply the integral mean value theorem generalized to many variables functions (demonstrated in Appendix \\ref{app:mean_val_th}) to the previous expression. We obtain\n\\begin{eqnarray}\n\\exists (\\beta_1,...,\\beta_{N-2}) \\hspace{1mm} :\n&& m \\frac{\\partial^{k+2}}{\\partial t^{k+2}} \\Big( \\rho_{int}(\\mathbf{r},t) - \\rho_{int}'(\\mathbf{r},t) \\Big) \\Big|_{t=t_0}\n\\nonumber\\\\\n&& \\hspace{5mm} = \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\nN \\Big(\\frac{N}{N-1}\\Big)^3 \\int d\\mathbf{\\xi}_1 ... d\\mathbf{\\xi}_{N-2}|\\psi_{int}(\\mathbf{\\xi}_1, ..., \\frac{N}{N-1}\\mathbf{r};t_0)|^2 \\Big]\n\\nonumber\\\\\n&& \\hspace{5mm} = \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\n\\rho_{int}(\\mathbf{r},t_0) \\Big]\n.\n\\label{eq:partial_j_int2}\n\\end{eqnarray}\nTo prove the one-to-one correspondence $V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)\\leftrightarrow \\rho_{int}(\\mathbf{r},t)$ it remains to show that (\\ref{eq:partial_j_int2})\ncannot vanish for $v_{int}$ and $v_{int}'$ that lead to the relation (\\ref{eq:Vint}).\nThen the internal densities $\\rho_{int}(\\mathbf{r},t)$ and $\\rho_{int}'(\\mathbf{r},t)$ would become different infinitesimally later than $t_0$.\nWe use the \\textit{reductio ad absurdum} method, in the spirit of Refs. \\cite{Run84,Gro94}. We suppose that (\\ref{eq:partial_j_int2}) vanishes, which implies:\n\\begin{eqnarray}\n0&=&\\frac{N-1}{N}\\int d\\mathbf{r} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0) \\mathbf{\\nabla_{\\mathbf{r}}} . \\Big[\n\\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0)\n\\rho_{int}(\\mathbf{r},t_0) \\Big]\n\\nonumber\\\\\n&=& \n-\\int d\\mathbf{r} \\Big[ \\mathbf{\\nabla}_\\frac{N\\mathbf{r}}{N-1} w_k(\\beta_1,...,\\beta_{N-2}, \\frac{N}{N-1}\\mathbf{r};t_0) \\Big]^2\n\\rho_{int}(\\mathbf{r},t_0)\n.\n\\label{eq:absurdum2}\n\\end{eqnarray}\nAs $w_k$ is a many-body function, the Eq. (\\ref{eq:Vint}) does not imply that\n$\\forall (\\beta_1,...,\\beta_{N-2}):\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)\\ne\\overrightarrow{0}$ in the general case.\nHowever, we check if this relation holds for the particular form (\\ref{eq:V_int}) we choose for $V$.\n\nInserting the results of Appendix \\ref{app:jacobi} in (\\ref{eq:V_int}) and (\\ref{eq:wk}), we obtain, if $N>2$ \n(the case $N=2$ will be discussed later on)\n\\begin{eqnarray}\n&& w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\n\\label{eq:w_k}\\\\\n&&\\quad\n\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(\\frac{N-1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0}\n+ \\sum_{i=1}^{N-2}\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(\\gamma_i-\\frac{1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0}\n+\\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\frac{1}{N}\\xi_{N-1};t\\big) \\Big|_{t=t_0} ,\n\\nonumber\n\\end{eqnarray}\nwhere we defined\n\\begin{eqnarray}\n\\gamma_{N-2}=\\frac{N-2}{N-1}\\beta_{N-2}\n\\quad\\quad \\text{and} \\quad\\quad\n\\forall i \\in [1,N-3]: \\gamma_{i}=\\frac{i}{i+1}\\beta_{i} - \\sum_{\\alpha=i+1}^{N-2} \\frac{1}{\\alpha+1}\\beta_{\\alpha}.\n\\label{eq:gamma}\n\\end{eqnarray}\nThe form of the third term of the right hand side of Eq. (\\ref{eq:w_k}) comes from the fact that $\\sum_{i=1}^N(\\mathbf{r}_i-\\mathbf{R})=0$, which implies, using the Appendix \\ref{app:jacobi}, that $-\\sum_{\\alpha=1}^{N-2} \\frac{1}{\\alpha+1}\\beta_{\\alpha}=-\\sum_{i=1}^{N-2}\\gamma_i$. \nWe see from Eq. (\\ref{eq:gamma}) that the set $(\\gamma_1,...,\\gamma_{N-2})$ is perfectly defined by the set $(\\beta_1,...,\\beta_{N-2})$ and vice versa.\nWe now can calculate\n\\begin{eqnarray}\n&&\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\n\\nonumber\\\\\n&&\\hspace{3cm}\n\\frac{N-1}{N} \\mathbf{D} \\big(\\frac{N-1}{N}\\xi_{N-1}\\big)-\\frac{1}{N}\\sum_{i=1}^{N-2} \\mathbf{D} \\big(\\gamma_i-\\frac{1}{N}\\xi_{N-1}\\big)\n-\\frac{1}{N} \\mathbf{D} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\frac{1}{N}\\xi_{N-1}\\big)\n\\label{eq:w}\n,\n\\end{eqnarray}\nwhere we introduced\n\\begin{eqnarray}\n\\mathbf{D}(\\mathbf{r})=\\mathbf{\\nabla}_\\mathbf{r} \\frac{\\partial^{k}}{\\partial t^{k}} \\Delta v_{int} (\\mathbf{r};t) \\big|_{t=t_0}\n.\n\\label{eq:D}\n\\end{eqnarray}\nfor simplicity.\nWe now check if\n$\\exists (\\beta_1,...,\\beta_{N-2}):$ $\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)=\\overrightarrow{0}$,\nwhich is equivalent, according to (\\ref{eq:gamma}) and (\\ref{eq:w}), to check if\n\\begin{eqnarray}\n\\exists (\\gamma_1,...,\\gamma_{N-2}):\\quad\n(N-1) \\mathbf{D} \\big((N-1)\\mathbf{r}\\big)=\\sum_{i=1}^{N-2} \\mathbf{D} \\big(\\gamma_i-\\mathbf{r}\\big)\n+\\mathbf{D} \\big(-\\sum_{i=1}^{N-2}\\gamma_i-\\mathbf{r}\\big)\n.\n\\end{eqnarray}\n\\end{widetext}\nSome mathematical considerations show that this equation cannot be fulfilled for all $\\mathbf{r}$ when $N>2$, whatever the set of $(\\gamma_1,...,\\gamma_{N-2})$,\ninstead if $\\mathbf{D} (\\mathbf{r})=\\overrightarrow{const}$.\nBut if $\\mathbf{D} (\\mathbf{r})=\\overrightarrow{const.}$, then $\\Delta v_{int} (\\mathbf{r};t)$ should be \nequal to $\\mathbf{b}(t).\\mathbf{r}+c(t)\/N$, according to (\\ref{eq:D}),\nwhich is forbidden by the condition (\\ref{eq:Vint}), cf. discussion of the \\S \\ref{sub:int}.\n\nIt remains to discuss the case $N=2$. It is easy to show that then, we have\n$\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\xi_{N-1};t_0)=\n-\\frac{1}{2}\\mathbf{D}\\big(-\\frac{1}{2}\\xi_{N-1}\\big)+\\frac{1}{2}\\mathbf{D}\\big(\\frac{1}{2}\\xi_{N-1}\\big)$,\nwhich is null if $\\mathbf{D}(\\mathbf{r})$\nis any par function of $\\mathbf{r}$.\nBut if $\\mathbf{D} (\\mathbf{r})$ is par, then $\\frac{\\partial^{k}}{\\partial t^{k}}\\Delta v_{int} (\\mathbf{r};t)$ should be an odd function of $\\mathbf{r}$\n(up to an additional time-dependnt function), according to (\\ref{eq:D}),\nwhich is also forbidden by the condition (\\ref{eq:Vint}), cf. discussion of the \\S \\ref{sub:int}.\n\nThus, we can conclude that, in our case\n\\begin{eqnarray}\n\\forall (\\beta_1,...,\\beta_{N-2}):\n\\mathbf{\\nabla}_{\\xi_{N-1}} w_k(\\beta_1,...,\\beta_{N-2}, \\xi_{N-1};t_0)\\ne\\overrightarrow{0}\n.\n\\nonumber\n\\label{eq:absurdum1}\n\\end{eqnarray}\nWe immediately deduce the incompatibility of\nthis relation,\nwhich is a consequence of (\\ref{eq:Vint}) and of the particular form (\\ref{eq:V_int}) of $V$, with (\\ref{eq:absurdum2}). Thus, the hypothesis we made is absurd: Eq. (\\ref{eq:partial_j_int2}) cannot vanish if $V[\\Delta v_{int}]\\ne c(t)$, so that the internal densities $\\rho_{int}(\\mathbf{r},t)$ and $\\rho_{int}'(\\mathbf{r},t)$ become different infinitesimally later than $t_0$. As a consequence, the map $G$, defined in (\\ref{eq:map}), is invertible (up to an additive time-dependent function) and $|\\psi^0_{int}(t))$ can be written as a functional of the internal density (we use the notation (\\ref{eq:v'})).\nThus, any expectation value of an operator $\\hat{O}$ which does not contain a time derivative can be written as a functional of $\\rho_{int}$ as the phase cancels out.\nThis achieves to prove the time-dependent Internal DFT theorem\n(which is a variant of the Runge-Gross theorem \\cite{Run84,Gro94} for self-bound systems and internal densities).\n\nMind that all the previous reasonings hold only for a fixed initial state $\\psi_{int}(t_0)$ (and a given type of particle), so that $\\psi^0_{int}$ is not only a functional of $\\rho_{int}$, but also depends on $\\psi_{int}(t_0)$. This will be discussed further.\n\n\n\n\n\\subsection{Link with traditional (time-dependent) DFT.}\n\nWe stress here the link and differences between the traditional DFT and internal DFT potentials.\nWe recall that the form of the potential $v_{ext}$ of traditional DFT can be fundamentally justified starting from the\nlaboratory Hamiltonian of\nan isolated molecule where the nuclei are treated explicitely.\nAs a molecule is a self-bound system, one can apply the Jacobi coordinates method.\nWe denote the N electronic coordinates related to \nthe laboratory frame\nas $\\mathbf{r}_i$, \nthe nuclear c.m. coordinate as $\\mathbf{R}^{nucl}$ and\nthe N electronic coordinates related to the c.m. of the nuclei as $\\mathbf{r}'_i=\\mathbf{r}_i - \\mathbf{R}^{nucl}$.\nA key point concerning the molecules is that, as the nuclei are much heavier than the electrons,\nthe c.m. of the whole molecule coincides with $\\mathbf{R}^{nucl}$,\nand it is an excellent approximation to apply the Jacobi coordinates to the nuclear coordinates only.\nAs a result, the c.m. motion will be described by a $\\Gamma(\\mathbf{R}^{nucl})$ wave function.\nThe redundant coordinate problem (thus the c.m.\\ correlations) will concern the nuclei only, and will be ``external'' to the electronic problem:\nthe N electrons are still described by N coordinates.\nThen, if one decouples the electronic motion from the nuclear one doing the clamped nuclei approximation,\nthe interaction of the electrons with the nuclear background is described by a potential of the form $\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i - \\mathbf{R}^{nucl})$,\nwhich becomes $\\sum_i v_{ext}(\\mathbf{r}'_i)$ when moving to the c.m. frame.\nWe then recover the form of the traditional DFT potential.\nThe potential $v_{ext}$, which is \\textit{internal} for the (self-bound) molecular problem, becomes \\textit{external} for the pure electronic problem.\nThose considerations also hold in the time domain, the difference being that the potential\n\\begin{eqnarray}\n\\sum_{i=1}^N v_{ext}(\\mathbf{r}_i - \\mathbf{R}^{nucl};t)\n\\label{eq:v_ext}\n\\end{eqnarray}\ncan then contain an explicit time dependence in addition to the part which\ndescribes the interaction of the electrons with the nuclear background.\nWe recover the traditional time-dependent DFT potential \\cite{Run84,Gro94,Gro90,Mar04} when moving in the c.m. frame.\n\n\nThose reasonings explicit the link between the traditional DFT potential expressed with the laboratory coordinates, Eq. (\\ref{eq:v_ext}),\nand the Internal DFT potential expressed with the laboratory coordinates, Eq. (\\ref{eq:v}).\nThey both act only on the internal properties, and not on the c.m. motion\n(because it is anyway not comparable to the experimental one).\nThe difference is that as, in the molecular case, some particules are much heavier than the other,\nit is a very good approximation to assimilate the c.m. of the whole molecule with $\\mathbf{R}^{nucl}$,\nwhich permits to neglect the c.m. correlations for the electronic system,\nand to justify the clamped nuclei approximation.\nThis simplifies greatly the electronic problem and the traditional DFT can be used to study it.\nWhen the particules constituting the self-bound system have nearly the same masses, as it is the case for the nuclei or the He droplets, \nthe total c.m. ($\\mathbf{R}$) should be calculated with \\textit{all} the particules,\nso that the c.m. correlations will concern all the particules, and no clamped approximation can be justified.\nThen, we should use the formalism proposed here.\n\n\n\n\n\n\n\\section{Time-dependent Internal Kohn-Sham scheme.}\n\nWe now provide a practical scheme to calculate the internal density $\\rho_{int}$, which consists in the generalization of the stationary Internal KS scheme of \\cite{Mes09} to the time-dependent case.\nFirst, we note that for any normalizable initial state $|\\psi_{int}(t_0))$, which are the only allowed, the ``internal'' Schr\\\"odinger equation (\\ref{eq:schro}) stems\nfrom a variational principle\non the ``internal'' quantum action \\cite{ker76,Run84,Vig08}\n\\begin{eqnarray}\nA_{int} = \\int_{t_0}^{t_1} dt (\\psi_{int}(t)|i\\hbar\\partial_t-H_{int}|\\psi_{int}(t)) .\n\\label{eq:action}\n\\end{eqnarray}\nAs the function $c(t)$ possibly contained in the potential $V_{int}$ is perfectly canceled by the time derivative of the corresponding phase $e^{-i\\alpha(t)\/\\hbar}$ of $\\psi_{int}$, see (\\ref{eq:v'}),\nwe have $A_{int}=\\int_{t_0}^{t_1} dt (\\psi_{int}^0[\\rho_{int}](t)|i\\hbar\\partial_t-H_{int}|\\psi_{int}^0[\\rho_{int}](t))$\nif $V_{int}$ is chosen so that no additive time-dependent function can be split.\nThus, the internal quantum action can be considered as a functional of $\\rho_{int}$.\nIts\n$\\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha} - U[u]|\\psi_{int}^0(t))$\npart is a universal functional of $\\rho_{int}$ in the sense that, for a given type of particle (a given interaction $u$), the same dependence on $\\rho_{int}$ holds for every $V[v_{int}]$, thus $v_{int}$ (see (\\ref{eq:V_int})).\n\nUsing the Eq. (\\ref{eq:H_int}), we develop the ``internal'' quantum action as\n\\begin{eqnarray}\nA_{int}[\\rho_{int}]&=& \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n\\label{eq:action2}\\\\\n&& - \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|U[u](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1})|\\psi_{int}^0(t))\n\\nonumber\\\\\n&& - \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|V[v_{int}](\\mathbf{\\xi}_1, ..., \\mathbf{\\xi}_{N-1};t)|\\psi_{int}^0(t))\n\\nonumber .\n\\end{eqnarray}\nTo rewrite its last two terms in a more convenient way, we establish a useful relation.\nFor any function $f(\\vec{r}_1,...,\\vec{r}_N;t)$ of the laboratory coordinates,\nexpressible with the Jacobi coordinates [we denote $F(\\xi_1,...,\\xi_{N-1};t)$], we have\n\\begin{eqnarray}\n\\label{eq:rel}\n\\lefteqn{(\\psi_{int}^0(t)| F(\\xi_1,...,\\xi_{N-1};t) |\\psi_{int}^0(t))}\n\\\\\n& = & \\int \\! d\\mathbf{\\xi}_1 \\cdots d\\mathbf{\\xi}_{N-1} F(\\xi_1,...,\\xi_{N-1};t) \n\\big| \\psi_{int}^0 (\\xi_1,...,\\xi_{N-1};t) \\big|^2\n\\nonumber\\\\\n& = & \\int \\! d\\mathbf{R} d\\mathbf{\\xi}_1 \\cdots d\\mathbf{\\xi}_{N-1} \\delta(\\mathbf{R}) F(\\xi_1,...,\\xi_{N-1};t)\n\\nonumber\\\\\n&& \\hspace{4.5cm} \\times \\big| \\psi_{int}^0 (\\xi_1,...,\\xi_{N-1};t) \\big|^2\n\\nonumber\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_{N} \\delta(\\mathbf{R}) f(\\vec{r}_1,...,\\vec{r}_N;t) \\big| \\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t) \\big|^2\n\\nonumber\n\\, .\n\\end{eqnarray}\nWe see that the \"internal mean values\" calculated with $\\psi_{int}$\nexpressed as a function of the ($N-1$) coordinates $\\xi_\\alpha$, can also be calculated with $\\psi_{int}$\nexpressed as a function of the $N$ coordinates $\\mathbf{r}_i$.\nAs one of them is redundant, a $\\delta(\\mathbf{R})$ which\nrepresents the dependence of the redundant coordinate on the others appears \\cite{foot3}.\n\nThe relation (\\ref{eq:rel}) leads to\n\\begin{eqnarray}\n\\label{eq:Eext}\n\\lefteqn{\n(\\psi_{int}^0(t)|V[v_{int}](\\xi_1,...,\\xi_{N-1};t)|\\psi_{int}^0(t)) \n} \\nonumber\\\\\n& = & \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \n \\delta(\\mathbf{R}) \\sum_{i=1}^N v_{int}(\\vec{r}_i - \\vec{R};t) |\\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)|^2 \\, \n \\nonumber\\\\\n& = & \\sum_{i=1}^N \n \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \n \\int \\! d\\vec{r}_1 \\cdots d\\vec{r}_N \\; \\delta(\\mathbf{R})\n \\nonumber\\\\\n& & \\hspace{2cm} \\times |\\psi_{int}^0(\\vec{r}_1, \\ldots, \\vec{r}_{N};t)|^2 \\delta \\big( \\vec{r}-(\\vec{r}_i-\\vec{R}) \\big) \n \\nonumber\\\\\n& = & \\sum_{i=1}^N \n \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \n \\, \\frac{\\rho_{int}(\\vec{r},t)}{N} \n \\nonumber\\\\\n& = & \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \\rho_{int}(\\vec{r},t)\n,\n\\end{eqnarray}\nwhere we used (\\ref{eq:rho_int0}) to obtain the penultimate equality.\nWe see that the potential $\\sum_{i=1}^N v_{int}(\\vec{r}_i - \\vec{R};t)$ that is $N$ body \nwith respect to the laboratory coordinates (and $(N-1)$ body when \nexpressed with Jacobi coordinates), becomes one body (and local) when\nexpressed with the c.m. frame coordinates\n(mind that $\\rho_{int}$ is defined in the c.m. frame, i.e. that $\\mathbf{r}$ is measured in the c.m. frame, cf. \\S \\ref{par:def}).\n\nApplying (\\ref{eq:rel}) to the second term of the action integral (\\ref{eq:action2}) gives\n$(\\psi_{int}^0(t)| U[u](\\xi_1,...,\\xi_{N-1}) |\\psi_{int}^0(t))=\\frac{1}{2} \\int \\! d\\vec{r} \\, d\\vec{r'} \\gamma_{int}(\\vec{r},\\vec{r'};t) u(\\vec{r}-\\vec{r'})$,\nwhere $\\gamma_{int}$ is defined in (\\ref{eq:gamint0}).\n\nThe action integral (\\ref{eq:action2}) can thus be rewritten\n\\begin{eqnarray}\nA_{int}[\\rho_{int}] &=& \\int_{t_0}^{t_1} dt (\\psi_{int}^0(t)|i\\hbar\\partial_t-\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n\\nonumber\\\\\n&& - \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\gamma_{int}(\\vec{r},\\vec{r'};t) u(\\vec{r}-\\vec{r'})\n\\nonumber\\\\\n&& - \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\rho_{int}(\\vec{r},t)\n.\n\\label{eq:action3}\n\\end{eqnarray}\n\nUp to now we did not do any hypothesis.\nTo recover the \\old{associated} Internal time-dependent KS scheme, we assume, as to obtain the\ntraditional time-dependent KS scheme \\cite{Run84,Gro94}, that there exists, \\textit{in the c.m.\\ frame}, a $N$-body non-interacting system\n(i.e. a local single-particle potential $v_S$)\n\\begin{eqnarray}\n\\label{eq:td_KS}\n\\Big( -\\frac{\\hbar^2\\Delta}{2m} + v_S(\\mathbf{r},t) \\Big)\\varphi^i_{int}(\\mathbf{r},t) = i\\hbar\\partial_t \\varphi^i_{int}(\\mathbf{r},t)\n\\end{eqnarray}\nwhich reproduces \\textit{exactly} the density $\\rho_{int}$ of the interacting system\n(mind that $\\rho_{int}$ is defined in the c.m. frame)\n\\begin{eqnarray}\n\\rho_{int}(\\mathbf{r},t) =\\sum_{i=1}^N |\\varphi^i_{int}(\\mathbf{r},t)|^2 .\n\\label{eq:rho_int}\n\\end{eqnarray}\nEven if only ($N-1$) coordinates are sufficient to describe the internal properties,\nthey still describe a system of $N$ particles. Thus, we have to introduce $N$ orbitals in the KS scheme (as we did)\nif we want them to be interpreted (to first order only) as single-particle orbitals\nand obtain a scheme comparable (but not equivalent) to mean-field like calculations with effective interactions.\n\nIn (\\ref{eq:td_KS}) we implicitely supposed that the particles are Fermions (a KS scheme to describe Boson condensates can be set similarly equalling all the $\\varphi^i_{int}$).\nUniqueness of the potential $v_S(\\mathbf{r},t)$ for a given density $\\rho_{int}(\\mathbf{r},t)$\n(and initial $|\\varphi^i_{int}(t_0))$ which yield the correct initial density $\\rho_{int}(\\mathbf{r},t_0)$)\nis ensured by a direct application of the traditional time-dependent DFT formalism \\cite{Run84,Gro94}.\nOf course, the question of the validity of the KS hypothesis, known as the \\textit{non-interacting v-representability} problem,\nremains, as in traditional (time-dependent) DFT \\cite{Dre90,Gro94}.\n\nTo use similar kinds of notations than the traditional DFT ones, we add and substract to the internal action integral (\\ref{eq:action3}) the internal Hartree term\n\\\\\n$\nA_{H}[\\rho_{int}] = \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\, \n\\rho_{int}(\\vec{r},t) \\, \\rho_{int}(\\vec{r'},t) \\, u(\\vec{r}-\\vec{r'})\n$,\nthe non-interacting kinetic energy term\\\\\n$\\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t))$\nand the\n$\\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t))$\nterm.\nThis permits to rewrite the ``internal'' action integral (\\ref{eq:action3}) as\n\\begin{eqnarray}\nA_{int}&=& \\int_{t_0}^{t_1} dt \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t-\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t)) - A_{H}[\\rho_{int}]\n\\nonumber\\\\\n&& - A_{XC}[\\rho_{int}] - \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\; v_{int}(\\vec{r};t) \\, \\rho_{int}(\\vec{r},t) \n\\label{eq:action4}\n\\end{eqnarray}\nwhere the internal exchange-correlation part is defined as\n\\begin{widetext}\n\\begin{eqnarray}\nA_{XC}[\\rho_{int}]\n&=& \\frac{1}{2} \\int_{t_0}^{t_1} dt \\int \\! d\\vec{r} \\, d\\vec{r'} \\, \n \\Big( \\gamma_{int}(\\vec{r},\\vec{r'};t) - \\rho_{int}(\\vec{r},t) \\, \\rho_{int}(\\vec{r'},t) \\Big) \\, \n u(\\vec{r}-\\vec{r'})\n\\nonumber\\\\\n&& + \\int_{t_0}^{t_1} dt \\Big( \n (\\psi_{int}^0(t)|\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t)) \n - \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|\\frac{\\vec{p}^2}{2m}|\\varphi^i_{int}(t)) \\Big)\n\\nonumber\\\\\n&& - \\int_{t_0}^{t_1} dt \\Big( (\\psi_{int}^0(t)|i\\hbar\\partial_t|\\psi_{int}^0(t))\n - \\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t)) \\Big)\n.\n\\label{eq:Axc}\n\\end{eqnarray}\n\\end{widetext}\nWe see that it contains the exchange-correlation which comes from the interaction $u$ (first line of (\\ref{eq:Axc})), but also the correlations contained in the interacting\nkinetic energy (second line of (\\ref{eq:Axc})) and in the interacting ``$i\\hbar\\partial_t$'' term (third line of (\\ref{eq:Axc})).\nA key point is that, as the KS assumption implies $\\varphi^i_{int}[\\rho_{int}]$ \\cite{Run84,Gro94,Dre90},\n$A_{XC}[\\rho_{int}](t)$ can be written as a functional of $\\rho_{int}$\n(for given $|\\psi_{int}^0(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$ which yield the same initial density $\\rho_{int}(\\mathbf{r},t_0)$).\n\nIt remains to vary the ``internal'' quantum action (\\ref{eq:action4}) to obtain the equations of motion (which define $\\rho_{int}$).\nVignale, see Ref. \\cite{Vig08}, showed recently that the correct formulation of the variational principle is not to stationarize the quantum action, i.e. $\\delta A_{int}[\\rho_{int}]=0$ as done so far \\cite{ker76,Run84,Gro94}, but\n\\begin{eqnarray}\n\\delta A_{int}[\\rho_{int}]=&&\ni\\big(\\psi_{int}[\\rho_{int}](t_1)\\big|\\delta \\psi_{int}[\\rho_{int}](t_1)\\big)\n\\nonumber\\\\\n&& - i \\big(\\psi^S_{int}[\\rho_{int}](t_1)\\big|\\delta \\psi^S_{int}[\\rho_{int}](t_1)\\big)\n\\label{eq:action}\n\\end{eqnarray}\n(where $\\psi^S_{int}$ is the Slater determinant constructed from the $\\varphi^i_{int}$).\nThe two formulations lead to identical final results for theorems derived form symmetries of the action functional because\ncompensations occur \\cite{Vig08}, but Vignales's formulation permits to solve the causality paradox of the previous formulation.\n\nVarying (\\ref{eq:action}) with respect to the $\\varphi^{i*}_{int}(\\mathbf{r},t)$, with $t\\in[t_0,t_1]$,\nleads straightforwardly to the Internal time-dependent KS equations for the $\\varphi^i_{int}$\n\\begin{equation}\n\\label{eq:varphi_i}\n\\Big(\n- \\frac{\\hbar^2}{2m}\\Delta \n+ U_H[\\rho_{int}] \n+ U_{XC}[\\rho_{int}] \n+ v_{int}\n\\Big) \\varphi^i_{int} = i\\hbar\\partial_t \\varphi^i_{int}\n\\end{equation}\nwith the potentials\n\\begin{widetext}\n\\begin{eqnarray}\n&& U_{H}[\\rho_{int}](\\vec{r},t) \n= \\frac{\\delta A_{H}[\\rho_{int}]}{\\delta \\rho_{int}(\\vec{r},t)}\n\\nonumber\\\\\n&& U_{XC}[\\rho_{int}](\\vec{r},t) \n= \\frac{\\delta A_{XC}[\\rho_{int}]}{\\delta \\rho_{int}(\\vec{r},t)}\n-i\\big(\\psi_{int}[\\rho_{int}](t_1)\\big|\\frac{\\delta \\psi_{int}[\\rho_{int}](t_1)}{\\delta \\rho_{int}(\\vec{r},t)}\\big)\n+ i \\big(\\psi^S_{int}[\\rho_{int}](t_1)\\big|\\frac{\\delta \\psi^S_{int}[\\rho_{int}](t_1)}{\\delta \\rho_{int}(\\vec{r},t)}\\big)\n\\label{eq:Uxc}\n\\end{eqnarray}\n\\end{widetext}\nwhich are local as expected ($v_S=U_H[\\rho_{int}]+ U_{XC}[\\rho_{int}] + v_{int}$ with the notations of Eq. (\\ref{eq:td_KS})).\nNote that the variational formulation of Vignale \\cite{Vig08} leads to the addition of the last\ntwo terms in the definition of $U_{XC}[\\rho_{int}](\\vec{r},t)$, see Eq. (\\ref{eq:Uxc}),\ncompared to the traditional result obtained by stationarization of the action. It are those terms which permit to solve the causality paradox \\cite{Vig08}.\n\nEquations~(\\ref{eq:varphi_i}) have the same form as the traditional time-dependent KS \nequations formulated \\old{in the laboratory frame} for non-translationally \ninvariant Hamiltonians \\cite{Koh65,Run84,Gro94}\nand permit to define $\\rho_{int}$ through (\\ref{eq:rho_int}).\nHere, we have justified their form \\textit{in the c.m. frame} for \nself-bound systems described with translationally invariant Hamiltonians.\n\nBut there is a major difference with the traditional DFT formalism.\nFollowing similar steps as in Eq.~(\\ref{eq:rel}), one can show that the\ninteracting kinetic energy term and the interacting ``$i\\hbar\\partial_t$'' term can be rewritten \\cite{foot3}\n\\begin{widetext}\n\\begin{eqnarray}\n&& (\\psi_{int}^0(t)|\\sum_{\\alpha=1}^{N-1} \\frac{\\tau_\\alpha^2}{2\\mu_\\alpha}|\\psi_{int}^0(t))\n= \\int d\\vec{r}_1 \\cdots d\\vec{r}_N \\delta(\\mathbf{R}) \n\\psi_{int}^{0*}(\\vec{r}_1,...,\\vec{r}_N;t)\n\\sum_{i=1}^N \\frac{\\mathbf{p}_i^2}{2m}\\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)\n\\nonumber\\\\\n&& (\\psi_{int}^0(t)|i\\hbar\\partial_t|\\psi_{int}^0(t)) =\n\\int d\\vec{r}_1 \\cdots d\\vec{r}_N \\delta(\\mathbf{R}) \\psi_{int}^{0*}(\\vec{r}_1,...,\\vec{r}_N;t)\ni\\hbar\\partial_t \\psi_{int}^0(\\vec{r}_1,...,\\vec{r}_N;t)\n,\n\\label{eq:cm_cor}\n\\end{eqnarray}\n\\end{widetext}\nwhich makes it clear that the differences with the non-interacting kinetic energy term\n$\\sum_{i=1}^{N} \\int d\\vec{r} \\varphi^{i*}_{int}(\\vec{r})\\frac{\\vec{p}^2}{2m}\\varphi^i_{int}(\\vec{r})$\nand the non-interacting ``$i\\hbar\\partial_t$ term'' $\\sum_{i=1}^{N} (\\varphi^i_{int}(t)|i\\hbar\\partial_t|\\varphi^i_{int}(t))$\n(found in the exchange-correlation functional (\\ref{eq:Axc}))\ncome, on the one hand, from the correlations neglected in the traditional independent-particle framework,\nbut also from the c.m. correlations described by the $\\delta(\\mathbf{R})$ term in (\\ref{eq:cm_cor}), which does not appear in traditional time-dependent DFT \\cite{Run84,Gro94}.\nThe inclusion of the c.m. correlations in the exchange-correlation functional (\\ref{eq:Axc}) and potential (\\ref{eq:Uxc})\nis the main difference with the traditional KS scheme, and is a key issue for self bound-systems as atomic nuclei.\n\nMind that all the previous considerations only hold for fixed initial states $|\\psi_{int}(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$ which should of course give the same initial density $\\rho_{int}(\\mathbf{r},t_0)$ (and also for a fixed type of particle).\nAs a consequence, $\\psi_{int}^0$ is not only a functional of $\\rho_{int}$, but also depends on the initial state $|\\psi_{int}(t_0))$, and $U_{XC}$, Eq. (\\ref{eq:Uxc}),\nalso depends on the initial orbitals $\\{|\\varphi^i_{int}(t_0))\\}$.\nAn important difference to the ground state Internal DFT formalism \/ KS scheme presented in \\cite{Mes09}\nis that $|\\psi_{int}(t_0))$ and the $\\{|\\varphi^i_{int}(t_0))\\}$\ncannot necessarily be written as functionals of $\\rho_{int}(\\mathbf{r},t_0)$.\nHowever, as underlined in \\cite{Run84,Gro94}, if one starts from initial states $|\\psi_{int}(t_0))$ and $\\{|\\varphi^i_{int}(t_0))\\}$\nthat are non-degenerate ground states, i.e. that can be written as functionals of $\\rho_{int}(\\mathbf{r},t_0)$ \\cite{Mes09},\n$\\psi_{int}$ and $U_{XC}$ become functionals of $\\rho_{int}(\\mathbf{r},t)$ alone.\nThen, in the limit of stationary ground states, the theory reduces to the stationary Internal DFT \/ KS.\n\nWe recall that, as in traditional DFT, the previously discussed functionals are defined only for internal densities $\\rho_{int}$ which correspond to some internal potential $v_{int}$, called \\textit{v-representable} internal densities \\cite{Run84,Gro94}.\nUp to now, we do not know exactly how large the set of v-representable densities is.\nThis has to be kept in mind when variations with arbitrary densities are done, as to obtain the time-dependent KS equations.\n\n\n\n\n\n\\section{Conclusion.}\n\nIn summary, we have shown that, for a fixed initial state, the internal wave function,\nwhich describes the internal properties of a time-dependent self-bound system, can be written (up to a trivial phase) as a functional of the internal density.\nThis implies that the \"internal\" expectation values of any observable (which does not contain a time derivative),\nthat are of experimental interest, can be regarded as functionals of the internal density.\nThen, we set up, in the c.m.\\ frame, a practical scheme which permits to calculate the internal density and whose form is similar to the traditional time-dependent KS equations,\nthe difference being that the exchange-correlation functional contains the c.m. correlations.\n\nThis work is a first step towards the justification to the use of density functionals\nfor time-dependent nuclear mean-field like calculations with effective interactions \\cite{tdhf_nucl,Neg82},\nproving that there exists an ultimate functional which permits to reproduce the exact internal density\n(up to the non-interacting v-representability question).\nIf this functional was known, there would be no need for a c.m.\\ correction.\n\nPractically speaking, the time-dependent Internal KS scheme can describe, for instance in the nuclear case,\nthe collision of two nuclei in the frame attached to the total c.m.\\ of the nuclei.\nThen, $v_{int}$ is zero but the dependency to the initial state allows to start from\na state which corresponds to two nuclei with different velocities, or ``boosts''\n(choosen such as the total kinetic momentum is zero because we are in the c.m. frame). According to the choice of the boosts, \nwe can describe a wide variety of physical phenomena, from nuclear fusion \\cite{tdhf_nucl} to\nCoulomb excitation \\cite{Ald56}. One of the nuclei can also simply consist in a particule as a proton,\nto describe the excitation of a nucleus by diffusion.\n\nA case where a non-zero $v_{int}$ would be interesting could be\nthe case of the laser irradiation ($v_{int}$ would then contain a laser potential switched on at $t>t_0$).\nThis is not of major interest in the nuclear case because, experimentally speaking,\nwe do not yet have lasers that are suited to the study of the laser irradiation of a nucleus.\nHowever, this could be interesting in view of a generalization of this work to the whole molecule\n(following from the generalization to different types of particules, which is underway).\n\nMany questions remain open.\nIn particular, the question of the form of the potential which describes the c.m.\\ correlations;\nin addition to its practical interest, this question would also give interesting arguments concerning the non-interacting v-representability question.\nGeneralization to different types of particles (Fermions or Bosons) appears desirable.\nFinally, the same reasoning should be applied\nto rotational invariance to formulate the theory in term of the so-called \"intrinsic\" one-body density \\cite{Gir08a}\n(which is not directly observable). This is more complicated because rotation\ndoes not decouple from internal motion, but it should be interesting concerning the symmetry breaking question.\n\n\n\n\n\n\n\\begin{acknowledgments}\n\nThe author is particularly grateful to M. Bender, E.K.U. Gross, and E. Suraud for enlightening discussions and\nreading of the manuscript,\nand thanks the referee for his pertinent remarks.\nThe author thanks the Centre d'Etudes Nucl\\'eaires de Bordeaux-Gradignan for warm hospitality,\nand the Institut Universitaire de France and the\nAgence Nationale de la Recherche (ANR-06-BLAN-0319-02) for financial support. \n\n\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHamel and Prahalad \\cite{2005Competing} point out that the rapid changes in technology and the non-stationary nature of market demand create dynamic behavioural characteristics of the environment, and that in today's society, where technological innovations are proliferating and transformations are taking place rapidly, the cumulative effects of technological innovation in firms are more likely to take on the characteristics of a complex, irregular and non-periodic non-linear dynamical system. This non-linear characteristic seems to be chaotic, but in fact it is in order. From the perspective of non-linear theory, the evolution of a system can be summarised as shown in Figure \\ref{fig:fig1} . McBride \\cite{2010Chaos} defines chaos as a qualitative study of the unstable, acyclic behaviour of a deterministic nonlinear dynamical system, and Sardar and Abrams \\cite{2004Introducing} argue that 'order and chaos coexist, order in chaos and chaos in order. The presence of chaos in a non-linear system should be categorised and discussed. There are situations where chaos should be avoided, times when it should be controlled, and times when it should be exploited. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.7]{figure1.png}\n \\caption{System evolution illustration.}\n \\label{fig:fig1}\n\\end{figure}\n\nSo as of now, research on models of the cumulative effects of technological innovation in the textile industry is still very limited and far from systematic. The textile industry, as an indispensable part of the population, plays an important role in clothing, food, housing and transport, and has experienced a long period of development. The cumulative effect of technological innovation in the industry is also prominent, so the textile industry is chosen as the subject of this study. This paper discusses the following: 1. the application of a model of the cumulative effect of technological innovation in the textile industry from the perspective of chaos theory; and 2. the provision of countermeasures for technological innovation in the textile industry to appropriately control and manage chaos and increase the effective innovation rate of the industry.\n\n\n\\section{A chaotic model of the cumulative effect of technological innovation}\n\\subsection{Chaos theory}\nThe basic characteristics of chaotic motion are as follows: deterministic, non-linear, sensitive dependence on initial conditions and non-periodic. The connotation of chaos is reflected in the fact that it is a seemingly random dynamic behaviour generated by a non-linear system \\cite{R1976Simple} . A deterministic system is said to be chaotic if, in the absence of external stochastic influences, 1. the state of motion of the system is irregular and complex, similar to the Brownian motion of small molecules; 2. the system has a sensitive dependence on the initial conditions; and 3. some characteristic of the system (e.g. positive Lyapunov exponent, positive topological entropy, attractor of fractional dimension, etc.) has little to do with the choice of initial conditions.\nIn this paper, on top of the basic framework of previous studies, a chaotic economic model of the cumulative effect of technological innovation in the textile industry is developed using the worm mouth model in chaos theory as an example, with appropriately given parameters so that it can be used to describe the dynamic evolution of the textile industry.\n\n\\subsection{The textile industry is chaotic in nature}\nThe evolution of the textile industry is irregular and complex. The reason is that it is impossible to predict which textile company will launch a new garment that will be liked by the people, or to judge success or failure by its industry innovations. The system of cumulative effects of technological innovation in the textile industry is sensitive to the initial conditions - the cumulative effects of the initial innovation - and is dependent on them. Moreover, Mensh \\cite{1979Stalemate}, Houstein et al. \\cite{1982Long}, Kleinknecht \\cite{1987Are} and Silverberg et al. \\cite{1993Long} have successively conducted empirical studies on the distribution function of the temporal occurrence pattern of innovation using relevant statistical data, and the results show that the realisation of innovation is a similar Poisson distribution and has an exponential growth trend of stochastic process. Therefore, both the textile industry itself and its innovation accumulation are chaotic in nature.\n\n\\subsection{A chaotic model of the cumulative effect of innovation in the textile industry}\nLet $X_t$ be the marginal contribution of the accumulation of innovative technology in the textile industry system to the growth rate of the total economic volume of the textile industry system at a certain moment $t$, i.e. when the accumulation of innovation in the textile industry increases by $1\\%$, the total economic volume of the textile industry increases by $X_t\\%$, and call $X_t$ the accumulation effect of innovation. It is easy to see that $X_t$ is a function about time, that is, the value of $X_t$ is time-dependent. Similar to the variables used to describe the changing state of economic growth, such as capital output rate, labour productivity and capital-labour ratio, $X_t$ should also be a state variable to describe the evolution of the textile system.\nThe contribution of innovation to economic growth in the textile industry is much more than a simple linear accumulation of individual innovation contributors, and therefore $X_t$ is a more complex variable, as are variables such as the capital output rate, labour productivity and the capital-labour ratio. According to the definition of $X_t$, its variation mainly reflects fluctuations in the structure of productivity within the textile industry due to the continuous generation and accumulation of innovation. Silverberg \\cite{1993Long} developed a model of technological progress and its evolutionary chaos, which not only explains the existence of the impact of innovation, but also shows that innovation makes productivity move in an irregular cycle, i.e. chaos. This is the theoretical basis for this paper's $X_t$. \\\\\nIntroducing the model more commonly used in chaos economics for studying chaos in economic growth, as shown in equation \\ref{equ1}.\n\n\\begin{equation}\\label{equ1}\n X_{t+1} = X_{t}^{\\beta} \\frac{\\sigma A}{1+\\lambda} ( s-X_t )^{\\gamma}\n\\end{equation}\n\nwhere $X_t$ is the capital-labour ratio, $\\sigma (\\sigma >0)$ is the savings rate, $\\lambda (0<\\lambda <1)$ is the natural growth rate of labour, $A(A>0)$ is the technological progress factor, $\\beta (\\beta >0)$ is the elasticity of the capital-labour ratio, $\\gamma$ is a constant greater than zero, and $s$ is the maximum capital-labour rate. \\\\\nWhen $\\beta=\\gamma=s=1$, let $\\mu=\\sigma A\/(1+\\gamma) $, then \\ref{equ1} can be transformed into \\ref{equ2}.\n\n\\begin{equation}\\label{equ2}\n X_{t+1} = \\mu X_{t}(1-X_{t})\n\\end{equation}\n\nequation \\ref{equ2} is the general form of the insect-population model. The purpose of introducing the transformation of equation \\ref{equ1} into equation \\ref{equ2} is to construct a chaotic economic model of the cumulative effect of technological innovation by analogy with a chaotic initialization \\cite{2018Improving}. Drawing on the insect-population model, the following assumptions are introduced: 1) Technological innovation does not arise out of thin air, and existing innovations are usually the basis for subsequent innovations, similar to the relationship between parent and offspring insects in nature. 2) Technological innovation is measured by its level of innovation, and usually there is no mixture of parent and offspring, i.e., there is a certain amount of substitution of the firm's new product for the old one. 3) The technological innovation of an enterprise is limited by its own economic resources, just like the survival environment of a worm in nature. Therefore, this paper considers that technological innovation in textile enterprises meets the prerequisites of the insect-population model. According to the previous paper, the cumulative effect of technological innovation is a chaotic economic variable, which evolves in much the same way as the capital-labour ratio, with a non-linear evolution mechanism. Therefore, a chaotic economic model of the cumulative effect of technological innovation can be defined similarly - equation \\ref{equ3}.\n\n\\begin{equation}\\label{equ3}\n X_{t+1} = T \\epsilon X_{t}(1-X_{t})\n\\end{equation}\n\nwhere $X_t \\in (0,1)$, $\\epsilon \\in (0,10)$, $T\\epsilon \\in (0,4)$, $X_t$ represents the proportion of the cumulative effect of technological innovation at time $t$, as the state variable of the textile system; $\\epsilon$ denotes the government regulation parameter; $T$ is the specific coefficient of the textile firm (i.e. the combined coefficient of the growth rate of technological inputs $\\alpha$, the proportion of technological content of output $\\beta$ and the annual growth rate of labour force $n$ at a point in time, which The relationship between them is $T=\\frac{\\alpha + \\beta}{1+n}$, where $\\alpha, \\beta, n \\in (0,1)$, $T\\epsilon$ together constitute the innovation control parameters of the textile enterprises themselves.\n\n\\subsection{Chaos of the model}\nThe chaotic state of \\ref{equ3} can be determined using the Li-Yorke theorem. Constructing the function,\n\n\\begin{equation}\\label{equ4}\n f(x)=T\\epsilon x(1-x)\n\\end{equation}\n\nIt is not difficult to determine where $f(0)=0$ and $f(x)$ is a single-peaked function. \\\\\nFirst, determine the point $x^*$ at which $f(x)$ reaches its maximum value. $x^*$ represents the maximum technological innovation accumulation effect value of $f(x)$. $x^*$ can be obtained by solving the following first order partial derivative equation.\n\n\\begin{equation}\\label{equ5}\n \\frac{\\partial f(x^*)}{\\partial x^*}=0\n\\end{equation}\n\nThe mapping point $x^*$ corresponding to the value of the maximum reachable function is obtained,\n\n\\begin{equation}\\label{equ6}\n x^*=1\/2\n\\end{equation}\n\nThe maximum innovation accumulation effect value $X_{max}$ is,\n\n\\begin{equation}\\label{equ7}\n x_{max}=f(x^*)=\\frac{T\\epsilon}{2}\n\\end{equation}\n\nFrom the above definition, it follows that the value of the maximum innovation accumulation effect should satisfy,\n\n\\begin{equation}\\label{equ8}\n x_{max} \\le s\n\\end{equation}\n\nNext, determine the initial image point $x_i$ , and from equations \\ref{equ4} and \\ref{equ6}, we have,\n\n\\begin{equation}\\label{equ9}\n T\\epsilon x(1-x)=1\/2\n\\end{equation}\n\nThe left end of equation \\ref{equ9} is equation \\ref{equ4}, a single-peaked function, so the smaller root is taken to be $x_i$, \n\n\\begin{equation}\\label{equ10}\n x_i = \\frac{T\\epsilon - \\sqrt{(T\\epsilon)^2-T\\epsilon}}{2T\\epsilon}\n\\end{equation}\n\nFinally, determine its third-order mapping $f(x_{max})$,\n\n\\begin{equation}\\label{equ11}\n f(x_{max}) = \\frac{T^2\\epsilon ^2}{2}(1-T\\epsilon \/2)\n\\end{equation}\n\nprovided that the first-, second- and third-order mappings $x^*$,$x_{max}$, $f(x_{max})$ generated by the initial preimage $x_i$ are mapped from the interval $[0,s]$ to its own interval $[0,s]$ and satisfy the sufficient conditions for chaos in the Li-Yorke theorem, i.e.,\n\n\\begin{equation}\\label{equ12}\n 0 \\le f(x_{max}) \\le x_i \\le x^* \\le x_{max}\n\\end{equation}\n\nThen the textile industry system \\ref{equ3} appears chaotic, as shown in Figure \\ref{fig:fig2}, which is a chaotic phase diagram regarding the proportion of the cumulative effect of two adjacent generations of innovation in the textile industry innovation system. Where the curve is the image of equation (3) and the straight line is $X_{t+1} = X_t$, which represents the process of converting a preimage point to the next preimage point.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.7]{figure2.png}\n \\caption{Chaotic phase diagrams in textile industry.}\n \\label{fig:fig2}\n\\end{figure}\n\n\n\\section{Chaotic nature of the model}\n\\subsection{Local structural stability and cyclic bifurcation propertiest}\nFor model \\ref{equ3}, specifically, when $T\\epsilon$ has a determined value, if the corresponding $x_t$ also has only one value (indicating that the firm's innovation reaches a uniquely determined proportion, i.e. the level of intensification). That is, the technological innovation effect of a textile firm reaches a uniquely determined level. Then the period of $x_t$ is said to be $1$; if at this point $x_t$ has two values corresponding to it, then the period of $x_t$ is said to be $2$; if at this point $x_t$ has n values corresponding to it, then the period of $x_t$ is said to be $n$. In particular, the structure of $x_t$ is unstable when there is a bifurcation and the steady state can only be maintained when the period of $x_t$ is determined. This also means that a stable accumulation of technological innovation in textile companies can only be guaranteed when $T\\epsilon$ takes on a value within a particular interval. In other words, a good accumulation of technological innovation in the textile industry can be maintained to a certain extent when the role of policy innovation and industry-specific coefficients are well controlled. A diagram of the iterative process for varying values of the control parameter $T\\epsilon$ from $0$ to large is shown in Figure \\ref{fig:fig3}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.7]{figure3.png}\n \\caption{Bifurcations.}\n \\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Effect of parameter values on the initial innovation ratio in the textile industry}\nBased on its definition of firm intensification in a chaotic economic model, the effect of the $T\\epsilon$ parameter on the initial innovation ratio of textile firms is derived by analogy. \n\\begin{itemize}\n\t\\item When $03.5441$, the local structure of $x_t$ is stable and the period of $x_t$ is $8$. The rest of the process continues until $T\\epsilon=3.5699=T\\epsilon_{\\infty}$, when the period of $x_t$ is infinite, i.e. chaos emerges. In other words, there is no relationship between the firm's innovation accumulation and the control parameters at this point.\n\\end{itemize}\n\n\\subsection{Sexual analysis of different parameter values}\nFirstly, the values of $\\alpha$, $\\beta$, and $n$ are taken and defined as shown in Table \\ref{tab1}, and all data are obtained from the findings of the \"China Statistical Yearbook\".\n\n\\begin{table}[H]\n \\caption{\\label{tab1} Variable definition and value}\n \\centering\n \\resizebox{\\linewidth}{32pt}{\n \\begin{tabular}{ccc}\n \\hline\n Symbol & Description & Fetching method \\\\\n \\hline\n $\\alpha$ & Growth rate of investment in science and technology & Log difference of R\\&D inputs \\\\\n $\\beta$ & Proportion of technological content of output & Log difference in the number of active patents in the textile industry \\\\\n $n$ & Annual growth rate of the labour force & Log difference of the average number of workers employed \\\\\n $\\epsilon$ & Government regulation parameters & By taking the product of $T\\epsilon$ backwards or assuming that given \\\\\n \\hline\n \\end{tabular}}\n\\end{table}\n\nFrom an econometric perspective, log-differencing helps to enhance the significance of the main regression to some extent. As a corollary, the significance of log-differencing the variables in this paper is to smooth out the effects of unreasonable outliers and reduce the \"misleading\" effect of episodic events on the overall judgment. In addition, as the average number of workers before 14 years could not be found directly, the calculation of business income\/per capita business income was used, and the use of logarithmic difference to calculate the growth rate can also reduce the subtle influence of different quantiles on the conclusion. The log-difference results are shown in Table \\ref{tab2} \\footnote{All data are obtained from the China Statistical Yearbook, and are rounded to four decimal places after calculation by stata. The values of $\\alpha$ range from 1.46\\% to 47.44\\%, $\\beta$ from 9.58\\% to 51.69\\% and $n$ from -16.46\\% to 4.79\\%, so the values of $\\alpha+\\beta$ and $1+n$ are inferred from this.}.\n\n\\begin{table}[H]\n\t\\caption{\\label{tab2} Log-differential results}\n\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\hline\n\t\t\tYear & $\\alpha$ & $\\beta$ & $n$ & $\\alpha+\\beta$ & $1+n$ \\\\\n\t\t\t\\hline\n\t\t\t2009&\/&\/&\/&\/&\/\t\\\\\n\t\t\t2010&.2017&.5088&.0479&.7105&1.0479\\\\\n\t\t\t2011&.4744&.1671&-.0947&.6416&.9053\\\\\n\t\t\t2012&.0146&.1347&-.0566&.1494&.9434\\\\\n\t\t\t2013&.1382&.1418&-.0614&.2800&.9386\\\\\n\t\t\t2014&.1144&.5169&-.0653&.6313&.9347\\\\\n\t\t\t2015&.1558&.0958&-0.540&.2516&.9460\\\\\n\t\t\t2016&.0574&.2867&-.0627&.3441&.9373\\\\\n\t\t\t2017&.0584&.2688&-.1090&.3273&.8910\\\\\n\t\t\t2018&.0912&.2629&-.1646&.3541&.8354\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab2} shows that the values of $\\alpha+\\beta$ range from 14.94\\% to 71.05\\%, and the values of $1+n$ range from 83.54\\% to 104.79\\%. Considering that the value of the parameter $T$ in equation \\ref{equ3} is proportional to $(\\alpha+\\beta)$ and inversely proportional to $(1+n)$, and that $T$ is related to the innovation accumulation $x_t$ of the textile enterprises in equation \\ref{equ3}, the sexual attitude of $x_t$ can be analysed through this. To facilitate the calculation, the three fixed cases of the lower, middle and upper limits of these elements are taken separately, and only $\\epsilon$ is adjusted to analyse the nature of $x_t$.\n\n\\begin{itemize}\n\t\\item Lower bound case. $\\alpha+\\beta=0.1494$ , $1+n=0.8354$ . When $\\epsilon=5.5917$, $T\\epsilon=1.0000$ . From equation \\ref{equ3} and Figure \\ref{fig:fig3}, $x_t=0$. when $\\epsilon=10$, $T\\epsilon=1.7884$ and $x_t=0.4408$ . This means that at $(\\alpha+\\beta=0.1494)$, the degree of innovation accumulation achieved by textile firms remains zero when the parameter of government regulation of textile firms is 5.5917. When adjusted to the maximum value $\\epsilon=10$, $T\\epsilon=1.7884$ (satisfying $10)$ is the savings rate, $\\lambda (0<\\lambda <1)$ is the natural growth rate of labour, $A(A>0)$ is the technological progress factor, $\\beta (\\beta >0)$ is the elasticity of the capital-labour ratio, $\\gamma$ is a constant greater than zero, and $s$ is the maximum capital-labour rate. \\\\\nWhen $\\beta=\\gamma=s=1$, let $\\mu=\\sigma A\/(1+\\gamma) $, then \\ref{equ1} can be transformed into \\ref{equ2}.\n\n\\begin{equation}\\label{equ2}\n X_{t+1} = \\mu X_{t}(1-X_{t})\n\\end{equation}\n\nequation \\ref{equ2} is the general form of the insect-population model. The purpose of introducing the transformation of equation \\ref{equ1} into equation \\ref{equ2} is to construct a chaotic economic model of the cumulative effect of technological innovation by analogy with a chaotic initialization \\cite{2018Improving}. Drawing on the insect-population model, the following assumptions are introduced: 1) Technological innovation does not arise out of thin air, and existing innovations are usually the basis for subsequent innovations, similar to the relationship between parent and offspring insects in nature. 2) Technological innovation is measured by its level of innovation, and usually there is no mixture of parent and offspring, i.e., there is a certain amount of substitution of the firm's new product for the old one. 3) The technological innovation of an enterprise is limited by its own economic resources, just like the survival environment of a worm in nature. Therefore, this paper considers that technological innovation in textile enterprises meets the prerequisites of the insect-population model. According to the previous paper, the cumulative effect of technological innovation is a chaotic economic variable, which evolves in much the same way as the capital-labour ratio, with a non-linear evolution mechanism. Therefore, a chaotic economic model of the cumulative effect of technological innovation can be defined similarly - equation \\ref{equ3}.\n\n\\begin{equation}\\label{equ3}\n X_{t+1} = T \\epsilon X_{t}(1-X_{t})\n\\end{equation}\n\nwhere $X_t \\in (0,1)$, $\\epsilon \\in (0,10)$, $T\\epsilon \\in (0,4)$, $X_t$ represents the proportion of the cumulative effect of technological innovation at time $t$, as the state variable of the textile system; $\\epsilon$ denotes the government regulation parameter; $T$ is the specific coefficient of the textile firm (i.e. the combined coefficient of the growth rate of technological inputs $\\alpha$, the proportion of technological content of output $\\beta$ and the annual growth rate of labour force $n$ at a point in time, which The relationship between them is $T=\\frac{\\alpha + \\beta}{1+n}$, where $\\alpha, \\beta, n \\in (0,1)$, $T\\epsilon$ together constitute the innovation control parameters of the textile enterprises themselves.\n\n\\subsection{Chaos of the model}\nThe chaotic state of \\ref{equ3} can be determined using the Li-Yorke theorem. Constructing the function,\n\n\\begin{equation}\\label{equ4}\n f(x)=T\\epsilon x(1-x)\n\\end{equation}\n\nIt is not difficult to determine where $f(0)=0$ and $f(x)$ is a single-peaked function. \\\\\nFirst, determine the point $x^*$ at which $f(x)$ reaches its maximum value. $x^*$ represents the maximum technological innovation accumulation effect value of $f(x)$. $x^*$ can be obtained by solving the following first order partial derivative equation.\n\n\\begin{equation}\\label{equ5}\n \\frac{\\partial f(x^*)}{\\partial x^*}=0\n\\end{equation}\n\nThe mapping point $x^*$ corresponding to the value of the maximum reachable function is obtained,\n\n\\begin{equation}\\label{equ6}\n x^*=1\/2\n\\end{equation}\n\nThe maximum innovation accumulation effect value $X_{max}$ is,\n\n\\begin{equation}\\label{equ7}\n x_{max}=f(x^*)=\\frac{T\\epsilon}{2}\n\\end{equation}\n\nFrom the above definition, it follows that the value of the maximum innovation accumulation effect should satisfy,\n\n\\begin{equation}\\label{equ8}\n x_{max} \\le s\n\\end{equation}\n\nNext, determine the initial image point $x_i$ , and from equations \\ref{equ4} and \\ref{equ6}, we have,\n\n\\begin{equation}\\label{equ9}\n T\\epsilon x(1-x)=1\/2\n\\end{equation}\n\nThe left end of equation \\ref{equ9} is equation \\ref{equ4}, a single-peaked function, so the smaller root is taken to be $x_i$, \n\n\\begin{equation}\\label{equ10}\n x_i = \\frac{T\\epsilon - \\sqrt{(T\\epsilon)^2-T\\epsilon}}{2T\\epsilon}\n\\end{equation}\n\nFinally, determine its third-order mapping $f(x_{max})$,\n\n\\begin{equation}\\label{equ11}\n f(x_{max}) = \\frac{T^2\\epsilon ^2}{2}(1-T\\epsilon \/2)\n\\end{equation}\n\nprovided that the first-, second- and third-order mappings $x^*$,$x_{max}$, $f(x_{max})$ generated by the initial preimage $x_i$ are mapped from the interval $[0,s]$ to its own interval $[0,s]$ and satisfy the sufficient conditions for chaos in the Li-Yorke theorem, i.e.,\n\n\\begin{equation}\\label{equ12}\n 0 \\le f(x_{max}) \\le x_i \\le x^* \\le x_{max}\n\\end{equation}\n\nThen the textile industry system \\ref{equ3} appears chaotic, as shown in Figure \\ref{fig:fig2}, which is a chaotic phase diagram regarding the proportion of the cumulative effect of two adjacent generations of innovation in the textile industry innovation system. Where the curve is the image of equation (3) and the straight line is $X_{t+1} = X_t$, which represents the process of converting a preimage point to the next preimage point.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.7]{figure2.png}\n \\caption{Chaotic phase diagrams in textile industry.}\n \\label{fig:fig2}\n\\end{figure}\n\n\n\\section{Chaotic nature of the model}\n\\subsection{Local structural stability and cyclic bifurcation propertiest}\nFor model \\ref{equ3}, specifically, when $T\\epsilon$ has a determined value, if the corresponding $x_t$ also has only one value (indicating that the firm's innovation reaches a uniquely determined proportion, i.e. the level of intensification). That is, the technological innovation effect of a textile firm reaches a uniquely determined level. Then the period of $x_t$ is said to be $1$; if at this point $x_t$ has two values corresponding to it, then the period of $x_t$ is said to be $2$; if at this point $x_t$ has n values corresponding to it, then the period of $x_t$ is said to be $n$. In particular, the structure of $x_t$ is unstable when there is a bifurcation and the steady state can only be maintained when the period of $x_t$ is determined. This also means that a stable accumulation of technological innovation in textile companies can only be guaranteed when $T\\epsilon$ takes on a value within a particular interval. In other words, a good accumulation of technological innovation in the textile industry can be maintained to a certain extent when the role of policy innovation and industry-specific coefficients are well controlled. A diagram of the iterative process for varying values of the control parameter $T\\epsilon$ from $0$ to large is shown in Figure \\ref{fig:fig3}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.7]{figure3.png}\n \\caption{Bifurcations.}\n \\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Effect of parameter values on the initial innovation ratio in the textile industry}\nBased on its definition of firm intensification in a chaotic economic model, the effect of the $T\\epsilon$ parameter on the initial innovation ratio of textile firms is derived by analogy. \n\\begin{itemize}\n\t\\item When $03.5441$, the local structure of $x_t$ is stable and the period of $x_t$ is $8$. The rest of the process continues until $T\\epsilon=3.5699=T\\epsilon_{\\infty}$, when the period of $x_t$ is infinite, i.e. chaos emerges. In other words, there is no relationship between the firm's innovation accumulation and the control parameters at this point.\n\\end{itemize}\n\n\\subsection{Sexual analysis of different parameter values}\nFirstly, the values of $\\alpha$, $\\beta$, and $n$ are taken and defined as shown in Table \\ref{tab1}, and all data are obtained from the findings of the \"China Statistical Yearbook\".\n\n\\begin{table}[H]\n \\caption{\\label{tab1} Variable definition and value}\n \\centering\n \\resizebox{\\linewidth}{32pt}{\n \\begin{tabular}{ccc}\n \\hline\n Symbol & Description & Fetching method \\\\\n \\hline\n $\\alpha$ & Growth rate of investment in science and technology & Log difference of R\\&D inputs \\\\\n $\\beta$ & Proportion of technological content of output & Log difference in the number of active patents in the textile industry \\\\\n $n$ & Annual growth rate of the labour force & Log difference of the average number of workers employed \\\\\n $\\epsilon$ & Government regulation parameters & By taking the product of $T\\epsilon$ backwards or assuming that given \\\\\n \\hline\n \\end{tabular}}\n\\end{table}\n\nFrom an econometric perspective, log-differencing helps to enhance the significance of the main regression to some extent. As a corollary, the significance of log-differencing the variables in this paper is to smooth out the effects of unreasonable outliers and reduce the \"misleading\" effect of episodic events on the overall judgment. In addition, as the average number of workers before 14 years could not be found directly, the calculation of business income\/per capita business income was used, and the use of logarithmic difference to calculate the growth rate can also reduce the subtle influence of different quantiles on the conclusion. The log-difference results are shown in Table \\ref{tab2} \\footnote{All data are obtained from the China Statistical Yearbook, and are rounded to four decimal places after calculation by stata. The values of $\\alpha$ range from 1.46\\% to 47.44\\%, $\\beta$ from 9.58\\% to 51.69\\% and $n$ from -16.46\\% to 4.79\\%, so the values of $\\alpha+\\beta$ and $1+n$ are inferred from this.}.\n\n\\begin{table}[H]\n\t\\caption{\\label{tab2} Log-differential results}\n\t\\centering\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\hline\n\t\t\tYear & $\\alpha$ & $\\beta$ & $n$ & $\\alpha+\\beta$ & $1+n$ \\\\\n\t\t\t\\hline\n\t\t\t2009&\/&\/&\/&\/&\/\t\\\\\n\t\t\t2010&.2017&.5088&.0479&.7105&1.0479\\\\\n\t\t\t2011&.4744&.1671&-.0947&.6416&.9053\\\\\n\t\t\t2012&.0146&.1347&-.0566&.1494&.9434\\\\\n\t\t\t2013&.1382&.1418&-.0614&.2800&.9386\\\\\n\t\t\t2014&.1144&.5169&-.0653&.6313&.9347\\\\\n\t\t\t2015&.1558&.0958&-0.540&.2516&.9460\\\\\n\t\t\t2016&.0574&.2867&-.0627&.3441&.9373\\\\\n\t\t\t2017&.0584&.2688&-.1090&.3273&.8910\\\\\n\t\t\t2018&.0912&.2629&-.1646&.3541&.8354\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab2} shows that the values of $\\alpha+\\beta$ range from 14.94\\% to 71.05\\%, and the values of $1+n$ range from 83.54\\% to 104.79\\%. Considering that the value of the parameter $T$ in equation \\ref{equ3} is proportional to $(\\alpha+\\beta)$ and inversely proportional to $(1+n)$, and that $T$ is related to the innovation accumulation $x_t$ of the textile enterprises in equation \\ref{equ3}, the sexual attitude of $x_t$ can be analysed through this. To facilitate the calculation, the three fixed cases of the lower, middle and upper limits of these elements are taken separately, and only $\\epsilon$ is adjusted to analyse the nature of $x_t$.\n\n\\begin{itemize}\n\t\\item Lower bound case. $\\alpha+\\beta=0.1494$ , $1+n=0.8354$ . When $\\epsilon=5.5917$, $T\\epsilon=1.0000$ . From equation \\ref{equ3} and Figure \\ref{fig:fig3}, $x_t=0$. when $\\epsilon=10$, $T\\epsilon=1.7884$ and $x_t=0.4408$ . This means that at $(\\alpha+\\beta=0.1494)$, the degree of innovation accumulation achieved by textile firms remains zero when the parameter of government regulation of textile firms is 5.5917. When adjusted to the maximum value $\\epsilon=10$, $T\\epsilon=1.7884$ (satisfying $1\\epsilon$, the exponential decay term of warp factor \\eqref{Warp_factor} is dominant in the whole bulk. It is clear that $V_I>0$ and $V_{II}<0$ in this case, namely, there is a positive tension brane located at the origin and a negative tension brane at $y_\\pi$. This is the model we will focus on in the rest of the paper. Especially, in the limit $\\epsilon e^{k y_\\pi}\\ll 1$, the warp factor \\eqref{Warp_factor} approaches the exponential decay form of RS model. Correspondingly, the fine-turning conditions reduce to the case of RS model as well,\n\\beq\nV_I\\approx-V_{II}\\approx6 k M^3, ~~~ \\Lambda=-6k^2.\n\\eeq\n\nBesides, there is another interesting case that both brane tensions are positive in case of $e^{-k|y_\\pi|}<\\epsilon<1$. It would generate a new brane configuration distinct from the RS1 model. Especially, the scenario that our universe is confined on the positive tension brane may potentially provide a possible model with better properties \\cite{Csaki1999,Shiromizu2000,Yang2012a}. This model will be studied in detail in our another work.\n\n\t\n\\section{Linear perturbations}\\label{Perturbation}\n\nIn order to investigate the stability of the model, we consider the full linear perturbations against the background. The perturbed metric is written as\n\\beq\nds^2=\\lt(g_{MN}+h_{MN}\\rt)dx^Mdx^N,\n\\label{Metric_NC}\n\\eeq\nwhere $h_{MN}$ represents the perturbations against the background metric $g_{MN}$ given in \\eqref{Brane_Metric}. Due to the 4D Lorentz symmetry of our background spacetime, it is convenient to decompose the perturbed metric into the scalar, transverse vector and transverse-traceless tensor modes, and to rewrite it by\n\\beqn\nh_{55}&=&-2\\xi,\\\\\nh_{\\mu 5}&=&-a \\lt(S_{\\mu}+\\pt_\\mu \\beta \\rt),\\\\\nh_{\\mu\\nu}&=&a^2\\lt[ D_{\\mu\\nu}+2\\eta_{\\mu\\nu}\\psi+\\fc{1}{2}\\lt(\\pt_{\\mu}F_{\\nu}+\\pt_{\\nu}F_{\\mu} \\rt)+2\\pt_\\mu\\pt_\\nu E \\rt],\n\\eeqn\nwhere the vector modes satisfy the transverse condition $\\pt^\\mu S_{\\mu}=\\pt^\\mu F_{\\mu}=0$, and the tensor mode satisfies the transverse-traceless (TT) condition $\\pt^\\mu D_{\\mu\\nu}=0$. \n\nCorrespondingly, the perturbed scalar fields are \n\\beq\n\\phi^a=x^a+\\pi^a,\n\\eeq \nwhere $\\pi^a=\\delta^a_\\mu\\pi^\\mu$ is the Goldstone excitation of the condensation. In order to maintain the scalar condensation \\eqref{Scalar_vacuum} under the general coordinate transformation $x^M \\to x^M+\\epsilon^M$, the Goldstone excitation has to transform opposite to the 4D coordinates simultaneously, i.e., $\n\\pi^\\mu\\to\\pi^\\mu-\\epsilon^\\mu$. This St\\\"uckelberg trick non-linearly restores the general covariance of the theory. Therefore, it behaves like a vector field and can be decomposed as $\\pi^\\mu=\\eta^{\\mu\\nu}\\lt(\\pt_\\nu \\varphi+A_\\nu \\rt)$, with $\\varphi$ a scalar field and $A_\\mu$ a transverse vector field.\n\nTherefore, there are 1 TT tensor, 3 transverse vector and 5 scalar modes in total. By decomposing $\\epsilon_\\mu =a^2(\\epsilon^{V}_{\\mu}+\\pt_\\mu \\epsilon^s)$ with $\\epsilon^{V}_{\\mu}$ a transverse vector and $\\epsilon^s$ a scalar, these 9 perturbed modes transform as follows under the general coordinate transformation,\n\\beqn\nD_{\\mu\\nu} &\\to& D_{\\mu\\nu}, \\quad S_{\\mu } \\to S_{\\mu } + a \\epsilon^{V}_{\\mu }{}' ,\\quad F_{\\mu } \\to F_{\\mu }-2 \\epsilon^{V}_{\\mu } ,\\nn\\\\\nA_{\\mu } &\\to& A_{\\mu }-{\\epsilon }^{V}_{\\mu }, \\quad \\psi \\to \\psi -H\\epsilon _5, \\quad E \\to E-\\epsilon^s, \\nn\\\\\n\\beta &\\to& \\beta+a {\\epsilon }^s{}'+\\frac{\\epsilon _5}{a}, \\quad \\xi \\to \\xi +\\epsilon '_5, \\quad \\varphi \\to \\varphi -{\\epsilon }^s.\n\\label{Gauge_transformation}\n\\eeqn\n\nNaively, we have 5 gauge freedoms to eliminate one vector and two scalar modes by fixing ${\\epsilon }^{V}_\\mu$, ${\\epsilon}^s$ and $\\epsilon _5$. A commonly used gauge choice is the so-called unitary gauge, in which the Goldstone excitations $\\pi^a$ of scalar fields are closed, i.e., $A_\\mu=\\varphi=0$. Moreover, we have another gauge freedom $\\epsilon _5$ to set $\\psi=0$. Then from the fact that $Z_\\mu-\\pi_\\mu$, with $Z_\\mu\\equiv a^2\\lt(F_\\mu\/2+\\pt_\\mu E \\rt)$, is a gauge invariant quantity, one observes that the 4 Goldstone excitations $\\pi^\\mu$ are ``eaten\" by the 5D graviton in the unitary gauge. Consequently, the 5D massless spin-2 graviton with 5 DOF gets weight and becomes massive, with 9 DOF on the spectrum. After gauge fixing, we have 1 TT tensor, 2 transverse vector and 3 scalar modes left. However, not all modes among them are physical, since some of them can be eliminated by some constraint equations, which can be easily extracted by utilizing the Arnowitt-Deser-Misner (ADM) formalism. \n\nThe RS metric \\eqref{Brane_Metric} in ADM formalism reads \n\\beq\nds^2=N^2dy^2+\\gamma _{\\mu \\nu } \\left(dx^{\\mu }+N^{\\mu }dy\\right) \\left(dx^{\\nu }+N^{\\nu }dy \\right),\n\\eeq\nwith $N=1$, $N^\\mu=0$ and $\\gamma_{\\mu\\nu}=a^2(y)\\eta_{\\mu\\nu}$. Correspondingly, the bulk action of \\eqref{Main_Action} is rewritten as \n\\beqn\nS&=&\\frac{M^3}{2}\\int d^5x\\sqrt{-\\gamma }\\Big[N \\lt(R^{(4)}-2 \\Lambda -m^2 \\gamma ^{\\mu \\nu } \\partial_\\mu\\phi ^a \\partial_\\nu\\phi ^a \\rt) -N^{-1}\\lt(E_{\\mu \\nu } E^{\\mu \\nu }-E^2\\rt)\\nn\\\\\n&& -m^2 N^{-1}\\left(\\phi^{a}{}'-N^{\\mu }\\partial_{\\mu }\\phi ^a\\right)\\left(\\phi^{a}{}'-N^{\\nu }\\partial_{\\nu }\\phi ^a\\right)\\Big], \\label{ADM_action}\n\\eeqn\nwhere $E_{\\mu \\nu }=\\frac{1}{2} \\left(\\gamma '_{\\mu \\nu }-\\nabla _{\\nu }N_{\\mu }-\\nabla _{\\mu }N_{\\nu }\\right)$. The constraint equations are yielded by varying with respect to the lapse function $N$ and shift vector $N_\\mu$, i.e.,\n\\beqn\nR^{(4)}-2 \\Lambda-m^2 \\gamma ^{\\mu \\nu } \\partial_\\mu\\phi^a \\partial_\\nu\\phi^a +N^{-2}\\lt(E_{\\mu \\nu } E^{\\mu \\nu }-E^2\\rt)&&\\nn\\\\\n+m^2N^{-2}\\left(\\phi^{a}{}'-N^{\\mu }\\partial _{\\mu }\\phi ^a\\right)\\left(\\phi^{a}{}'-N^{\\nu }\\partial_{\\nu }\\phi ^a\\right)&=&0, \\label{Constraint_Eq_I}\\\\\n\\nabla _{\\nu }\\lt[N^{-1}\\lt(E^{\\mu}_{\\nu }-\\delta^{\\mu}_{\\nu}E \\rt)\\rt]-m^2N^{-1}\\partial_{\\mu }\\phi ^a\\lt(\\phi^a{}'-N^{\\alpha } \\partial_\\alpha\\phi^a\\rt)&=&0. \\label{Constraint_Eq_II}\n\\eeqn\nBy perturbing the constraint equations, one has a constraint equation for vector modes and two constraint equations for scalar modes. Thus, one vector mode and two scalar modes can be worked out algebraically, leaving us one vector and one scalar modes eventually. \n\nAfter scalar-vector-tensor decomposition and expanding the action \\eqref{Main_Action} to the quadratic order of fluctuations, the TT tensor, transverse vector and scalar modes are decoupled with each other, so they can be treated separately.\n\n\\subsection{Tensor mode} \nBy including only the tensor perturbation in the perturbed metric and dropping the boundary terms, the quadratic action for tensor perturbation is read as\n\\beq\nS^{(2)}_T=\\frac{M^3}{8}\\int d^4xdya^2\\left[-{a^{2}} D_{ \\alpha \\beta }' D{}'^{\\alpha \\beta } -{\\partial_\\lambda D_{\\alpha \\beta }}\\partial ^{\\lambda }D ^{\\alpha \\beta }-2 m^2 D_{\\alpha \\beta } D^{\\alpha \\beta } \\right],\n\\eeq\nwhere the indices are raised and lowered by the 4D Minkowski metric $\\eta_{\\mu\\nu}$. Further, after a coordinate transformation $dy=adz$ that turns the background metric into a conformal form and a rescaling $\\tilde{D }={D }\/{2}$, a canonical form is obtained as\n\\beq\nS^{(2)}_T=\\frac{M^3}{2}\\int d^4y dz a^3 \\left[-\\dot{\\tilde{D }}_{\\alpha \\beta }\\dot{\\tilde{D }}^{\\alpha \\beta }-\\partial_\\lambda \\tilde D_{\\alpha \\beta } \\partial ^{\\lambda }\\tilde D ^{\\alpha \\beta }-2 m^2\\tilde D_{\\alpha \\beta} \\tilde D^{\\alpha \\beta } \\right],\n\\eeq\nwhere the dot denotes the derivative with respect to the extra dimension coordinate $z$. The tensor mode is free from the ghost instability due to the correct sign of kinetic term. Moreover, the action is expressed more concisely in the momentum space, where the d'Alembert operator $\\partial ^{\\alpha }\\partial_{\\alpha } $ is replaced by $ -k^2\\equiv -k^\\alpha k_\\alpha$, with $k^\\alpha$ the four-momentum of tensor mode. Then, it yields\n\\beq\nS^{(2)}_T=\\frac{M^3}{2}\\int d^4 k dz a^3 \\left[-\\dot{\\tilde{D }}_{\\alpha \\beta }\\dot{\\tilde{D }}^{\\alpha \\beta }-\\left({k^2}+{2 m^2}\\right)\\tilde D_{\\alpha \\beta }\\tilde D^{\\alpha \\beta } \\right].\n\\eeq\nBy variation with respect to $\\tilde{D }^{\\alpha \\beta }$, we have the equation of motion\n\\beq\n\\ddot{\\tilde{D} }_{\\alpha\\beta }+3H\\dot{\\tilde{D} }_{\\alpha\\beta }=(k^2+2m^2){\\tilde{D}_{\\alpha\\beta } }.\n\\label{eom_tensor}\n\\eeq\nAfter redefining $\\tilde{D}_{\\alpha\\beta } = a^{-\\fc{3}{2}}\\mathcal{D}_{\\alpha\\beta }$, a Schr\\\"odinger-like equation is obtained,\n\\beq\n-\\ddot{\\mathcal{D} }_{\\alpha\\beta } +\\left(\\frac{3}{2}\\dot H+\\frac{9}{4}H^2\\right) {\\mathcal{D} }_{\\alpha\\beta } =M_T^2{\\mathcal{D}}_{\\alpha\\beta } ,\n\\label{SE_Tensor}\n\\eeq\nwhere $M_T^2 \\equiv -k^2-2m^2$. The Hamiltonian can be further factorized as a supersymmetric quantum mechanics form, with $H_T=A_T^\\dag A_T=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$.\nThen with the boundary condition $\\pt_z \\tilde{D}_{ \\alpha \\beta } |_{z=0,z_b} =0$, the self-adjoint Hamiltonian gives non-negative eigenvalues \\cite{Yang2017}, i.e., $M_T^2\\geq 0$. Thus, with a positive 5D graviton mass $m$, the four-momentum of tensor mode $k^2=-M_T^2-2m^2<0$, i.e., the tensor excitations are all time-like particles. Thus, the model is also irrelevant to tachyonic instability. If $m$ is set to zero, then $M_T^2=-k^2\\geq 0$ gives us the well-known result that there is a massless graviton and a tower of massive gravitons in RS1 model.\n\n\\subsection{Vector modes} \n\nBy including only the vector perturbations in the perturbed metric and dropping the boundary terms, the quadratic action for vector perturbations is read\n\\beqn\nS_{V}^{(2)}&=&\\frac{M^3}{16}\\int d^4xdy a^2\\Big[-a^2\\pt_\\beta F_\\alpha{}'\\pt^\\beta F^\\alpha{}'-2m^2\\pt_\\beta F_\\alpha \\pt^\\beta F^\\alpha -8a^2m^2 A_\\alpha{}' A^\\alpha{}'-8m^2 \\pt_\\beta A_\\alpha \\pt^\\beta A^\\alpha \\nn\\\\\n&& -4\\partial_{\\beta }S_{\\alpha }\\partial^\\beta S^{\\alpha }-8m^2S_\\alpha S^\\alpha+8m^2 \\pt_\\beta F_\\alpha \\pt^\\beta A^\\alpha -4a\\pt_\\beta F_\\alpha{}'\\pt^\\beta S^\\alpha-16 a m^2 S_\\alpha A^\\alpha{}'\\Big].\n\\label{Action_SV}\n\\eeqn\nThe constraint equation can be obtained by counting the first order perturbations of \\eqref{Constraint_Eq_II}, or simplify by varying the above quadratic action with respect to $S^\\alpha$, i.e.,\n\\beq\n2\\partial _{\\beta }{\\partial^\\beta S_{\\alpha }}-4 m^2 S_{\\alpha }+a \\partial_{\\beta }{\\partial^\\beta F_{\\alpha }'}-4am^2A_\\alpha'=0.\n\\label{Vector_Constraint_Eq}\n\\eeq\nThen, $S^{\\alpha }$ can be worked out in momentum space, namely,\n\\beq\nS_{\\alpha }=-\\frac{a k^2 F_{\\alpha }' +4 a m^2 A_\\alpha '}{2 (k^2+2 m^2)}. \n\\label{Vector_Constraint}\n\\eeq\nWorking in the unitary gauge, $A_\\alpha$ is gauged away. Then, substituting the relation into the action \\eqref{Action_SV} yields\n\\beq\nS_V^{(2)}= M^3\\int d^4 k dy \\left[-\\frac{a^4 k^2 m^2 F_{\\alpha }{}' F^{\\alpha }{}'}{8 \\left(k^2+2 m^2\\right)}-\\frac{a^2 k^2 m^2}{8} F_{\\alpha } F^{\\alpha } \\right].\n\\eeq\nAfter a coordinate transformation into the coordinate $z$, a canonical normalized form can be obtained by redefining the vector perturbation as $\\tilde{F}_{\\alpha }=\\frac{k m F_{\\alpha } }{2 \\sqrt{k^2+2 m^2}}$, i.e.,\n\\beq\nS^{(2)}_V=\\frac{M^3}{2}\\int d^4 k dz a^3 \\left[-\\dot{\\tilde{F }}_{\\alpha} \\dot{\\tilde{F }}^{\\alpha}-\\left({k^2}+{2 m^2}\\right)\\tilde{F}_{\\alpha } \\tilde{F}^{\\alpha }\\right].\n\\eeq\nThe correct sign of the kinetic term ensures that the vector perturbation is free from the ghost instability. Then the equation of motion reads\n\\beq\n\\ddot{\\tilde{F} }_{\\alpha }+3H\\dot{\\tilde{F} }_{\\alpha }=(k^2+2m^2){\\tilde{F}_{\\alpha } }.\n\\eeq\nA Schr\\\"odinger-like equation is given by redefining $\\tilde{F}_{\\alpha }\\to a^{-\\frac{3}{2}} \\mathcal{F}_{\\alpha }$,\n\\beq\n-\\ddot{\\mathcal{F}}_\\alpha+\\left(\\frac{3}{2}\\dot H+\\frac{9}{4}H^2\\right) \\mathcal{F}_\\alpha=M_V^2\\mathcal{F}_\\alpha,\n\\label{SE_Vector}\n\\eeq\nwhere $M_V^2 \\equiv -k^2-2m^2$. The Hamiltonian can also be factorized as a supersymmetric quantum mechanics form, with $H_V=A_V^\\dag A_V=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$. With the boundary condition $\\pt_z \\tilde{F }_{\\alpha} |_{z=0,z_b} =0$, the eigenvalues are non-negative $M_V^2\\geq 0$. Thus, the vector excitations are also time-like particles and irrelevant to tachyonic instabilities for the cases of $M_V^2> 0$. \n\nHowever, for the case of $M_V^2=-k_0^2-2m^2=0$, the formula \\eqref{Vector_Constraint} is invalid, and the constraint equation \\eqref{Vector_Constraint_Eq} leads to \n\\beq\nF_\\alpha'=2A_\\alpha'.\n\\eeq\nThis implies that $F_\\alpha-2A_\\alpha \\equiv f_\\alpha(x)$ is a purely 4D field. Especially, the field $f_\\alpha(x)$ is a gauge invariant quantity under the general coordinate transformation by the observation from \\eqref{Gauge_transformation}. Then, the quadratic action \\eqref{Action_SV} reduces to\n\\beqn\nS^{(2)}_{V0} &=& -\\frac{M^3}{8}\\int dz a^3 m^2 k_0^2 \\left(F_{\\alpha }-2 A_{\\alpha }\\right) \\left(F^{\\alpha }-2 A^{\\alpha }\\right)=-\\frac{M^3}{8}\\int dz a^3 m^2 k_0^2 f_\\alpha f^\\alpha .\n\\eeqn\nThe equation of motion reads $k_0^2f_\\alpha=0$. Since $k_0^2=-2m^2$, it leads to $f_\\alpha=0$. This implies that the lightest vector mode does not exist in the mass spectrum. This is curial to recover the mass spectrum of RS1 model in the massless limit $m=0$, where no massless vector mode exists due to the lack of continuous isometries of the higher dimension in the presence of 3-branes \\cite{Randall1999}. Moreover, all the massive vector modes are gauge dependent in the massless limit $m=0$, therefore, they can be gauged away by gauge fixing.\n\n\\subsection{Scalar modes} \nBy including only the scalar perturbations in the perturbed metric and dropping the boundary terms, the quadratic action for scalar perturbations is read as\n\\beqn\nS^{(2)}_{S}&=&M^3\\int d^4xdy a^2\\Big(6 a^2 H^2 \\xi ^2+4 m^2 \\psi ^2-4 m^2 \\xi \\psi+6 a^2 \\psi '{}^2 +12 a^2 H \\xi \\psi ' +3\\pt_\\alpha \\psi \\pt^\\alpha \\psi\\nn\\\\\n&& -\\fc{m^2}{2} \\pt_\\alpha \\beta \\pt^\\alpha \\beta +3\\xi \\pt_\\alpha \\pt^\\alpha \\psi+3 aH\\xi \\pt_\\alpha \\pt^\\alpha \\beta + m^2 \\xi \\pt_\\alpha \\pt^\\alpha\\varphi-m^2 \\xi \\pt_\\alpha \\pt^\\alpha E -2m^2 \\psi \\pt_\\alpha \\pt^\\alpha \\varphi \\nn\\\\\n&& +2m^2\\psi \\pt_\\alpha \\pt^\\alpha \\varphi +3 a^2 H \\xi \\pt_\\alpha \\pt^\\alpha E' +3 a\\beta \\pt_\\alpha \\pt^\\alpha\\psi' + m^2 a \\beta \\pt_\\alpha \\pt^\\alpha \\varphi' -\\fc{m^2}{2} a^2 \\pt_\\alpha \\varphi' \\pt^\\alpha \\varphi'\\nn\\\\\n&&+3a^2 \\psi'\\pt_\\alpha \\pt^\\alpha E' -\\fc{m^2}{2} \\pt_\\alpha \\pt^\\alpha E \\pt_\\lambda \\pt^\\lambda E -\\fc{m^2 }{2} \\pt_\\alpha \\pt^\\alpha \\varphi \\pt_\\lambda \\pt^\\lambda \\varphi +m^2 \\pt_\\alpha \\pt^\\alpha \\varphi \\pt_\\lambda \\pt^\\lambda E \\Big).\n\\label{Scalar_full_action}\n\\eeqn\nFrom the first order perturbation of Eqs.~\\eqref{Constraint_Eq_I} and \\eqref{Constraint_Eq_II} or simply varying the above action respect to the modes $\\beta$ and $\\xi$ respectively, the constraint equations are obtained as \n\\beqn\n12 a^2 H^2 \\xi-4m^2\\psi+12 a^2 H \\psi'+3 \\pt_\\alpha \\pt^\\alpha \\psi +3aH\\partial _{\\alpha }\\partial ^{\\alpha }\\beta +m^2 \\partial _{\\alpha }\\partial ^{\\alpha }\\varphi\\nn\\\\\n-m^2\\partial _{\\alpha }\\partial ^{\\alpha }E +3a^2H\\partial _{\\alpha }\\partial ^{\\alpha }E'&=&0,\\\\\nm^2\\partial_\\alpha \\beta + 3aH\\partial _{\\alpha }\\xi + 3a \\pt_\\alpha \\psi' +m^2 a \\pt_\\alpha \\varphi' &=&0.\n\\eeqn\nAfter closing the scalar perturbations $\\varphi$ and $\\psi$ in the unitary gauge, we only have to take the remaining perturbations $\\xi$, $\\beta$ and $E$ into account in the quadratic action, i.e., \n\\beqn\nS^{(2)}_{S}&=&M^3\\int d^4xdy a^2\\Big(6 a^2 H^2 \\xi ^2 -\\frac{m^2}{2}\\partial _\\alpha \\beta \\partial ^{\\alpha }\\beta +3 a H \\xi\\partial _{\\alpha }\\partial ^{\\alpha }\\beta-m^2\\xi \\partial _{\\alpha }\\partial ^{\\alpha }E\\nn\\\\\n&&+3a^2H\\xi \\partial _{\\alpha }\\partial ^{\\alpha }E'-\\frac{m^2}{2}\\partial _{\\alpha }\\partial ^{\\alpha }E\\partial _{\\lambda }\\partial ^{\\lambda }E\\Big).\n\\label{Scalar_action}\n\\eeqn\nCorrespondingly, the constraint equations are rewritten as \n\\beqn\n12 a^2 H^2 \\xi+3aH\\partial _{\\alpha }\\partial ^{\\alpha }\\beta -m^2\\partial _{\\alpha }\\partial ^{\\alpha }E +3a^2H\\partial _{\\alpha }\\partial ^{\\alpha }E'&=&0,\\\\\nm^2\\partial_\\alpha \\beta+3aH\\partial _{\\alpha }\\xi &=&0.\n\\eeqn\nIn momentum space, $\\beta$ and $\\xi $ can be worked out from the constraint equations as \n\\beqn\n\\beta &=&-\\frac{3 aH}{m^2}\\xi ,\\\\\n\\xi &=&\\frac{k^2m^2(3 a^2 HE'- m^2 E)}{3 a^2 H^2 \\left(3 k^2+4 m^2\\right)}.\n\\eeqn\nThus, the perturbations $\\beta$ and $\\xi$ can be eliminated by substituting these relations into the action \\eqref{Scalar_action}, then a quadratic action for scalar perturbation $E$ is achieved,\n\\beq\nS^{(2)}_{S}\\supset M^3\\int dk^4 dy \\fc{3a^2k^4 m^2}{6 k^2+8 m^2} \\left[-a^2E' E' -\\lt(k^2+2 m^2\\rt)E^2\\right].\n\\eeq\nThe correct sign of the kinetic term ensures that the scalar perturbation is free from the ghost instability as well. Further, after redefining the scalar perturbation as $\\tilde E= \\sqrt{\\frac{3 k^4 m^2}{3 k^2+4 m^2}}E$, the above action can be rewritten as a canonical normalized form in the conformal coordinate $z$, i.e.,\n\\beq\nS^{(2)}_{S}\\supset\\frac{M^3}{2}\\int d^4 k dz a^3 \\left[-\\dot{\\tilde{E} }\\dot{\\tilde{E}} -\\left({k^2}+{2 m^2}\\right)\\tilde{E}^2\\right].\n\\eeq \nFurther, the equations of motion of scalar mode $\\tilde{E}$ is given by\n\\beq\n\\ddot{\\tilde{E} }+3H\\dot{\\tilde{E} }=(k^2+2m^2){\\tilde{E} }.\n\\eeq\nBy redefining $\\tilde{E}=a^{-\\fc{3}{2}}{\\varepsilon}$, it can be rewritten as a Schr\\\"odinger-like equation,\n\\beq\n-\\ddot{{\\varepsilon }}+\\left(\\frac{9 }{4}H^2+\\frac{3 }{2}\\dot H\\right){\\varepsilon } = M_S^2{\\varepsilon},\n\\label{SE_Scalar}\n\\eeq\nwhere $M_S^2\\equiv-k^2-2m^2$. The Hamiltonian can be factorized as a supersymmetric quantum mechanics form as well, $H_S=A_S^\\dag A_S=\\lt(\\pt_z+\\fc{3}{2}H\\rt)\\lt(-\\pt_z+\\fc{3}{2}H\\rt)$, so with the boundary condition $\\pt_z \\tilde E |_{z=0,z_b} =0$, the eigenvalues are non-negative $M_S^2\\geq 0$. Since the excitations of scalar mode are all time-like particles, it is also irrelevant to the tachyonic instability in scalar perturbations. \n\nIn RS1 model, the presence of IR brane abruptly ends AdS space, so it spontaneously breaks the conformal invariance of AdS bulk in the IR. The massless radion is just the Goldstone boson associated with the broken dilatation invariance \\cite{Arkani-Hamed2001a,Rattazzi2001}. For the current model, the scalar curvature reads $R=-20k^2+\\fc{4m^2e^{2k y}}{1+\\epsilon^2 e^{2ky}}$ in the bulk. Thus, the scalar condensation deforms the bulk geometry a little bit, which is not a pure AdS bulk anymore. Consequently, the radion acquires a tiny mass due to this explicit symmetry breaking. In the massless limit $m=0$, the quadratic action \\eqref{Scalar_full_action} will reduce to that of RS model, with only a massless radion in the mass spectrum \\cite{Goldberger1999a,Charmousis2000,Callin2005}. \n\n\n\n\\section{Mass Spectra and Gauge Hierarchy}\\label{Hierarchy}\n\nFor the lightest tensor and scalar mode corresponding to $M_{T,S}=0$, their wave functions can be easily solved from Eqs.~\\eqref{SE_Tensor} and \\eqref{SE_Scalar}. After returning to the coordinate space, and utilizing the KK decompositions, $\\mathcal{D}_{\\alpha\\beta }(x,z) =d_{\\alpha\\beta }(x)\\Psi(z)$ and $\\varepsilon(x,z) =e(x)\\Psi(z)$, Eqs.~\\eqref{SE_Tensor} and \\eqref{SE_Scalar} can be further reduced to the 4D Klein-Gordon equations, $\\Box^{(4)}d_{\\mu\\nu}=m_T^2 d_{\\mu\\nu}$ and $\\Box^{(4)}e(x)=m_S^2 e(x)$, with $m_{T,S}$ the effective mass of KK particles in 4D point of view, and the Schr\\\"odinger-like equation,\n\\beq\n-\\ddot{\\Psi} +\\left(\\frac{3}{2} \\dot{H}+\\frac{9}{4}H^2\\right)\\Psi =M_{T,S}^2\\Psi,\n\\eeq\nwhere $M_{T,S}^2=m_{T,S}^2-2m^2$. The wave function of ground states is easily achieved by setting $M_{T,S}=0$ or $m_{T,S}=\\sqrt{2}m$ in above Schr\\\"odinger-like equation, \n\\beq\n\\Psi^{(0)}(z)=N_0a(z)^{\\fc{3}{2}},\n\\eeq \nwith $N_0$ a normalization factor. The boundary conditions $\\pt_z \\tilde{D}_{ \\alpha \\beta } |_{z=0,z_b} =0$ and $\\pt_z \\tilde E |_{z=0,z_b} =0$ lead to $\\lt.\\lt( \\Psi-\\frac{3}{2}H \\rt)\\rt|_{z=0,z_b}=0$. Obviously, the wave function $\\Psi^{(0)}(z)$ satisfies the boundary condition. All KK particles are massive in this model, which is a significant difference from the RS1 model, where the lightest KK particles are massless spin-2 graviton and spin-0 radion. Especially, from the redefinitions $\\tilde{D}_{\\alpha\\beta } = a^{-\\fc{3}{2}}\\mathcal{D}_{\\alpha\\beta }$ and $\\tilde{E}=a^{-\\fc{3}{2}}{\\varepsilon}$, their canonical normalized field configurations are given by $\\tilde{D}^{(0)}_{\\alpha\\beta }=a^{-\\fc{3}{2}}\\mathcal{D}^{(0)}_{\\alpha\\beta }=d^{(0)}_{\\alpha\\beta }(x)$ and $\\tilde{E}^{(0)}=a^{-\\fc{3}{2}} {\\varepsilon}^{(0)} =e^{(0)}(x)$. Therefore, the lightest graviton and radion propagate only on the brane. \n\nHowever, the effective mass of 4D graviton is severely constrained by the gravitational experiments \\cite{Rham2017}, e.g., the bound of the graviton mass is $m_g\\leq 4.7\\times 10^{-23}$eV from the detection of gravitational waves \\cite{Abbott2019}, thus the parameter $m$ must be tinier than the experimental constraints, i.e., $m<3.3\\times 10^{-23}$eV. Since the lightweight radion has the same mass as the lightest graviton, exchanging such nearly massless scalar particle would cause a fifth force and violate experimental observations. Therefore, similar to the RS1 model, the radion must gain weight to meet the experimental expectations, which is realized through GW mechanism \\cite{Goldberger1999a} in the next section.\n\nWith the normalization condition $\\int^{z_b}_{-z_b}\\Psi_0^2dz=1$, the normalization factor is worked out as \n\\beqn\nN_0^{-2}=\\frac{1}{k}\\lt[1-e^{-2 k y_\\pi}+4\\epsilon^2 k y_\\pi -\\epsilon^4(1-e^{2 k y_\\pi})\\rt].\n\\label{Normalization_factor}\n\\eeqn\nBy shutting down the 5D graviton mass $m$, one recovers the result of RS1 model. However, if we remove the visible brane, i.e., $y_\\pi\\to\\infty$, the quasi-massless graviton is no longer normalizable. This means that the effective 4D gravity theory can not be recovered on the brane. So the RS2-like single brane model \\cite{Randall1999a} is not a physically available one in current massive gravity.\n\nThe braneworld scenario provides a natural way to solve the gauge hierarchy problem, which is a crucial motivation of the well-known ADD model \\cite{Arkani-Hamed1998} and RS1 model \\cite{Randall1999}. In our toy model, the low-energy effective theory is obtained by including the nearly-massless gravitons, i.e.,\n\\beq\nds^2=a^2(y)\\bar g_{\\mu\\nu}(x)dx^\\mu dx^\\nu+dy^2=a^2(y)\\lt[\\eta_{\\mu\\nu}+\\gamma_{\\mu\\nu}(x)\\rt]dx^\\mu dx^\\nu+dy^2,\n\\label{Metric_Lowenergy}\n\\eeq\nthen the 4D effective gravitational mass scale $M_{\\text{eff}}$ is read from the curvature term in the action \\eqref{Main_Action},\n\\beq\nM_{\\text{eff}}^2=M^3\\int^{y_\\pi}_{-y_\\pi} a^2 dy={N_0^{-2}}{M^3}. \\label{Mass_scale_relation}\n\\eeq\n\nOn the other hand, in order to produce a large hierarchy between the Planck scale and the electroweak scale, our Universe should be embedded on the IR brane located at $y_\\pi$. Then the Higgs field action on the brane is non-canonically normalized, \n\\beq\nS_\\text{H}\\supset\\int{ d^4x \\sqrt{| g_\\text{II}|}\\lt[-{g}_\\text{II}^{\\mu\\nu}D_{\\mu}H^{\\dag}D_{\\nu}H-\\lambda(H^{\\dag}H-v_0^2)^2 \\rt] },\n\\eeq\nwhere $g_{\\text{II}\\mu\\nu}=a(y_\\pi)\\tilde{g}_{\\mu\\nu}$ is the induced metric on the brane at $y_\\pi$, and $v_0$ the fundamental Higgs vacuum expectation value (VEV). Writting the warp factor explicitly, it is \n\\beq\nS_\\text{H}\\supset\\int{ d^4x \\sqrt{|\\tilde g|}\\lt[-a(y_\\pi)^{2}\\tilde{g}^{\\mu\\nu}D_{\\mu}H^{\\dag}D_{\\nu}H-a(y_\\pi)^4\\lambda(H^{\\dag}H-v_0^2)^2 \\rt] },\n\\eeq\n With a field renormalization, $H \\to \\tilde{H}\/a(y_\\pi)$, the effective action of Higgs on the brane is \n \\beq\n S_\\text{H}\\supset\\int{ d^4x \\sqrt{|\\tilde g|}\\lt[-\\tilde{g}^{\\mu\\nu}D_{\\mu}\\tilde{H}^{\\dag}D_{\\nu}H-\\lambda(\\tilde{H}^{\\dag}H-v_\\text{eff}^2)^2 \\rt] },\n \\eeq\n where the effective Higgs VEV, $v_\\text{eff}=a(y_\\pi) v_0$, sets the electroweak scale on the brane. \n \n Therefore, in order to solve the gauge hierarchy problem, the fundamental parameters $M$, $k$, $v_0$ are all set to be the order of Plank scale $M_\\text{Pl}$. Then, the warp factor has to provide enough redshift to recover a TeV scale of Higgs VEV on the brane, namely, $a(y_\\pi)\\sim10^{-16}$. In the condition $e^{-k |y_\\pi|}>\\epsilon$ and $m<3.3\\times 10^{-23}$eV, it leads to $e^{-ky_\\pi} \\approx a(y_\\pi) -\\fc{\\epsilon^2}{a(y_\\pi)} \\sim10^{-16}$. Thus, it requires $y_\\pi\\approx 37\/k$. \n\n\nSince the mass splitting scale of massive KK modes is inversely proportional to the conformal size of extra dimension $z_b$, i.e., $\\Delta m_T \\propto 1\/z_b$. From the coordinate transformation $dy=adz$, one has\n\\beq\nz_\\pi=\\frac{1}{k \\epsilon} \\left[\\text{arctan}\\left(\\epsilon e^{k y_\\pi}\\right)-\\text{arctan} \\left(\\epsilon \\right)\\right],\n\\eeq\nwhere the integral constant has been chosen so that $z(y=0)=0$. Since $e^{-ky_\\pi} \\sim10^{-16}$, $m<3.3\\times 10^{-23}$eV and $k\\sim M_\\text{Pl}$, one has $\\epsilon e^{k y_\\pi} \\ll 1$. Thus, the formula can be rewritten approximately as $z_\\pi \\approx \\fc{e^{ky_\\pi}}{k}\\lt[1+\\mathcal{O}(\\epsilon^2)\\rt]$. So the mass splitting scale is similar to the RS model, i.e., $\\Delta m_T \\sim k e^{-ky_\\pi} \\sim \\mathcal{O}(\\text{TeV})$.\n\n\t\n\\section{Radius stabilization}\\label{Radius_stabilization}\n\nIn RS1 model, the radius is not dynamically fixed, so there is a massless radion in the effective theory, which corresponds to the fluctuations of the radius of compact extra dimensions. However, the massless radion is phenomenologically unacceptable since it would contribute to Newton's law and cause a fifth force. It is well-known that the GW mechanism \\cite{Goldberger1999a} can be introduced to stabilize the size of the extra dimension and to increase the mass of radion. In our model, the light weight radion and the lightest graviton have equal mass, which is unacceptable in phenomenology as well. Therefore, we also utilize the GW mechanism here to increase the mass of radion. \n\nBy adding a bulk scalar field $\\Phi$ into the model, which has interaction with the two branes, the action is given by\n\\beqn\nS_\\Phi&=&\\fc{1}{2}\\int d^4x\\int^\\pi_{-\\pi}d\\theta\\sqrt{-g}\\lt(-g^{MN}\\pt_M\\Phi\\pt_N\\Phi-m_\\Phi^2\\Phi^2 \\rt)-\\int{}d^4x\\sqrt{-g_\\text{I}}\\lambda_\\text{I}(\\Phi^2-v_\\text{I}^2)^2\\nn\\\\\n&&-\\int{}\\sqrt{-g_{\\text{II}}}\\lambda_{\\text{II}}(\\Phi^2-v_{\\text{II}}^2)^2, \n\\eeqn\nwhere $g_{MN}$ is the background metric \\eqref{Brane_Metric} with the radius $r_c$ of compactified extra dimension given by $y=r_c \\theta$, $\\lambda_{\\text{I\/II}}$ is the coupling parameters, and $V_{\\text{I\/II}}=\\lambda_{\\text{I\/II}} v_{\\text{I\/II}}^4$.\n\nThe equation of motion of the scalar filed $\\Phi$ is achieved by varying with respect to $\\Phi$, \n\\beqn\n\\frac{1}{r_c}{\\partial_\\theta }\\left(a^4 {\\partial_\\theta \\Phi }\\right)-a^4 m_{\\Phi }^2\\Phi-{4 a^4 \\lambda_\\text{I} \\Phi \\left(\\Phi ^2-v_\\text{I}^2\\right)}\\delta (\\theta)-4 a^4\\lambda _\\text{II} \\left(\\Phi ^2-v_\\text{II}^2\\right) \\Phi \\delta (\\theta -\\pi ) =0.\n\\label{EoM_Phi}\n\\eeqn\nAway from the two branes at $\\theta=0,\\pi$, the general solution of this equation reads\n\\beq\n\\Phi(\\theta)=e^{2 \\sigma } \\left[A {}_2\\text{F}_1\\left(2,\\nu +2,\\nu +1, -\\epsilon ^2e^{2 \\sigma} \\right) e^{ \\nu \\sigma }+B {}_2\\text{F}_1\\left(2,2-\\nu ,1-\\nu ,-\\epsilon ^2e^{2 \\sigma } \\right) e^{-\\nu \\sigma }\\right],\n\\eeq\nwhere $\\sigma(\\theta)=k r_c \\theta$, $\\nu=\\sqrt{4+m_\\Phi^2\/k^2}$, ${}_2\\text{F}_1$ is the hypergeometric function, $A$ and $B$ are integration constants. Especially, under the condition $\\epsilon e^{\\sigma} \\ll 1$, up to the first order of correction, the solution can be approximately written as\n\\beq\n\\Phi(\\theta)\\simeq e^{2 \\sigma } \\left[A \\left(1-2 \\epsilon^2\\frac{\\nu +2}{\\nu +1}e^{2 \\sigma } \\right)e^{ \\nu \\sigma }+B \\left(1-2\\epsilon^2\\frac{\\nu -2}{\\nu -1} e^{2 \\sigma }\\right)e^{ -\\nu \\sigma }\\right].\n\\eeq\n If we close the mass of 5D graviton, i.e., $\\epsilon=0$, the solution reduces to the one in general relativity \\cite{Goldberger1999a}. The integration constants $A$ and $B$ can be fixed by the boundary conditions on the branes, which are obtained by inserting the approximate solution into the equation of motion \\eqref{EoM_Phi} and matching the delta functions,\n \\beqn\n k \\left[ (\\nu +2) \\left(1+2\\epsilon ^2\\frac{\\nu -2}{\\nu +1}\\right)A- (\\nu-2 ) \\left(1+2\\epsilon ^2\\frac{\\nu +2}{\\nu -1}\\right)B\\right]\\nn\\\\\n -2 \\lambda _\\text{I} \\left(1+4 \\epsilon ^2\\right) \\Phi(0)\\left( \\Phi(0)^2 -v_\\text{I}^2\\right)=0,~~~\\\\\n k e^{2 \\sigma(\\pi) } \\left[(\\nu +2) e^{\\nu \\sigma(\\pi)} \\left(1+2\\epsilon ^2\\frac{\\nu-2}{\\nu +1} e^{2 \\sigma(\\pi)}\\right)A -(\\nu-2) e^{ - \\nu \\sigma(\\pi) } \\left(1+2\\epsilon ^2\\frac{\\nu +2}{\\nu -1} e^{2 \\sigma(\\pi) }\\right) B \\right]\\nn\\\\\n +2 \\lambda _v \\left(1+4 \\epsilon ^2 e^{2 \\sigma(\\pi)}\\right)\\Phi(\\pi )\\left(\\Phi(\\pi)^2 -v_v^2\\right) =0.~~~\n \\eeqn \n Rather than solving the above equations, we employ the trick of Ref.~\\cite{Goldberger1999a} to simplify the calculation. By substituting the approximate solution back into the action and integrating over $\\theta$, it yields the effective potential of compactified radius $r_c$, \n \\beqn\n V_\\Phi(r_c)&\\approx& k A^2 (\\nu +2) e^{2 \\nu k r_c \\pi } \\left(1-\\frac{8 \\epsilon^2 e^{2 \\pi k r_c}}{\\nu +1}\\right)+\\lambda _\\text{I} \\left( \\Phi(0)^2 -v_\\text{I}^2\\right)^2\\nn\\\\\n&& +e^{-4 k r_c \\phi }\\lambda _\\text{II} \\left(1+4 \\epsilon ^2 e^{2 \\pi k r_c}\\right)\\left( \\Phi(\\pi)^2 -v_\\text{II}^2\\right)^2,\n\\label{Potential_Phi}\n \\eeqn\n where the limit of $e^{k r_c \\pi}\\gg 1$ has been used in the calculation. Assuming that the interaction parameters $\\lambda_\\text{I}$ and $\\lambda_\\text{II}$ are very large \\cite{Goldberger1999a}, this effective potential implies the solution $\\Phi(0)=v_\\text{I}$ and $\\Phi(\\pi)=v_\\text{II}$. Then, the integration constants $A$ and $B$ can be solved approximately as \n \\beqn\n A&\\approx & v_\\text{II} e^{-(\\nu +2) \\sigma(\\pi) } \\left[1-2\\epsilon^2\\frac{\\nu +2}{\\nu +1} e^{2 \\sigma(\\pi) } \\right]-v_\\text{I} e^{-2 \\nu \\sigma(\\pi) } \\left[1+4\\epsilon^2\\frac{ \\nu }{\\nu ^2-1} e^{2 \\sigma(\\pi) }\\right],\\\\\n B&\\approx & v_\\text{I} \\left[1+2\\epsilon^2\\frac{\\nu +2}{\\nu +1} \\left(e^{\\nu \\sigma(\\pi) }-e^{2 \\sigma(\\pi) }\\right)\\right]-v_\\text{II} e^{-(\\nu +2) \\sigma(\\pi) } \\left[1+2\\epsilon^2\\frac{\\nu +2}{\\nu +1} e^{\\nu \\sigma(\\pi) }\\right].\n \\eeqn\nFurther, assuming that the mass of the scalar filed is a small quantity, i.e., ${m_\\Phi}\/{k}\\ll 1$, so that $\\nu =\\sqrt{4+\\frac{m_\\Phi^2}{k^2}}\\simeq 2+\\delta$ with $\\delta=\\frac{m_\\Phi^2}{4 k^2}$ a tiny quantity. Then the effective potential can be rewritten as\n\\beqn\nV_\\Phi(r_c)&\\approx& (4+\\delta) k e^{-4 k r_c \\pi}\\left(1-\\frac{8 \\epsilon ^2e^{2 k r_c \\pi } }{\\delta +3}\\right) \\nn\\\\\n&&\\times \\left[\\left(1+\\frac{4 (\\delta +2) \\epsilon ^2e^{2 k r_c \\pi } }{\\delta ^2+4 \\delta+3 }\\right)e^{-\\delta k r_c \\pi} v_\\text{I} -\\left(1-\\frac{2 (\\delta +4) \\epsilon ^2e^{2k r_c \\pi }}{\\delta +3}\\right)v_\\text{II} \\right]^2.\n\\eeqn \nThus the effective potential has a minimum at \n\\beq\nr_c \\approx \\frac{1}{\\pi k\\delta }\\ln\\left(\\frac{v_h}{v_v}\\right)-\\frac{4\\epsilon ^2}{3\\pi k} \\left(\\frac{v_h}{v_v}\\right)^{\\fc{2}{\\delta }}\\left(1+\\frac{v_v}{v_h}\\right).\n\\eeq\nThe first leading term is just the result of GW mechanism in general relativity \\cite{Goldberger1999a}, and the second term proportional to $\\epsilon^2$ is a tiny correction stemming from the 5D graviton mass. The mechanism provides a dynamical way to stabilize the compactified radius of the extra dimension without introducing another large hierarchy. For instance, if $m_\\Phi\/k=0.1$ and ${v_h}\/{v_v}=1.34$, then one obtains $ky_\\pi=k r_c\\pi\\approx 37$ to generate a proper hierarchy. After the radius stabilization, the radion acquires a mass roughly $\\mathcal{O}(\\delta^2)$ TeV \\cite{Goldberger2000}, which is somewhat smaller than the TeV scale. \n\n \n\\section{Conclusions}\\label{Conclusions}\n\n\nIn this work, we generalized the RS1 model in a 5D extension of the Lorentz-violating massive gravity. It is found that the theory supports two distinct brane configurations. The configuration possessing both positive and negative tension branes is similar to the RS1 model, while the other possessing only two positive tension branes is distinct from RS1 model. The full linear perturbations against the background metric were also analyzed. It is found that the models are free from the ghost and tachyonic instabilities, and all KK particles are massive. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=9cm]{Mass_spectra.eps}\n\\end{center}\n\\caption{The mass spectra of RS1 model and current model, where the blue lines refer to the tensor modes, red lines to vector modes, and green lines to scalar modes.}\n\\label{Mass_Spectra}\n\\end{figure}\n\nAs shown in Fig.~\\ref{Mass_Spectra}, the tensor and scalar modes have similar mass spectra which start from $\\sqrt{2}m$ with a mass splitting of TeV scale, nevertheless, the lightest vector mode does not exist in the mass spectrum. The ground states of tensor and scalar modes propagate only along the brane. The graviton mass $\\sqrt{2}m$ has to be tiny enough to fit the experimental constraints. However, the light weight radion would lead to a fifth force to violate experimental observations, so the GW mechanism was considered to stabilize the size of the extra dimension and to weight the mass of radion. After radius stabilization, the 4D effective theory on the brane includes a nearly massless graviton plus three towers of non-pathologic very massive spin-2, spin-1 and spin-0 particles. \n\nThe gauge hierarchy problem was also solved as an application of the model. Furthermore, due to the very distinct KK mass spectra between current model and RS1 model, the new KK towers of vector and scalar modes introduce some new reaction channels and suggest some new signals in colliders. This is potentially an interesting property of current model. However, it is beyond the scope of current work and left for our future consideration. \n\n\n\n\n\\section*{ACKNOWLEDGMENTS}\n\nWe would like to especially thank Prof.~Yu-Xiao Liu for a very helpful discussion of our paper. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12005174 and 12165013. K. Yang acknowledges the support of Natural Science Foundation of Chongqing, China under Grant No. cstc2020jcyj-msxmX0370.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}