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+{"text":"\\section{Introduction}\n\nDynamical chiral symmetry breaking and its partial \nrestoration in finite density systems is one of the important subjects of \nhadron physics. Recently, spectroscopy of \ndeeply bound pionic atom of Sn~\\cite{Suzuki:2002ae} and \nlow-energy pion-nucleus scattering~\\cite{Friedman:2004jh},  \nwith helps of theoretical analyses~\\cite{Kolomeitsev:2002gc}, \nhave suggested that the partial restoration does take place in nuclei \nwith order of 30\\% reduction of the quark condensate. \nThe reduction of the quark condensate in nuclear medium also leads to\nvarious phenomena, for instance, \nattractive enhancement of scalar-isoscalar $\\pi\\pi$ \ncorrelation in nuclei\nand\nthe suppression of the mass difference between the chiral partners.\nMass reduction of the $\\eta^{\\prime}$ meson\nis also induced by partial restoration of chiral symmetry~\\cite{Jido:2011pq}.\nThe experimental observations of these phenomena, such as\nthe reduction of the $N$-$N(1535)$\nmass difference in the $\\eta$ mesonic \nnuclei formation~\\cite{Jido:2002yb},\ncan be further confirmation \nof partial restoration of chiral symmetry in\nnucleus.\n\n\n\n\\section{$\\eta^{\\prime}$ mass under chiral symmetry restoration}\n\nExperimentally, a strong mass reduction of $\\eta'$ ($\\gtrsim 200$ MeV)\nhas been reported in Ref.~\\cite{Csorgo:2009pa} at RHIC.  On the other\nhand, a small scattering length ($\\sim 0.1$ fm) has been suggested in\nRef.~\\cite{Moskal:2000pu} which indicates small mass reduction around 10\nMeV at normal saturation density in the linear density approximation.  \nThe transparency ratio of the $\\eta^{\\prime}$ meson in nuclei\nhas suggested the absorption \nwidth of the $\\eta^{\\prime}$ \nmeson in nuclei is around 30 MeV~\\cite{NanovaTalk}. \nTheoretically, NJL model calculations suggested around 200 MeV \nmass reduction\nat the saturation \ndensity~\\cite{Costa:2002gk,Nagahiro:2006dr}. In the instanton \npicture, rapid decrease of the effects of instantons in finite energy \ndensity hadronic matter induces a reduction of the $\\eta^{\\prime}$\nmass~\\cite{Kapusta:1995ww}.\nAn effective model which is consistent to the $\\eta' p$ scattering\nlength data~\\cite{Moskal:2000pu} was also proposed\nrecently~\\cite{Oset:2010ub}.  \n\nThe basic idea of the present work is that, if density dependence of\nthe U(1)$_{A}$ anomaly is moderate, a relatively\nlarge mass reduction of the $\\eta^{\\prime}$ meson is expected \nat nuclear density due to the partial restoration of chiral symmetry~\\cite{Jido:2011pq}.\nThis is based on the following symmetry argument. \nBoth the flavor single and octet pseudoscalar mesons composed of \na $\\bar q$-$q$ pair belong to the same \n$(\\bf{3},\\bf{\\bar 3})\\oplus (\\bf{\\bar 3},\\bf{3})$ \nchiral multiplet of the SU(3)$_{L}\\otimes$SU(3)$_{R}$ group. Therefore,\nwhen the SU(3)$_{L}\\otimes$SU(3)$_{R}$ chiral symmetry is manifest,\nthe flavor singlet and octet mesons should degenerate,\nno matter how the U(1)$_{A}$ anomaly effect depends on the density.\nIn other words, the chiral singlet gluonic current, which makes the \n$\\eta^{\\prime}$ mass lift up, cannot couple to the chiral pseudoscalar state \nwithout breaking chiral symmetry.\nHence, the $\\eta$ and $\\eta^{\\prime}$ mass splitting can take place \nonly with (dynamical and\/or explicit) chiral symmetry breaking, meaning that  \nthe U(1)$_{A}$ anomaly effect does push the $\\eta^{\\prime}$ mass up \nbut necessarily with the chiral symmetry breaking.\nIn this way the mass splitting of the $\\eta$-$\\eta^{\\prime}$ mesons is a \nconsequence of the interplay of the U(1)$_{A}$ anomaly effect and the \nchiral symmetry breaking. \nAssuming 30\\% reduction of the quark condensate in nuclear medium, for instance,\nand that the mass difference of $\\eta$ and $\\eta^{\\prime}$ comes \nfrom the quark condensate linearly, one could expect an order of 100 MeV \nattraction for the $\\eta^{\\prime}$ meson coming from partial restoration \nof chiral symmetry in nuclear medium. \n\nThe present mechanism of the $\\eta^{\\prime}$ mass reduction in finite \ndensity has a unique feature. \nAlthough some many-body effects introduce an absorptive potential\nfor the $\\eta^{\\prime}$ meson in medium,  \nthe mass reduction mechanism does not involve hadronic intermediate \nstates and, thus, the attraction dose not accompany an additional imaginary part. \nFurthermore, in the present case, since the suppression of the U(1)$_{A}$ \nanomaly effect in nuclear medium induces the attractive interaction, \nthe influence acts selectively on the $\\eta^{\\prime}$ meson and, thus, \nit does not induce inelastic transitions of the $\\eta^{\\prime}$ meson into \nlighter mesons in nuclear medium.  \nConsequently \nthe $\\eta^{\\prime}$ meson bound state may have a smaller width\nthan the binding energy. \n\n\n\\section{Formation spectrum of the $\\eta^{\\prime}$ mesonic nuclei}\n\nNow we discuss the $\\eta^{\\prime}$ bound states in a nucleus \nbased on the above observation and show expected spectra\nof the $\\eta^{\\prime}$ mesonic nucleus formation in a \n$^{12}$C($\\pi^{+},p)^{11}$C$\\otimes\\eta^{\\prime}$ \nreaction~\\cite{Jido:2011pq,Nagahiro:2010zz}. \nWe perform a simple estimation of the $\\eta^{\\prime}$  \nbound states and, thus, assume a phenomenological optical potential of \nthe $\\eta^{\\prime}$ meson in nuclei as\n$\n    V_{\\eta^{\\prime}}(r) = V_{0} \\rho(r)\/\\rho_{0}, \n$    \nwith the Woods-Saxon type density distribution $\\rho(r)$ for nucleus and \nthe saturation density $\\rho_0=0.17$ fm$^{-3}$. \nThe depth of the attractive potential is an order of 100 MeV at the normal nuclear \ndensity as discussed above and the absorption width is\nexpected to be less than 40 MeV~\\cite{NanovaTalk} which\ncorresponds to the 20 MeV imaginary part of the optical potential. \nThe formation spectrum is calculated in the approach developed  \nin Ref.~\\cite{Jido:2002yb,Nagahiro:2005gf}\nusing the impulse approximation and the Green's function method.\n\n\\begin{figure}\n   \\includegraphics[width=0.95\\linewidth]{green_fig.eps}\n\\caption{{Calculated spectra of the\n $^{12}$C($\\pi^+,p)^{11}$C$\\otimes\\eta'$ at $p_\\pi=1.8$ GeV as functions\n of the exitation energy $E_{\\rm ex}$ with (a) $V_0=-(0+20i)$ MeV, (b)\n $V_0=-(100+20i)$ MeV and (c) $V_0=-(150+20i)$ MeV.  The thick solid lines\n show the total spectra, and the dominant subcomponents are labeled\n by the neutron-hole state $(n\\ell_j)_n^{-1}$ and the $\\eta'$ state $\\ell_{\\eta'}$.\n}}\n\\label{fig:spec}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:spec}, we show the\ncalculated $^{12}$C$(\\pi^+,p)^{11}$C$\\otimes\\eta'$ cross\nsections with three different potential parameters. \nIn the figure, the vertical line \nindicates \nthe $\\eta^{\\prime}$ production threshold in vacuum. \nIn the case of no attractive potential, there is no structure in the \n$\\eta^{\\prime}$-binding  region but some bump in the quasi-free region. \nFinding so prominent peaks in the $\\eta^{\\prime}$-binding region\nas to be possibly observed in future experiments, we conclude that \nwith an order of 100 MeV mass reduction and a 40 MeV absorption width \nat the saturation density we have a chance to observe \nthe $\\eta^{\\prime}$-nucleus bound states in the $^{12}$C$(\\pi^{+},p)$ reaction.\nWe see also clear peaks around the $\\eta^{\\prime}$ production threshold,\nfor instance $(0p_{3\/2})_{n}^{-1}\\otimes d_{\\eta^{\\prime}}$ in plot (b)\nand $(0p_{3\/2})_{n}^{-1}\\otimes f_{\\eta^{\\prime}}$ in plot (c). They are \nnot signals of the bound states, \nhowever,\nthese are \nremnants of the bound states which could be formed if the attraction \nwould be stronger. Therefore, such peak structure also can be \nsignals of the strong attractive potential. \n\n\\section{Conclusion}\nWe point out that partial restoration of chiral symmetry in a nuclear medium \ninduces suppression of the U(1)$_{A}$ anomaly effect to the $\\eta^{\\prime}$ mass.\nConsequently, we expect a large mass reduction of the $\\eta^{\\prime}$ meson \nin nuclear matter with a relatively smaller absorption width. The mass reduction \ncould be observed as $\\eta^{\\prime}$-nucleus bound states in the formation reactions. \nThe interplay between the chiral symmetry restoration \nand the U(1)$_{A}$ anomaly effect can be a clue \nto understand the $\\eta^{\\prime}$ mass generation mechanism. Therefore,\nexperimental observations of the deeply $\\eta^{\\prime}$-nucleus bound states, or \neven confirmation of nonexistence of such deeply bound states,\nis important to solve the U(1)$_{A}$ problem.\n\n\n\n\\acknowledgements{%\nThis work was partially supported by the Grants-in-Aid for Scientific Research (No. 22740161, No. 20540273, and No. 22105510). This work was done in part under the Yukawa International Program for Quark- hadron Sciences (YIPQS).\n}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe separation and control of the electron spins in the two-dimensional electron gas (2DEG) has been a subject of intense investigation in the field of spintronics \\cite{Wolf2001}. \nIn the external magnetic field the focusing \nof the cyclotron trajectories can be detected in a set-up with  quantum point contact (QPC) source and drain terminals \\cite{Sharvin1964, Tsoi1974, Houten1989,Hanson2003,Aidala2007,Dedigama2006, Lo2017, Yan2017,Rokhinson2004, Chesi2011, Rokhinson2006}.\nIn this work we consider the spin-dependent trajectories that could be resolved in the magnetic focusing experiment\n\\cite{Sharvin1964, Tsoi1974, Houten1989,Hanson2003,Aidala2007,\nDedigama2006, Lo2017, Yan2017,Rokhinson2004, Chesi2011, Rokhinson2006}\nby the scanning gate microscopy \\cite{Sellier2011}. \nThe focusing of electron  trajectories for carriers injected across the QPC with spins \nseparated by spin-orbit interaction (SOI)\nwas considered theoretically \\cite{Usaj2004, Zulicke2007, Reynoso2007, Schliemann2008, Reynoso2008b, Kormanyos2010, Bladwell2015} and \nstudied experimentally \\cite{Rokhinson2004, Dedigama2006, Chesi2011, Lo2017, Yan2017, Yan2018}.\nThe spin separation by the strong spin-orbit interaction is achieved by splitting the magnetic focusing peaks with the orthogonal spin polarization for electrons that pass across the quantum point contacts.\nThe spin-orbit coupling alone in the absence of the external magnetic field has also been proposed\nfor the spin-separation in InGaAs QPCs \\cite{Kohda2012} and \nin U- \\cite{Zeng2012} or Y-shaped \\cite{Gupta2014} junctions of topological insulators.\n However, for strong spin-orbit coupling the electron spin precesses in the \neffective momentum-dependent spin-orbit magnetic field \\cite{Meier2007,Reynoso2008} that is oriented within the plane of confinement of the carrier gas.\nIn this work we indicate a possibility of imaging the spin-resolved electron trajectories\nfor which the electron spin is fixed and the spin-precession in the spin-orbit field is frozen\nby strong Zeeman effect due to an in-plane magnetic field. \n  For that purpose instead of the spin-orbit coupling \\cite{Usaj2004, Schliemann2008, Kormanyos2010,Rokhinson2004, Dedigama2006, Chesi2011, Lo2017, Yan2017}  we use an in-plane magnetic field \\cite{Watson2003,Li2012,Yan2018a} component\nthat introduces the spin-dependence of the cyclotron trajectories by the Zeeman splitting. \nWe demonstrate that for the indium antimonide -- a  large Land\\'e factor material --the spin-dependent electron trajectories can be clearly resolved by the scanning gate microscopy technique. \n\nIn the focusing experiments with the 2DEG the electrons are injected and gathered by QPCs \\cite{Wharam1988, Wees1988, Wees1991}. The constrictions formed in 2DEG by electrostatic gates depleting the electron gas lead to the formation of transverse quantized modes. By applying sufficiently high potential on the gates only one or few modes can adiabatically pass through the QPC. The quantized plateaus of conductance of such constrictions have been recently reported in InSb \\cite{Qu2016}.\n\nThe scanning gate microscopy (SGM) is an experimental technique in which a charged tip of atomic force microscope is raster-scanned over a sample while measuring the conductance \\cite{Sellier2011}. The tip acts as a movable gate that can locally deplete the 2DEG, with a possible effect on the conductance. \nThe SGM technique has been used in 2DEG confined in III-V nanostructures for example to image the branching of the current trajectories in systems with QPC and the interference of electrons backscattered between the tip and the QPC \\cite{LeRoy2005, Jura2009, Paradiso2010, Brun2014}, the scarred wave functions in quantum billiards \\cite{Crook2003, Burke2010}, \nand electron cyclotron trajectories \\cite{Aidala2007, Crook2010}. It has been used for imaging the cyclotron motion also in two-dimensional materials like graphene \\cite{Morikawa2015, Bhandari2016}. \n\n\n\n\n\n\\section{Model and theory}\n \nWe consider quantum transport at the Fermi level in 2DEG confined within an InSb quantum well.\nThe model system depicted in Fig.~\\ref{system} contains two QPCs on the left-hand side, and is open on the right-hand side. The electrostatically defined two quantum point contacts are separated by a distance $L$. The terminals are numbered as indicated in Fig.~\\ref{system}. The electrons entering from the lead 1 are injected trough the first (lower) QPC into the system in a narrow beam that is steered by the transverse magnetic field. Whenever the cyclotron diameter (or its integer multiple) fits the separation $L$, electrons can enter the second QPC which serves as a collector. \nElectrons that do not get to the collector exit the system through the\nlead 3, which is used as open boundary conditions. Hard wall boundary conditions are introduced on the perpendicular edges of the computational box. The size of the computational box (width $W=2400$ nm and length 1800 nm) \nis large enough to make the effects of the scattering by the hard wall boundaries negligible for the drain (lead 2) currents. \n\n \n\\begin{figure}[tb!]\n \\includegraphics[width=\\columnwidth]{scheme_nr.pdf}\n  \\caption{The scheme of the focusing system.\nThe dark blue shaded area is the gate-induced potential defining the two QPCs, separated by the distance $L$. The spin up (spin down) is parallel (antiparallel) to the total magnetic field. Due to the in-plane magnetic field (and hence Zeeman splitting) the spin-up and spin-down electrons have different momenta and get spatially separated due to difference in the cyclotron radii. The red and blue arrows correspond to spin-up and spin-down electron trajectories, respectively. The gray rectangles indicate the open boundary conditions. The terminals are numbered by integers from 1 to 3. Terminal 1 (2) is the source (drain) of the currents. Terminal 3 plays a role of an open boundary. \n  } \\label{system}\n\\end{figure}\n\nFor the transport modeling, we\nassume that the vertical confinement in the InSb quantum well\nis strong enough to justify the two-dimensional approximation for\nthe electron motion. The 2D effective mass Hamiltonian reads\n\\begin{eqnarray}\nH=& \\left[\\frac{\\hbar^2}{2m_{eff}}\\mathbf{k}^2 + eV(\\mathbf{r}) \\right]\\mathbf{1} +\\frac{1}{2}\\mu_B \\boldsymbol{B}^T \\boldsymbol{g}^* \\boldsymbol{\\sigma} +H_{SO}, \n\\label{eq:dh}\n\\end{eqnarray}\nwhere $\\mathbf{k}=-i\\boldsymbol{\\nabla}-e\\mathbf{A}$, with $\\mathbf{A}$ being the vector potential, $\\mathbf{B}=(B_x,B_y,B_z)$, $\\boldsymbol{\\sigma}$ is the vector of Pauli matrices, $\\mu_B$ is the Bohr  magneton, $\\mathbf{g}^*$ is the diagonal Land\\'e tensor, and $m_{eff}$ is the electron effective mass in InSb. \n\n\n \n\\begin{figure}[tb!]\n \\includegraphics[width=0.6\\columnwidth]{gates.pdf}\n  \\caption{The scheme of the gates inducing the potential of the two QPCs. \nThe figure is not to scale. The values of the geometrical parameters are:  $l=300$ nm, $r=500$ nm, $b_1=-600$ nm, $t_1=547$ nm, $t_2=652$ nm, $b_2=1747$ nm, $b_3=1852$ nm, and $t_3=3000$ nm.}.\n   \\label{gates}\n\\end{figure}\n\n\nThe external potential as seen by the Fermi level electrons is a superposition of the QPC and the potential induced by the charged SGM tip \n\\begin{equation}\nV(\\mathbf{r}) = V_{QPC}(\\mathbf{r})+V_{tip}(\\mathbf{r})  , \n\\label{eq:Vext}\n\\end{equation}\nwhere we model the QPC using the analytical formulas developed in \\cite{Davies1995} with electrostatic potential of a finite rectangular gate given by \n\\begin{eqnarray}\n\\begin{aligned}\n V_{r}(\\mathbf{r};l,r,b,t)=\\\\\n V_g &\\left[g( x-l,y-b  ) + g( x-l,t-y ) \\right. \\\\\n +& \\left. g( r-x,y-b ) +g( r-x,t-y)  \\right],\n \\end{aligned}\n\\end{eqnarray}\nwhere $g(u,v) = \\tfrac{1}{2\\pi} \\arctan\\left( \\tfrac{uv}{u^2+v^2+d^2} \\right)$ with $d=50$ nm, and $V_g$ is the potential applied to the gates. The QPC potential is a superposition of potentials of three such gates\n\\begin{equation}\n V_{QPC} = V_{r}(\\mathbf{r};l,r,b_1,t_1) + V_{r}(\\mathbf{r};l,r,b_2,t_2) + V_{r}(\\mathbf{r};l,r,b_3,t_3).\n \\label{eq:qpc_gates}\n\\end{equation}\n The gates and their labeling used in Eq.~(\\ref{eq:qpc_gates}) are schematically shown in Fig.~\\ref{gates}. \nThe splitting of the gates is $d_{QPC}=105$ nm defining the QPC width. The QPCs are separated by $L=1200$ nm. \n\nFor modeling the tip potential we use a Gaussian profile\n\\begin{equation}\n V_{tip}(\\mathbf{r})= V_t \\exp \\left[ -\\frac{(x-x_{tip})^2+(y-y_{tip})^2}{d_{tip}^2} \\right],\n\\end{equation}\nwith $V_{t}$ being the maximum tip potential, $d_{tip}$ its width, and $x_{tip}$, $y_{tip}$ the coordinates of the tip. \n\n\nThe spin-orbit interactions in InSb are strong, so we include them in the calculations.\nThe two last terms in (\\ref{eq:dh}) account for the SOI with $H_{SO} =H_{R} + H_{D}$, where \n\\begin{equation}\nH_{R} = \\alpha( - k_x\\sigma_y + k_y\\sigma_x )  \n\\label{eq:HR}\n\\end{equation}\ndescribes the Rashba interaction, and \n\\begin{equation}\nH_{D} = \\beta( k_x\\sigma_x - k_y\\sigma_y )  \n\\label{eq:HD}\n\\end{equation}\nthe Dresselhaus interaction. \nFor the Hamiltonian (\\ref{eq:dh}) we use the parameters for InSb quantum well, $\\alpha=-0.051$ eV\\AA, $\\beta=0.032$ eV\\AA, $g^*_{zz}=-51$ \\cite{Gilbertson2008}, $g^*_{xx}=\\tfrac{1}{2}g^*_{zz}$ \\cite{Qu2016}, $m_{eff}=0.018m_0$ \\cite{Qu2016}.\n\n\nWe perform the transport calculations in the finite difference formalism. For evaluation of the transmission probability, we use the wave function matching (WFM) technique \\cite{Kolacha}. The transmission probability from the input lead to mode $m$ with spin $\\sigma$ in the output lead is\n\\begin{equation}\nT^m_\\sigma = \\sum_{ n,\\sigma'} |t^{mn}_{\\sigma\\sigma'}|^2,\n\\label{eq:transprob}\n\\end{equation}\nwhere $t^{mn}_{\\sigma\\sigma'}$ is the probability amplitude for the transmission from the mode $n$ with spin $\\sigma'$ in the input lead to mode $m$ with spin $\\sigma$ in the output lead. \nWe evaluate the conductance as $G={G_0}\\sum_{m, \\sigma} T^{m}_\\sigma$, with $G_0={e^2}\/{h}$.\n\n\nThe considered system presented in Fig.~\\ref{system} has the width $W=2400$ nm, and the narrow leads numbered 1 and 2 have equal width $W'=1146$ nm. The spacing between the centers of the QPCs is $L=1200$ nm. We take the gate potential $V_g=62$ meV, for which at  $E_F=26$ meV\nin the absence of the external magnetic field the QPC conductance is close to $2\\tfrac{e^2}{h}$. For the SGM we use the tip parameters $V_t=260$ meV, and $d_{tip}=60$ nm.\n\n\\section{Results}\n\n\\subsection{No in-plane magnetic field}\n\n\\begin{figure}[tb!]\n \\includegraphics[width=\\columnwidth]{crossBx0.pdf}\n  \\caption{The conductance from the left bottom to the left top lead $G$ as a function of magnetic field and the lower QPC conductance $G_{QPC}$. The inset shows semi-classical trajectories of the electrons for $B_z<0$, and at the three focusing peaks $B_{z}^{(i)}$ with $i$=1,2,3.\n  } \\label{fig:onlyBz}\n\\end{figure}\n\nLet us first consider the transport in the system with the out-of-plane magnetic field only (i.e. $B_x=0$, $B_y=0$, $B_z \\ne 0$). \nIn Fig.~\\ref{fig:onlyBz} we present the conductance $G=G_{21}$ from the lead 1 to lead 2 as a function of the applied transverse magnetic field, and the summed conductance from the lead 1 to the leads 2 and 3, which is essentially the conductance of the lower QPC $G_{QPC}=G_{21}+G_{31}$.\nFor $B_z<0$ no focusing peaks occur because the electrons are deflected in the opposite direction than the collector, propagate along the bottom edge of the system and finally exit through the right lead. For $B_z>0$ conductance peaks almost equidistant in magnetic field appear. The first three maxima occur at $B_z^{(1)}=0.124$ T, $B_z^{(2)}=0.26$ T, $B_z^{(3)}=0.408$ T. Neglecting the SOI terms and the Zeeman term in (\\ref{eq:dh}), one obtains $|k_F|=\\sqrt{2m_{eff}E_F}=0.2148 \\tfrac{1}{\\mathrm{nm}}$. For the cyclotron diameter equal to \n\\begin{equation}\nD_c=\\frac{2\\hbar |k_F|}{|e| B_z}, \n\\label{eq:Dc}\n\\end{equation}\none obtains for the first three peaks $D_c^{(1)}=1176$ nm, $D_c^{(2)}=561$ nm, $D_c^{(3)}=358$ nm, respectively. This is close to the distance between the centers of the QPCs, $L=$1200 nm, its half , $L\/2=$600 nm, and one third, $L\/3=$400 nm, respectively. \nDespite the high spin-orbit interaction in the InSb quantum well, no spin splitting occurs. Let us denote the Fermi wave number of the subband of spin $\\sigma$ by $k_F^{\\sigma}$. For the adapted values of the SO parameters, the difference in momenta for both spins is small [see Fig.~\\ref{fig:disp2dBx}(a)]. For example for $k_{F,y}^{\\uparrow},k_{F,y}^{\\downarrow}=0$ and $E_F=26$ meV, the $x$ components extracted from the dispersion relation in Fig.~\\ref{fig:disp2dBx}(a) are $|k_{F,x}^\\uparrow|=0.11445$ nm$^{-1}$, and $|k_{F,x}^\\downarrow|=0.11142$ nm$^{-1}$, that for $D_c=1200$ nm yield transverse magnetic field $B_z^{(1)}=0.125$ T and $B_z^{(1)}=0.122$ T, respectively. \nThat is clearly too small difference to obtain a visible double peak. \n\n\n\\subsection{Enhancement of the Zeeman splitting with in-plane magnetic field}\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.49\\columnwidth]{Ek0.pdf}\n \\includegraphics[width=0.49\\columnwidth]{Ek4.pdf}\n  \\caption{Dispersion relation of the 2DEG with (a) $B_x=0$ and (b) $B_x=8$ T. The color map shows the dispersion relation of the spin-down band, and the contours show the isoenergetic lines for $E_f=26$ meV for spin up (black line) and spin down (red line) electrons.\n  } \\label{fig:disp2dBx}\n\\end{figure}\n\nIn the next step we apply an additional in-plane magnetic field. This leads to an increase of the Zeeman energy splitting for both spins leading to the increase of the momenta difference between both spin subbands. Fig.~\\ref{fig:disp2dBx} shows the momenta for both spins for $B_x=0$ and 8 T. Without in-plane magnetic field, the spin subbands are nearly degenerate. With $B_x$ of the order of a few tesla the difference in the momenta becomes significant. This induces a change of the cyclotron radii of the electrons with opposite spins. \n\nThe spins are oriented along the total magnetic field, $\\mathbf{B}+\\mathbf{B}_{SO}$, where $\\mathbf{B}_{SO}$ is the effective SO field. For $B_x$ of the order of a few tesla the out-of-plane magnetic field component and the SO effective field are small compared to the in-plane component. The spin is oriented nearly along the $x$ or $-x$ direction. We refer to these states as spin-up and spin-down.\n\n\\begin{figure}[tb!]\n \\includegraphics[width=\\columnwidth]{FocusBx.png}\n  \\caption{Transmission as a function of $B_x$ and $B_z$. The solid (dashed) lines are the analytically calculated positions of transmission peaks maxima for spin up (down) electrons.\n  } \\label{fig:transpBx}\n\\end{figure}\n\nFig.~\\ref{fig:transpBx} shows the conductance $G$ from the lead 1 to the lead 2 as a function of the in-plane (here $B_x$) and the transverse magnetic fields. For sufficiently high in-plane magnetic field the peaks split, with the splitting growing with increasing $B_x$. The lines plotted along the $n$-th pair of split peaks are calculated from the condition $B_{z,\\sigma}^{(n)}\\left(B_x\\right)=\\tfrac{2\\hbar |k_F^\\sigma|}{|e| D_c^{(n)}}$, with  $|k_F^\\sigma|$ obtained from \n\\begin{equation}\n E_F = \\tfrac{(\\hbar k_F^\\sigma)^2}{ 2m_{eff} } \\pm \\tfrac{1}{2}g^*_{xx}\\mu_B B_x,\n\\end{equation}\nwhere $\\sigma=\\uparrow,\\downarrow$, the $\\pm$ sign corresponds to spin down and up, respectively, and $D_c^{(n)}$ are extracted from Fig.~\\ref{fig:onlyBz}, using Eq.~(\\ref{eq:Dc}). Although the analytical lines are obtained neglecting the SOI and the Zeeman energy contribution from the transverse magnetic field, there is a good agreement between the obtained transport results and this simplified model. \n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.99\\columnwidth]{crossBx4.pdf}\n \\includegraphics[width=0.99\\columnwidth]{crossBx4b.pdf}\n  \\caption{(a) The cross section of the conductance and the lower QPC conductance for $B_x=8$ T. (b) The spin resolved conductance.\n   The first peak is split into two smaller peaks with $B_{z,\\downarrow}^{(1)}=0.11$ T for spin down electrons and $B_{z,\\uparrow}^{(1)}=0.137$ T for spin up electrons. \n   The inset in (a) shows semi-classical trajectories of the spin-up (red semi-circles) and spin-down (blue semi-circles) electron at the first focusing peak.\n  } \\label{fig:crossBx4}\n\\end{figure}\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.334\\columnwidth]{dysp0.pdf}\n \\includegraphics[width=0.283\\columnwidth]{dysp4.pdf}\n \\includegraphics[width=0.358\\columnwidth]{dysp6.pdf}\n  \\caption{Dispersion relation of an infinite channel with the lateral potential taken\nat the QPC constriction with $V_r=62$ meV, and (a) $B_x=0$ T (b) $B_x=8$ T and (c) 12 T. The color map shows the mean $x$ spin component of the subband. The spin-down subband shifts up in energy upon increasing $B_x$ and finally is raised above the Fermi energy. The opposite occurs for the spin-up electrons -- for increasing $B_x$ more and more subbands are available at the Fermi level.\n  } \\label{fig:dispQpcBx}\n\\end{figure}\n\n The cross section of the summed conductance and the spin-resolved conductance for $B_x=8$ T is shown in Fig.~\\ref{fig:crossBx4}. \nIn the pairs of focusing peaks, the spin-down (spin-up) conductance dominates\nfor the peak at lower (higher)  magnetic field [see Fig.~\\ref{fig:crossBx4}(b)]. \nInterestingly, in each pair of the peaks in Fig.~\\ref{fig:transpBx}, the lower one has smaller transmission than the upper one, and at $B_x\\approx 10$ T vanishes, while the transmission of the upper one slowly increases. The reason for this behavior is the strong Zeeman splitting due to the in-plane magnetic field and the spin-dependent conductance of the QPCs \\cite{Potok2002,Hanson2003}. \nFig.~\\ref{fig:dispQpcBx} shows the dispersion relation of an infinite channel with the lateral potential taken\nat the QPC constriction with applied $B_x=0$, 8 T and 12 T. For $B_x=8$ T at the Fermi level for spin up 3 transverse subbands are available, while for spin down only one. For higher $B_x=12$ T the spin-down subband is raised above the Fermi level, and only spin-up electrons can pass through the QPC. On the other hand, for growing $B_x$, the number of spin-up subbands increases. Thus in the focusing spectrum, the upper peak -- the spin-up peak -- becomes more pronounced, while the lower one -- the spin-down peak -- has lower value of transmission and finally disappears.\n\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.71\\columnwidth]{Dens1InSb.pdf}\n \\includegraphics[width=0.273\\columnwidth]{sx_upInSb.pdf}\n  \\caption{The density and average spins maps for the low-field peak in Fig.~\\ref{fig:crossBx4}. In (b) the average spin $x$ projection for a spin-down mode is shown, and in (c) for a spin-up mode. The spin in the $y$ and $z$ directions is negligibly small (not shown), and the average spin in the $x$ direction is preserved (cf Fig. 13 for the spin precession effects in the case where SOI dominates over the Zeeman interaction).\n  } \\label{fig:densInSb}\n\\end{figure}\n\n\n\nConcluding this section, we find that the in-plane magnetic field allows for a controllable separation of the electrons with opposite spins.\nIt is worth noting that in the systems that have strong SOI, without the in-plane magnetic field, only the odd focusing peaks get split \\cite{Usaj2004, Reynoso2008, Lo2017}, and in case of the in-plane magnetic field all of the peaks are split. This is caused by the spin precession due to SOI in those systems. In our case the spin is determined by the effective magnetic field, which is almost parallel to $x$ direction. Thus the spin in $x$ direction dominates and the fluctuation due to SOI is negligible. It is shown in a representative case of the density and average spins for the low-field focusing peak at $B_{z,\\downarrow}^{(1)}=0.11$ T in Fig.~\\ref{fig:densInSb}. The electron spins are nearly unchanged along the entire path. The $\\left\\langle s_y \\right\\rangle$ and $\\left\\langle s_z \\right\\rangle$ are negligibly small compared to the $\\left\\langle s_x \\right\\rangle$. \n\n\n\\subsection{Scanning gate microscopy of the trajectories}\n\n\\begin{figure}[tb!]\n\\begin{center}\n \\includegraphics[width=0.9\\columnwidth]{Pik1.pdf}\n \\includegraphics[width=0.9\\columnwidth]{Pik1up.pdf}\n \\includegraphics[width=0.9\\columnwidth]{Pik1down.pdf}\n\\end{center}\n  \\caption{The conductance maps for the spin-down (left column) and spin-up (right column) focusing peak in Fig.~\\ref{fig:crossBx4} at $B_{z,\\downarrow}^{(1)}=0.11$ T and $B_{z,\\uparrow}^{(1)}=0.137$ T, respectively. (a,b) the conductance summed over spins, (c,d) the spin-up conductance, (e,f) the spin-down conductance. The dashed semicircles show the semi-classical trajectory of an electron incident from the QPC with $k_x\\ne 0$ only. The tiny arrows in the upper right corner show which contribution of $\\Delta G$ is shown in the plot.\n  } \\label{fig:sgm1}\n\\end{figure}\nWe simulated the SGM conductance maps for the magnetic fields that correspond to the peaks of magnetic focusing in the absence of the tip. We used $B_x=8$ T. In the cross section for $B_x=8$ T in Fig.~\\ref{fig:crossBx4}(a) the dots show where the SGM scans were taken. Fig.~\\ref{fig:sgm1} presents the maps of \n$\\Delta G = G\\left(\\mathbf{r}_{tip}\\right) - G\\left(B_{z,\\sigma}^{(1)}\\right)$, and the spin-resolved conductances $\\Delta G_{\\sigma'} = G_{\\sigma'}\\left(\\mathbf{r}_{tip}\\right) - G_{\\sigma'}\\left(B_{z,\\sigma}^{(1)}\\right)$ with $\\sigma,\\sigma'=\\uparrow,\\downarrow$. The conductance maps exhibit semicircular pattern with a pronounced minimum along the semi-classical orbit of a carrier incident in the $x$ direction (indicated in Fig.~\\ref{fig:sgm1} with dashed semi-circles). For the spin-up focusing peak at $B_{z,\\downarrow}^{(1)}=0.11$ T, the scan [Fig.~\\ref{fig:sgm1}(a)] is slightly different than for the spin-down peak at $B_{z,\\uparrow}^{(1)}=0.137$ T [Fig.~\\ref{fig:sgm1}(b)]. In the first one there is a slight increase of conductance to the right of the dashed semi-circle [see the red blob in Fig.~\\ref{fig:sgm1}(a)]. Fig.~\\ref{fig:sgm1}(c,d) show the spin-up conductance, and Fig.~\\ref{fig:sgm1}(e,f) the spin-down conductance as a function of the tip position. One can see that in the spin-down peak (for $B_{z,\\downarrow}^{(1)}=0.11$ T) the $\\Delta G_{\\downarrow}$ is everywhere negative or zero [Fig.~\\ref{fig:sgm1}(e)], and $\\Delta G_{\\uparrow}$ -- positive or zero almost everywhere (except within the QPC) [Fig.~\\ref{fig:sgm1}(c)]. Examples of electron densities with the tip placed in two different points are shown in Fig.~\\ref{fig:densSGM}. In Fig.~\\ref{fig:densSGM}(a), the tip, when placed along the electron trajectory leads to the deflection of the beam and blocks the spin-down beam, preventing it from entering the collector. \nOn the other hand, in Fig.~\\ref{fig:densSGM}(b), the tip can deflect the beam of spin-up electrons into the collector.\n\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.49\\columnwidth]{Dens1min.pdf}\n \\includegraphics[width=0.49\\columnwidth]{Dens1max.pdf}\n  \\caption{The density maps for the tip placed in the points marked with diamonds in Fig.~\\ref{fig:sgm1}(c,e). (a) The tip blocking the beam with the tip at the point marked with green diamond in Fig.~\\ref{fig:sgm1}(c). (b) The tip enabling the spin-up beam to enter the collector  with the tip at the point marked with green diamond in Fig.~\\ref{fig:sgm1}(e).\n  } \\label{fig:densSGM}\n\\end{figure}\n\nThe situation is inverted in the peak at $B_z=0.137$ T. In the $\\Delta G_{\\uparrow}$ map [Fig.~\\ref{fig:sgm1}(d)], the values are smaller or equal to zero, and in the $\\Delta G_{\\downarrow}$ map [Fig.~\\ref{fig:sgm1}(f)], bigger or equal zero. In this case, the spin-up beam is blocked by the tip, thus $\\Delta G_{\\uparrow}$ drops along the semi-circle marked in Fig.~\\ref{fig:sgm1}(d). On the other hand, the spin-down electrons have a smaller cyclotron diameter [than the QPC spacing $L$], but they can be scattered by the tip to the collector, which leads to an increase of $\\Delta G$ at some points to the left of (or along) the dashed semi-circle.\n\n\n\n\\subsection{Magnetic focusing for heavy holes in GaAs\/AlGaAs heterostructure }\n\nWe consider  an experiment conducted for two-dimensional hole gas (2DHG) in GaAs\/AlGaAs, in Ref.~\\onlinecite{Rokhinson2004}, where the splitting of the first focusing peak was visible without an in-plane magnetic field, and was solely due to the spin-orbit interaction. For this problem we assume the distance between the two QPCs $L=800$ nm, the computational box of width $W=1608$ nm and length $3000$ nm, the QPC defined in the same manner as in Eq.~\\ref{eq:qpc_gates} with the geometrical parameters:  $l=500$ nm, $r=1100$ nm, $b_1=-600$ nm, $t_1=336$ nm, $t_2=468$ nm, $b_2=1140$ nm, $b_3=1272$ nm, $t_3=2208$ nm, and $d$=20 nm. We employ the effective mass of heavy holes $m_{eff}=0.17 m_e$ \\cite{Plaut1988}, Land\\'e factor $g_{zz}^*=-0.6$ \\cite{Arora2013},\nthe Dresselhaus SO parameter $\\beta=0.0477 $ eV{\\AA } \\cite{Rokhinson2004}, and zero Rashba SO. \n\nWe tune the lower QPC to $G_{QPC}=2e^2\/h$, with $V_g=18$ meV, and $E_F=3.2$ meV. Figure \\ref{fig:crossHole} shows the focusing conductance of the system. The focusing peaks are resolved, with the first peak split by 35 mT, remarkably close to the result in Ref.~\\onlinecite{Rokhinson2004}, with the measured splitting of 36 mT. The splitting is due to the Dresselhaus SOI, which leads to the spin-polarization in the direction dependent on the hole momentum, and the difference in the Fermi wavenumbers $k_F$ of the holes with opposite spins. The band structure in the injector QPC is shown in Fig. ~\\ref{fig:dispQpc_hole}. The hole spin in the injector QPC is in the $x$ direction.\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.99\\columnwidth]{crossBx0_holes_spins.pdf}\n  \\caption{(a) The summed and spin resolved conductance for a hole system in GaAs\/AlGaAs system. \n   The first peak is split into two smaller peaks with $B_{z,\\downarrow}^{(1)}=0.187$ T for spin down holes and \n   $B_{z,\\uparrow}^{(1)}=0.222$ T for spin up holes. \n    The peak splitting is 35 mT. \n  } \\label{fig:crossHole}\n\\end{figure}\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.5\\columnwidth]{dysp0_hole.pdf}\n  \\caption{Dispersion relation of an infinite channel with the lateral potential taken\nat the QPC constriction with $V_r=18$ meV, and $B_x=0$ T. The color map shows the mean $x$ spin component of the subbands. \n  } \\label{fig:dispQpc_hole}\n\\end{figure}\n\nThe difference in focusing magnetic field due to SOI can be evaluated by: \n\\begin{equation}\n B_{z,\\sigma}^{(1)} = \\frac{2\\hbar k_F^{\\sigma}  }{ e D_c^{(1)} } = \\frac{ \\sqrt{2m_{eff} E_F} \\mp m_{eff}\\beta\\hbar }{e D_c^{(1)} }.\n\\label{eq:dressBz}\n\\end{equation}\n\nThe density and the spin evolution in the peaks highlighted in Fig.~\\ref{fig:crossHole} by tiny triangles is shown in Fig.~\\ref{fig:densHole}. In the densities [Fig.~\\ref{fig:densHole}(a,d)] the contributions of both spins with slightly different cyclotron radii are visible. In the averaged spin $x$ component maps for the mode injected with spin up [Fig.~\\ref{fig:densHole}(b,e)] the precession is visible, but a little blurred due to the scattering from the gates' potential. For the mode injected with spin down [Fig.~\\ref{fig:densHole}(c,f)] the flip of the spin direction in the detector is clearly visible.\n\n\\begin{figure}[tb!]\n \\includegraphics[width=0.71\\columnwidth]{Dens1a.pdf}\n \\includegraphics[width=0.275\\columnwidth]{sx_a_up.pdf}\n \\includegraphics[width=0.71\\columnwidth]{Dens1b.pdf}\n \\includegraphics[width=0.275\\columnwidth]{sx_b_up.pdf}\n  \\caption{The density and average spin $x$ component maps for the focusing marked peaks in Fig.~\\ref{fig:crossHole}, for the low-field peak marked with a red triangle (upper row) and high-field peak marked with a blue triangle (lower row). (a) and (b) The densities, (b) and (e) the average spin for the injected spin-up mode, and (c) and (f) for the injected spin-down mode. The flip of the spin direction in the detector QPC is visible. \n  } \\label{fig:densHole}\n\\end{figure}\n\n\n\\section{Summary and Conclusions}\nWe have studied the spatial spin-splitting of the electron trajectories\n in the transverse focusing system.\nWe demonstrated that the in-plane magnetic field of a few tesla in \nInSb induces the Zeeman splitting which is large enough to \nseparate the conductance focusing peaks for the spin-down and spin-up\nFermi levels. The orientation of the spin is translated to the \nposition of the conductance peak on the magnetic field scale. \nThe focused trajectories for both the spin orientations can be resolved by\nthe scanning gate microscopy conductance maps. Moreover,\nthe SGM maps for opposite spin peaks contain qualitative differences \ndue to the spin dependence of the cyclotron radii. \nThe present finding \npaves the way for studies of the spin-dependent trajectories in the \nsystems with the two-dimensional electron gas with high Land\\'e factor materials. \n\n\\section*{Acknowledgments}\nThis work was supported by the National Science Centre (NCN) Grant No. DEC-2015\/17\/N\/ST3\/02266,\nand by AGH UST budget with the subsidy of the Ministry of\nScience and Higher Education, Poland with Grant No. 15.11.220.718\/6 for young researchers \nand Statutory Task No. 11.11.220.01\/2.\nThe calculations were performed on PL-Grid and ACK CYFRONET AGH Infrastructure. \n\n\n\n\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nScheduling is one of the fundamental research subjects, which is\ncentral to virtually all scientific domains that require any kind of\nresource sharing. Therefore, a large body of literature exists that\nintroduces basic scheduling algorithms for various scheduling\nproblems~\\cite{Blazewicz2019,brucker2004,Drozdowski09,handbook2004,pinedo2016}.\nMost theoretical works in scheduling research use the three-field\nnotation $\\alpha|\\beta|\\gamma$ of \\citet{graham1979optimization} for\nclassifying scheduling problems.  By using this notation, each\nscheduling problem can be described by the machine environment\n$\\alpha$, the job characteristics $\\beta$, and the optimality\ncriterion $\\gamma$. For example, \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace} defines the\nproblem of scheduling jobs on identical parallel machines, where the\nmaximum completion time (\\ensuremath{C_{\\text{max}}}\\xspace) should be minimized, while no further\njob characteristics are given. One example of such job characteristics\ncould be the \\emph{moldable} job model (denoted as $any$), i.e.\\xspace, in the\nproblem\n\\Igraham{P}{any}{\\ensuremath{C_{\\text{max}}}\\xspace}~\\cite{Drozdowski09,DBLP:journals\/concurrency\/Hunold15}. In\nthe moldable model, each job can not only be executed on \\num{1}\nmachine, but it may be allotted to several machines (between \\num{1}\nand $m$ machines). The number of machines is selected by the\nscheduler, but this number of machines will not change until a job has\nbeen completed. Another example of job characteristics are job\nprocessing times that are variable and depend on environmental factors\nsuch as the position $r$ of a job in a schedule. For example, in the\n\\Igraham{P}{in-tree, $p_{j,r} = \\varphi(r)$}{\\ensuremath{C_{\\text{max}}}\\xspace} problem, the\nprocessing times of jobs with in-tree precedence constraints are\ndescribed by the $\\varphi$ function\n\\cite{Przybylski2017,Przybylski2018a}.\n\n\nSince scheduling problems are so fundamental to many scientific\ndisciplines, thousands of algorithms exist for a seemingly endless\nlist of problem variations. Among them, a significant number of\nscheduling algorithms can be described in the three-field notation.\nSeveral surveys on specific scheduling problems, e.g.\\xspace,\n\\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}, have been conducted that compare the scheduling\nperformance of different algorithms via simulations. Although these\nstudies are very informative for the readers, they are often hard to\nreproduce, as many of the building blocks are imprecisely explained\nand because the source code is often not provided or has become\ninaccessible over the years. For that reason, many algorithms cannot\nbe compared fairly or in a scientifically sound manner, as too many\ndetails are missing.\n\nTo overcome this problem, we propose \\texttt{Scheduling.jl}\\xspace, which provides a\ngeneric and open scheduling platform, on top of which a large number\nof scheduling algorithms can be implemented.\n\nIn the remainder of the article, we introduce the core functionalities\nof the \\texttt{Scheduling.jl}\\xspace package and show an example of how to use them.\n\n\\section{Overview of \\texttt{Scheduling.jl}\\xspace}\n\n\\texttt{Scheduling.jl}\\xspace provides the main building blocks for implementing scheduling\nalgorithms in their most generic form, which are \\texttt{Job}\\xspace,\n\\texttt{Machine}\\xspace, \\texttt{JobAssignment}\\xspace, and \\texttt{Schedule}\\xspace.  A classical\n\\texttt{Job}\\xspace $J_j$ is defined by its processing time $p_j$ but can also be\ncharacterized by a weight $w_j$, a release date $r_j$, a due date\n$d_j$, or a deadline $\\bar{d_j}$. A \\texttt{Machine}\\xspace $M_i$ is mainly\ndefined by its speed. Of course, the sets of parameters used can be easily extended. The task of a scheduling algorithm is to find an\nassignment of jobs to machines, such that a given criterion is\noptimized. An assignment of jobs to machines defines the starting and\nthe completion time of a job $J_j$ on a machine $M_i$. The final\n\\texttt{Schedule}\\xspace is composed of a vector of jobs, a vector of machines,\nand a vector of job to machine assignments. It is worth noticing that \\texttt{Scheduling.jl}\\xspace is designed to operate on exact values (rational numbers) rather than inexact ones (floating point numbers).\n\nOnce a schedule has been obtained by executing an algorithm, the\npackage \\texttt{Scheduling.jl}\\xspace provides different optimization criteria that can be\ncomputed for a schedule, e.g., the makespan \\ensuremath{C_{\\text{max}}}\\xspace, the average\ncompletion time $\\sum_j C_j$, or the number of tardy jobs\n$\\sum_j U_j$.  Additionally, the package is shipped with\nimplementations of various scheduling algorithms, in particular, for\n\\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}.\n\nIn order to obtain \\texttt{Scheduling.jl}\\xspace, one needs to install the package from\n\\url{https:\/\/github.com\/bprzybylski\/Scheduling.jl}.  The stable\nversion can be installed by calling \\verb|Pkg.add(\"Scheduling\")| and\nthe development version by executing \\verb|Pkg.develop(\"Scheduling\")|.\n\nListing\\xspace~\\ref{lst:example} presents an example of how to leverage the\nbasic functionality of \\texttt{Scheduling.jl}\\xspace. Here, we create a set of jobs $J$ and\na set of machines $M$. Then, we can apply the LPT algorithm to obtain\na schedule.  On this schedule, we can compute various metrics like\n\\ensuremath{C_{\\text{max}}}\\xspace or $\\sum_j C_j$.\n\n\\begin{lstlisting}[float=t,caption={Example of using the basic functionality of \\texttt{Scheduling.jl}\\xspace; applying the LPT algorithm to a small scheduling problem and reporting the \\ensuremath{C_{\\text{max}}}\\xspace and the $\\sum_j C_j$ metrics.},label={lst:example}]\nusing Scheduling\nusing Scheduling.Algorithms\nusing Scheduling.Objectives\n\n# Generate a set of jobs with processing times\nJ = Jobs([27, 19, 19, 4, 48, 38, 29])\n# Generate a set of 4 identical machines\nM = Machines(4)\n# Generate a schedule using LPT list rule\nLPT = Algorithms.lpt(J, M)\nprintln(\"Cmax     = $(Int(cmax(LPT)))\")\nprintln(\"sum(C_j) = $(Int(csum(LPT)))\")\n\\end{lstlisting}\n\n\\texttt{Scheduling.jl}\\xspace also provides means to visualize the resulting schedules. For\nexample, scientists can choose to produce an image of a schedule,\nwhere it is also possible to animate the schedule creation. If\ndesired, the schedule can also be plotted as an TikZ image, which can\ndirectly be inserted into publications.\n\n\n\\section{Using \\texttt{Scheduling.jl}\\xspace: The Case of \\texttt{$P||\\ensuremath{C_{\\text{max}}}\\xspace$}}\n\nNow, we turn our attention to the NP-hard problem\n\\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}, for which \\citet{DBLP:conf\/afips\/Graham72}\ndevised two fundamental approximation algorithms, namely the LIST and\nthe LPT (Largest Processing Time) algorithm.\n\\citet{DBLP:conf\/afips\/Graham72} showed that for any instance of\n\\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}, LIST provides a $2-1\/m$ approximation, while\nLPT improves this bound to\n$4\/3-1\/(3m)$. \\citet{DBLP:journals\/jacm\/HochbaumS87} devised a PTAS\nfor this problem, by developing a dual approximation algorithm to\nsolve \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}, which internally relies on solving a\nbin packing problem.\n\nAlthough our list is far from exhaustive, we discuss several\nheuristics that have been proposed to solve \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}.\n\\citet{DBLP:journals\/cor\/FrancaGLM94} developed the 3-PHASE algorithm\nfor which the authors stated that it ``outperforms alternative\nheuristics'' on their respective test instances. Similarly,\n\\citet{DellAmico08} presented heuristics that are combined to compute\nexact solutions for various instances of \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}.\nLast, \\citet{Ghalami19} presented a parallel implementation of the\nalgorithm of \\citet{DBLP:journals\/jacm\/HochbaumS87}. In each of these\nworks, the authors implemented their own interpretation of existing\nalgorithms.  They also generated their own problem instances and\nreported results for a subset of the instances. Neither the\nimplementations of the algorithms nor the instances (or generators)\nare available anymore.  Such a lack of code and meta-information is a\ncommon problem in many scientific fields when looking at the\nreproducibility of results. Independent researchers, who would like to\ncontinue studying heuristics for \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}, will have to\nstart from scratch and create implementations and instances\nthemselves.\n\nWe have developed \\texttt{Scheduling.jl}\\xspace to improve the reproducibility in the\nscheduling domain. For the problem of \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace},\n\\texttt{Scheduling.jl}\\xspace contains several algorithms that can be used to solve given\ninstances.  Besides heuristics with guarantees (i.e.\\xspace, LIST and LPT),\n\\texttt{Scheduling.jl}\\xspace also contains an implementation of the algorithm of\n\\citet{DBLP:journals\/jacm\/HochbaumS87} as well as an exact algorithm\nthat was presented by \\citet{Drozdowski09}.\n\nIf independent researchers now set out to compare novel heuristics to\nalready established methods, they could simply use the source code\nprovided. Listing\\xspace~\\ref{lst:schedules} exemplifies how different\nalgorithms for one specific instance of a problem can be compared.  In\nthis particular case, we created three different TiKZ files containing\nthe Gantt charts of the schedules. Figure\\xspace~\\ref{fig:pcmax_schedules}\npresents these Gantt charts produced by different algorithms when\nsolving an instance of \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace}.\n\n\\begin{lstlisting}[float=t,caption={Comparing different algorithms for an instance of P$||\\ensuremath{C_{\\text{max}}}\\xspace$.},label={lst:schedules}]\nJ = Jobs([3, 3, 4, 5, 8, 5, 5, 7, 8, 9, 13, 8, 11, 7])\nM = Machines(4)\n\nS_LPT = Algorithms.lpt(J, M)\nS_OPT = Algorithms.P__Cmax_IP(J, M)\nS_HS  = Algorithms.P__Cmax_HS(J, M; eps=1\/\/10)\nScheduling.TeX(S_LPT, \"schedule_lpt.tex\")\nScheduling.TeX(S_OPT, \"schedule_opt.tex\")\nScheduling.TeX(S_HS, \"schedule_hs.tex\")\n\\end{lstlisting}\n\n\\begin{figure*}[t]\n  \\centering\n  \\begin{subfigure}[t]{.7\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig1}\n    \\subcaption{\\label{fig:alg_hs}Hochbaum\\,\\&\\,Shmoys algorithm}\n  \\end{subfigure}\\\\[1ex]\n  \\begin{subfigure}[t]{.7\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig2}\n    \\subcaption{\\label{fig:alg_lpt}LPT algorithm}\n  \\end{subfigure}\\\\[1ex]\n  \\begin{subfigure}[t]{.7\\linewidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig3}\n    \\subcaption{\\label{fig:alg_opt}OPT (exact)}\n  \\end{subfigure}\n  \\caption{\\label{fig:pcmax_schedules}Comparing schedules produced by\n    different algorithms for P$||\\ensuremath{C_{\\text{max}}}\\xspace$, where Gantt charts of\n    schedules have been created with \\texttt{Scheduling.jl}\\xspace.}\n\\end{figure*}\n\n\n\\section{Conclusions and Future Work}\n\nWe have introduced the Julia package \\texttt{Scheduling.jl}\\xspace, which is an effort to\nincrease the reproducibility in the scheduling community. By providing\nbasic building blocks for developing scheduling methods as well as\nseveral implementations of well-known scheduling algorithms, \\texttt{Scheduling.jl}\\xspace\ncan serve as a foundation for developing a large variation of\nscheduling algorithms. The package provides easy-to-use plotting\nfunctions to easily obtain Gantt charts of computed schedules.\n\nThe package is far from complete and it should serve as a starting\npoint for future work. So far, we have focused on providing a general\ndevelopment platform for implementing various algorithms. We made sure\nthat design choice are applicable by implementing multiple algorithms\nfor the classical problem of \\Igraham{P}{}{\\ensuremath{C_{\\text{max}}}\\xspace} ourselves.  Now,\nwe hope that the community will contribute new algorithms to this\npackage. We will also continue to integrating more algorithms into\n\\texttt{Scheduling.jl}\\xspace, starting with algorithms for which Julia code already exists,\ne.g., the algorithm of \\citet{DBLP:journals\/tpds\/BleuseHKMMT17} for\n\\Igraham{(P$m$,P$k$)}{mold}{$C_{\\max}$}.\n\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\n\\noindent MB acknowledges support by the Natural Environment Research Council (NERC) under training grant no. NE\/L002515\/1. PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under grants no. EP\/M006883\/1 and EP\/N014529\/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048 and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). PD is on leave from CNRS, Institut de Math\\'ematiques de Toulouse, France. MTW  acknowledges  partial  support  from  the  Austrian  Academy  of\nSciences under the New Frontier's grant NST-001 and the EPSRC under the First Grant EP\/P01240X\/1. \n\n\\section*{AMS subject classification}\n35Q84, 35Q91, 35J15, 35J57, 91A13, 91A23, 49N70, 34C60, 37M05\n\n\\section*{Key words}\nMean field games, best reply strategy, stationary Fokker-Planck equation\n\n\\section*{Data statement: no new data were collected in the course of this research.}\n\n\\section{Abstract}\n\\noindent Mean field games (MFGs) and the best reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system, see ~\\cite{Barker,Bertucci2019,Degond2017}. In this paper we present a systematic analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in a variety of modelling situations.\n\n\\section{Introduction}\n\n\\noindent Mean field games (MFGs) describe the dynamics of large interacting agent systems, in which individuals determine their optimal strategy by minimising a given cost functional. The extensive current literature is based on the original work of Lasry and Lions~\\cite{Lasry2006,Lasry2006a,Lasry2007} and Huang, Caines and Malham\\'e~\\cite{Huang2007a,Huang2006,Huang2006b}. MFGs have been used successfully in many different disciplines. A good overview is presented by Caines, Huang and Malham\\'e in~\\cite{Caines2017}, a detailed probabilistic approach by Carmona and Delarue in~\\cite{Delarue2018,Delarue2018a}.\n\nMFGs can be formulated as parabolic optimal control problems (under certain conditions on the cost). This connection can be used to construct approximations to MFGs. Degond, Liu and Ringhofer proposed a so-called best reply strategy (BRS) in~\\cite{Degond2014b}. It can be derived from the corresponding explicit in time discretisation of the respective optimal control problem, as in~\\cite{Barker,Degond2017}. More recently it has been derived in~\\cite{Bertucci2019} through considering a discounted optimal control problem and taking the discount factor to $\\infty$. Specifically, in~\\cite{Barker,Degond2017} the limit $\\Delta t\\rightarrow 0$ in the case of the following cost functional \n\\[ J^{\\Delta t}(\\alpha;m) = \\mathbb{E} \\left[ \\int_t^{t + \\Delta t} \\left(\\frac{\\alpha_s^2}{2} + \\frac{1}{\\Delta t} h(X_s,m(X_s))\\right)~ ds \\right] \\, , \\]\nis analysed. In~\\cite{Bertucci2019} the limiting behavior of MFG systems for cost functionals\n\\[ J^\\rho(\\alpha;m) = \\mathbb{E} \\left[ \\int_0^T \\left( \\frac{\\alpha_s^2}{2} + h(X_s,m(X_s)) \\right) e^{- \\rho s}~ds \\right] \\, , \\]\nas the temporal discount factor $\\rho$ tends to infinity is considered. In both cases the resulting dynamics depend instantaneously on the cost function $h$. Hence agents do not anticipate future dynamics in the respective limits, as they do in MFG approaches.\n\nAs far as the authors are aware, no systematic analysis and comparison of the two approaches has been done yet. However, the BRS is computationally less expensive and therefore more attractive in applications. Therefore it is important to understand under which circumstances the use of each model is appropriate and whether there are situations where the two models are comparable. This paper is a first step analysing the similarities and differences between the two models in a systematic way.\n\nThe existence and uniqueness of solutions to stationary MFGs has been studied extensively in previous literature ( c.f.~\\cite{Cardaliaguet2015,Ferreira2018,Gomes2017,Lasry2007}). However, apart from a small number of papers e.g.~\\cite{Cirant2015,Ferreira2019}, almost all results focus on problems posed on the torus in order avoid dealing with boundary conditions. In~\\cite{Benamou2017} the Dirichlet problem was motivated as a stopping time problem, and it was analysed in~\\cite{Ferreira2019}. In this paper we consider Neumann boundary conditions, which relate to a no-flux boundary. The only other paper we are aware of that deals with such a situation is~\\cite{Cirant2015}. In this paper the authors prove existence of solutions to the MFG problem with non-local dependence on the distribution using a Schauder fixed point argument. They then perturb the solutions to prove existence in the case of a local dependence on the distribution. Other typical methods of proof use continuation methods~\\cite{Evangelista2018,Ferreira2018,Gomes2014}, Schauder's fixed point theorem~\\cite{Boccardo2016,Cirant2016} or variational approaches through energy minimisation problems~\\cite{Cesaroni2018,Evangelista2018a}. In our proof we exploit the linear-quadratic nature of the control. This was done in the time-dependent case in~\\cite{Gueant2012} where the problem was reduced to a forward-backward system of heat equations, but we don't think our method has been considered in the stationary case. Our result sits nicely alongside the only other result for Neumann boundary conditions~\\cite{Cirant2015}. On the one hand the Hamiltonian used in~\\cite{Cirant2015} is more general than ours, however the regularity assumptions and the form of nonlinearity $h$ required in~\\cite{Cirant2015} is relaxed in our case.\n\nDue to assumption \\ref{a:hincrease}, which states that the running cost $h$ is an increasing function of density, we are in the setting of monotone stationary MFGs. Existence and uniqueness of such MFGs has been studied extensively by Gomes and collaborators in a number of papers e.g~\\cite{Evangelista2018a,Gomes2017,Gomes2016}. Although the setting in these papers focusses on domains with periodic boundary conditions, it is worth mentioning the types of techniques used and how they compare to the method in this paper. In \\cite{Gomes2017} a Hopf--Cole transformation is used to prove existence and uniqueness of minimisers of an energy functional related to a specific case of an MFG with periodic boundary conditions and a cost $h$ that is logarithmic in the density. The concepts used in our existence and uniqueness proof are similar to those used in \\cite{Gomes2017}, though we are able to generalise the density dependence and consider Neumann boundary value problems. In \\cite{Gomes2014} the results of \\cite{Gomes2017} are extended using a continuation method. There were further improved in \\cite{Ferreira2018} where a combination of a continuation method and Minty's method is used. In both cases the methods allow the authors to perturb a problem for which existence and uniqueness is known to prove existence and uniqueness of the problem of interest. The methods used there and in many subsequent works (e.g. \\cite{Ferreira2019}) rely on monotonicity properties of the operators. In our work presented here monotonicity similarly plays a central role in proving existence and uniqueness --- through both the use of the maximum principle to prove existence and uniqueness for strictly increasing functions $h$ and through the ability to uniformly perturb an increasing function into a strictly increasing function through the addition of a logarithmic congestion term.\n\nOur non-linear stationary BRS model~\\eqref{eq:brssystem} is an example of a stationary non-linear Fokker-Planck equation. Existence and uniqueness of solutions to non-linear Fokker-Planck equations have been studied extensively (see for example~\\cite{Carrillo2019,Carrillo2006} and references therein). Many results (e.g. in~\\cite{Carrillo2019,Chayes2010,Tugaut2014}) focus on non-local non-linear terms i.e. they consider Fokker-Planck equations of the form\n\\begin{equation} \\label{eq:intro-nonloc-fp}\n    - \\frac{\\sigma^2}{2} \\nabla^2 m - \\nabla \\cdot (m \\nabla W*m) = 0 \\, ,\n\\end{equation}\nwith suitable boundary conditions. Here $\\nabla W*m = \\int_{\\Omega} \\nabla W(x - y) m(y)~dy$ is the usual convolution operator. For our model, we consider a local function of density $h = h(x,m(x))$, rather than a convolution term. In this case there are a number of results of existence and uniqueness of solutions to the stationary model, as well as convergence of the dynamic model to the stationary version, see e.g.~\\cite{Biane2001,Carrillo2003,Carrillo2006,McCann1997}. These papers all consider the term $h$ to be either independent of $x$ i.e. $h = h(m)$, or of the form $h = h_1(x) + h_2(m)$. So our result extends this case to more general local functions of the density. In previous literature the proof for the local case, as in~\\cite{McCann1997}, relies on a related energy functional, for which minimisers can be proven to exist and be unique. Then these minimisers are also solutions to the Fokker-Planck equation. Our result takes a different approach, one that is more closely related to the case with a convolution term, as in~\\cite{Carrillo2019}. For the non-local case~\\eqref{eq:intro-nonloc-fp} it has been frequently shown (c.f.~\\cite{Carrillo2019,Tamura1984}) that solutions of the PDE are equivalent to fixed points of a non-linear map \n\\[ T(m) = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} W*m} \\, , \\quad \\text{where } Z = \\int_{\\Omega} e^{- \\frac{2}{\\sigma^2} W*m}~dx \\, . \\]\nWe approach existence and uniqueness of solutions to our PDE~\\eqref{eq:brssystem} in a similar vein, considering solutions to the implicit equation~\\eqref{eq:xu_brs_sys1}. While the proof in~\\cite{Carrillo2019} relies on Schauder's fixed point theorem, we are able to take advantage of $h$ being a local function of density so instead we use the implicit function theorem and intermediate value theorem to prove our result. \n\nThis paper is organised as follows. We start by briefly introducing the time-dependent MFG and BRS models in Section~\\ref{sec:dynamic_setup}, following a more detailed derivation presented in~\\cite{Barker,Degond2017}. We then describe how the dynamic problems relate to the stationary case. In Section~\\ref{sec:ex_unique_stat_sol} we present a proof of existence and uniqueness for the stationary BRS and MFG. Both proofs use similar arguments for proving existence and uniqueness, relying on the observation that both models involve a stationary Fokker-Planck equation with integral constraints. In Section~\\ref{sec:quad potential} we describe an explicit solution to the MFG and BRS model. The explicit solution allows us to analyse in which problem specific parameter ranges the solutions are compareable and in which not.  In Section~\\ref{sec:numerical_sim} we illustrate the different behavior of solutions to both models with numerical simulations. Finally, in Section~\\ref{sec:conclusion} we conclude with summarising the implications of the results found, specifically what they tell us about the similarities and differences between the two models. We also briefly comment on future directions for research. \n\n\\subsection{The dynamic MFG problem and the corresponding BRS}\\label{sec:dynamic_setup}\n\nFirst we briefly review the underlying modeling assumptions of MFGs and the respective BRS models. For the ease of presentation we consider a quadratic cost on the control and restrict ourselves to the $d$-dimensional torus $\\T^d$ in the introduction. However we will consider bounded domains with Neumann boundary conditions from Section~\\ref{sec:ex_unique_stat_sol} on. Note that some of the following arguments have not been proven for such a set-up. However it is not unreasonable to assume that the following results extend naturally to the bounded domain case. \nConsider a distribution of agents which is absolutely continuous with respect to the Lebesgue measure. We denote the density of the distribution by a function $m:\\T^d \\to [0,\\infty)$. We take a representative agent, with state $X_t \\in \\T^d$ moving in this distribution according to the following SDE:\n\\[ \\begin{aligned}\n        & dX_t = \\alpha_t dt + \\sigma dB_t \\\\\n        & \\mathcal{L}(X_0) = m_0 \\, ,\n    \\end{aligned} \\]\nwhere $\\mathcal{L}(X_0)$ denotes the law of the random variable $X_0$, $m_0$ is a given initial distribution of all agents, $\\sigma \\in (0,\\infty)$ denotes the size of idiosyncratic noise in the model and $B_t$ is a $d$-dimensional Wiener process. The function $\\alpha_t:[0,T] \\to \\T^d$ is a control chosen by the representative agent as a result of an optimisation problem, from a set of admissable controls $\\alpha \\in \\mathcal{A}$. The representative agent takes the distribution of other agents, $m$, to be given and attempts to optimise the following functional\n\\[ J(\\alpha;m) = \\mathbb{E} \\left[ \\int_0^T \\left(\\frac{\\alpha_s^2}{2} + h(X_s,m(X_s))\\right)~ds \\right] \\, . \\]\nThe functional $J$ consists of a quadratic cost for the control $\\alpha_t$ and a density dependent cost function $h:\\T^d \\times (0,\\infty) \\to \\R$ over a finite time horizon $T \\in (0,\\infty)$. Then the optimal cost trajectory $u(x,t)$ is\n\\[ u(x,t) = \\inf_{\\alpha \\in \\mathcal{A}} \\mathbb{E} \\left[ \\left. \\int_t^T \\left(\\frac{\\alpha_s^2}{2} + h(X_s,m(X_s))\\right)~ds \\right| X_t = x \\right] \\, . \\]\nThe optimal control is given in terms of $u$ by $\\alpha_t^* = - \\nabla u(X_t,t)$. The optimal cost trajectory evolves backwards in time according to\n\\begin{gather*}\n    \\partial_t u = \\frac{\\left| \\nabla u \\right|^2}{2} - h(x,m) - \\frac{\\sigma^2}{2} \\nabla^2 u \\\\\n        u(x,T) = 0 \\, .\n\\end{gather*}\nWe complete the model by assuming all agents act in the same way as the representative agent, and so the backward PDE is coupled to a forward Fokker-Planck PDE describing the evolution of agents. So the full MFG model is given by\n\\begin{subequations} \\label{eq:dynmfg}\n    \\begin{align}\n        & \\partial_t u = \\frac{\\left| \\nabla u \\right|^2}{2} - h(x,m) - \\frac{\\sigma^2}{2} \\nabla^2 u \\\\\n        & \\partial_t m = \\nabla \\cdot \\left[m \\nabla u \\right] + \\frac{\\sigma^2}{2} \\nabla^2 m\\\\\n        & m(x,0) = m_0 \\\\\n        & u(x,T) = 0 \\, . \n    \\end{align}\n\\end{subequations}\nFollowing the approach in~\\cite{Barker,Degond2017}, the corresponding BRS model arises through considering a rescaled cost functional over a short, rolling time horizon\n\\[ J^{\\Delta t}(\\alpha;m) = \\mathbb{E} \\left[ \\left. \\int_t^{t + \\Delta t} \\left( \\frac{\\alpha_s^2}{2} + \\frac{1}{\\Delta t} h(X_s,m(X_s)) \\right) ds \\right| X_t = x \\right] \\, . \\]\nGoing through a similar procedure to the MFG problem, approximating the result up to $O(\\Delta t)$ and taking the limit $\\Delta t \\to 0$, the optimal control is given by $\\alpha_t = - \\left. \\left[ \\nabla h(x,m(x)) \\right] \\right|_{x = X_t}$. Again we complete the model by assuming all agents act in the same way as the representative agent. Then the distribution of agents evolves according to the Fokker-Planck equation\n\\begin{subequations}\\label{eq:brs}\n    \\begin{align}\n        \\partial_t m &= \\nabla \\cdot \\left[ \\left( \\nabla h(x,m(x)) \\right) m  \\right] + \\frac{\\sigma^2}{2} \\nabla^2 m\\\\\n        m(x,0) &= m_0 \\, .\n    \\end{align}\n\\end{subequations}In \\eqref{eq:brs} the dynamics of agents is influenced by the current agent density only. Hence the anticipation behavior, which is characteristic for MFG, is `lost'. Only the current state drives the dynamics. We shall refer to equation~\\eqref{eq:brs} as the BRS strategy in the following.  \n\n\\subsection{From the dynamic problems to the stationary case} \\label{sec:stat_prob}\n\nFor the MFG the interpretation of the stationary problem is slightly subtle because in the dynamic case we are considering a problem set on a fixed time horizon so we cannot simply consider the stationary problem by setting $\\partial_t u,\\partial_t m = 0$ and interpreting it as the long-time behaviour of the dynamic case. Instead we follow the work by Cardaliaguet, Lasry, Lions and Porretta~\\cite{Cardaliaguet2012}. To highlight the dependence of the MFG on the time horizon $T$ we use the notation $\\bar{u}^T,\\bar{m}^T$ for solutions satisfying~\\eqref{eq:dynmfg}. Then we define the rescaled functions $u^T$ and $m^T$ by\n\\[ u^T(x,t) = \\bar{u}^T(x,tT) \\, , \\quad m^T(x,t) = \\bar{m}^T(x,tT) \\, . \\]\nThen Theorem 1.2 in~\\cite{Cardaliaguet2012} states that under some mild assumptions on the data we have that as $T \\to \\infty$:\n\\[ \\begin{aligned}\n    u^T - \\int_{\\T^d} u^T~dy \\to u \\, , \\quad & \\text{in} \\, L^2(\\T^d \\times (0,1)) \\\\\n    \\frac{1}{T} u^T \\to (1 - t) \\lambda \\, , \\quad & \\text{in} \\, L^2(\\T^d \\times (0,1)) \\\\\n    m^T \\to m \\, , \\quad & \\text{in} \\, L^p(\\T^d \\times (0,1)) \\, ,\n\\end{aligned} \\]\nwhere $p$ depends on the space dimension $d$. Then the triple $(m,u,\\lambda) \\in C^2(\\T^d) \\times C^2(\\T^d) \\times \\R$ satisfies the following stationary problem\n\\begin{subequations}\\label{eq:statmfg}\n    \\begin{align} \n        - \\frac{\\sigma^2}{2} \\nabla^2 m - \\nabla \\cdot (m \\nabla u) &= 0 \\\\\n        - \\frac{\\sigma^2}{2} \\nabla^2 u + \\frac{| \\nabla u |^2}{2} - h(x,m) + \\lambda &= 0 \\\\\n        \\int_{\\T^d} m~dx &= 1 \\\\\n        \\int_{\\T^d} u~dx &= 0 \\, .\n    \\end{align}\n\\end{subequations}\nThe corresponding stationary BRS model is obtained by setting $\\partial_t m=0$ in~\\eqref{eq:brs}. Hence we have\n\\begin{subequations} \\label{eq:statbrs}\n    \\begin{align}\n        \\nabla \\cdot \\left[ \\left( \\nabla h(x,m(x)) \\right) m  \\right] + \\frac{\\sigma^2}{2} \\nabla^2 m &= 0 \\\\\n        \\int_{\\T^d} m~dx &= 1 \\, .\n    \\end{align}\n\\end{subequations}\nEquation~\\eqref{eq:statbrs} can be understood as either the long-time behaviour of the dynamic BRS or, under suitable convexity conditions (c.f.~\\cite{McCann1997}), a competitive equilibrium distribution of the following minimisation problem\n\\begin{align}\\label{eq:Estat}\n\\min \\mathbb{E} \\left[ h(X_t,m(X_t)) \\right] \n\\end{align}\nBy competitive equilibrium we mean a stationary distribution $m$ for which $\\mathbb{E} \\left[ h(X,m(X)) \\right]$ is minimised when $\\mathcal{L}(X) = m$.\n\n\\section{Existence and Uniqueness of Stationary Solutions} \\label{sec:ex_unique_stat_sol}\nIn this section we will show that the MFG~\\eqref{eq:statmfg} and the BRS~\\eqref{eq:statbrs} admit unique solutions on a bounded domain $\\Omega$ with no flux boundary conditions. We make the following assumptions on the function ${h(x,m): \\Omega \\times (0,\\infty) \\to \\R}$ and the domain $\\Omega$:\n\\begin{enumerate}[label=(A$_\\arabic*$)]\n\\item  \\label{a:omega} $\\Omega \\subset \\R^d$ is an open bounded set with a $C^{2,\\alpha}$ boundary, for some $\\alpha \\in (0,1)$ and $d \\geq 1$.\n    \\item $h(x,\\cdot)$ is an increasing function for every $x \\in \\Omega$.\\label{a:hincrease}\n    \\item There exists a continuous function $g:(0,\\infty) \\to [0,\\infty)$ such that $\\sup_{x \\in \\Omega} |h(x,m)| \\leq g(m)$ for every $m \\in (0,\\infty)$. \\label{a:hbound}\n\\end{enumerate}\nSince $\\Omega$ is bounded, we can now define $|\\Omega| = \\int_{\\Omega} dx$, where this integral is with respect to the standard Lebesgue measure. Furthermore, we denote the unit outer normal vector by $\\nu$. \\\\\nFor the BRS we further assume:\n\\begin{enumerate}[label=(BRS$_\\arabic*$)]\n    \\item  $h \\in C^2 \\left( \\Omega \\times (0,\\infty) \\right) \\cap C^1 \\left( \\bar{\\Omega} \\times(0,\\infty) \\right)$. \\label{a:brs1}\n    \\item There exists a continuous function $f: (0,\\infty) \\to [0,\\infty)$ such that $\\sup_{x \\in \\Omega} | \\nabla_x h(x,m)| \\leq f(m)$ for every $m \\in (0,\\infty)$. \\label{a:brs2}\n\\end{enumerate}\nWhile for the MFG we will assume:\n\\begin{enumerate}[label=(MFG$_\\arabic*$)]\n    \\item $h \\in C \\left( \\Omega \\times (0,\\infty) \\right)$ \\label{a:mfg_hreg}\n    \\item $\\lim_{m \\to 0} \\sup_{x \\in \\Omega} h(x,m) < \\inf_{x \\in \\Omega} h \\left( x,\\frac{1}{|\\Omega|} \\right)$. \\label{a:mfg_mto0}\n    \\item $\\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{|\\Omega|} \\right) < \\lim_{m \\to \\infty} \\inf_{x \\in \\Omega} h(x,m)$. \\label{a:mfg_mtoinf}\n\\end{enumerate}\n\n\\subsection*{Discussion of assumptions:} Since we are interested in classical solutions (generally in $C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$) the above assumptions on the cost and domain ensure sufficient regularity and boundedness of $h$. It is worth mentioning why we need $h$ to be increasing. This assumption of an increasing function can be related to ``crowd aversion''. When $h$ is increasing in $m$ then areas of high density are more highly penalised than low density areas in the optimisation problem related to the MFG and BRS (see Section~\\ref{sec:dynamic_setup}). This prevents ``accumulation points\" occurring where higher density is preferable and a Dirac delta might be introduced into the solution. As well as being a problem for regularity, the position of the Dirac deltas would be sensitive on the data and so uniqueness could not be guaranteed.\n\nAssumptions~\\ref{a:mfg_mto0} and~\\ref{a:mfg_mtoinf} are also less intuitive than the rest of the assumptions. The MFG problem has two integral constraints related to it. We prove that these constraints can be satisfied using the intermediate value theorem. In doing so we show that two functions $m_1,m_2$ exist such that $m_1(x) \\leq \\frac{1}{|\\Omega|} \\leq m_2(x)$ for every $x \\in \\Omega$. The existence of these functions is guaranteed if assumptions~\\ref{a:mfg_mto0} and~\\ref{a:mfg_mtoinf} hold. As it may not be initially clear what kind of function $h$ satisfies our requirements, some sufficient conditions if $h(x,m) = h_1(x) + h_2(m)$ are:\n\\begin{itemize}\n    \\item $h_1 \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$\n    \\item $h_2 \\in C^2\\left((0,\\infty)\\right)$\n    \\item $h_2$ is increasing\n    \\item $\\lim_{m \\to 0} h_2(m) < h_2\\left( \\frac{1}{|\\Omega|} \\right) + \\inf_{x \\in \\Omega} h_1(x)$\n    \\item $\\lim_{m \\to \\infty} h_2(m) > h_2\\left( \\frac{1}{|\\Omega|} \\right) + \\sup_{x \\in \\Omega} h_1(x)$\n\\end{itemize}\n\n\\subsection{Best Reply Strategy} \\label{section:xu_brs}\nWe start by defining the notion of classical solutions we are aiming for. \n\\begin{definition}\n    Let assumptions~\\ref{a:omega}--\\ref{a:hbound} and~\\ref{a:brs1}--\\ref{a:brs2} be satisfied. Then the stationary BRS boundary value problem is to find a function $m: \\Omega \\to (0,\\infty)$ satisfying\n    \\begin{subequations}\\label{eq:brssystem}\n        \\begin{align}\n            m \\in C^2\\left(\\Omega\\right) \\cap C^1 \\left(\\bar{\\Omega}\\right)&\\\\\n            - \\frac{\\sigma^2}{2} \\nabla^2 m - \\nabla \\cdot (m \\nabla [h(x,m)]) &= 0 \\, , \\quad x \\in \\Omega \\label{eq:xu_brs}\\\\ \n            - \\frac{\\sigma^2}{2} \\nabla m \\cdot \\nu - m \\nabla [h(x,m)] \\cdot \\nu &= 0 \\, , \\quad x \\in \\partial \\Omega \\label{eq:xu_brs_bc} \\\\\n            \\int_{\\Omega} m\\, dx &= 1 \\, . \\label{eq:xu_brs_norm}\n        \\end{align}\n    \\end{subequations}\n\\end{definition}\nThroughout this subsection we will assume~\\ref{a:omega}--\\ref{a:hbound} and~\\ref{a:brs1}--\\ref{a:brs2} hold.\n\n\\begin{lemma} \\label{lm:xu_brs_1}\nFor any $Z \\in (0, \\infty)$ there exists a unique $m_Z: \\Omega \\to (0,\\infty)$ such that \n\\begin{equation} \\label{eq:xu_brs_mz}\n     m_Z(x) = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}h(x,m_Z(x))} \\, .\n\\end{equation}\nFurthermore, $m_Z \\in C^2 \\left( \\Omega \\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$.\n\\end{lemma}\n\\begin{proof}\n    Fix $Z \\in (0,\\infty)$ and $x \\in \\Omega$. Consider $G_{Z,x}: (0,\\infty) \\to \\R$ given by\n    \\[ G_{Z,x}(m) = \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2}h(x,m)} - m \\, . \\]\n    This is a strictly decreasing function of $m$. Furthermore, since $h$ is increasing and continuous with respect to $m$ and $\\sup_{x \\in \\Omega} h(x,m) \\leq g(m)$, we must have $\\lim_{m \\to 0} \\sup_{x \\in \\Omega} h(x,m) \\leq \\sup_{x \\in \\Omega} h(x,1) \\leq g(1) < \\infty$. So we get the following limit inequality as $m \\to 0$, which holds uniformly in $x$:\n    \\[ \\lim_{m \\to 0} G_{Z,x}(m) > 0 \\, . \\]\n    So there exists some $\\epsilon > 0$, independent of $x$, such that $G_{Z,x}(\\epsilon) > 0$. Furthermore, after defining a constant ${C = \\inf_{x \\in \\Omega} h(x,\\epsilon)}$, it is clear that\n    \\[ G_{Z,x}\\left( \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2}C} + \\epsilon \\right) \\leq \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2} h(x,\\epsilon)} - \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2}C} - \\epsilon \\leq - \\epsilon < 0 \\, .\\]\n    Therefore by the intermediate value theorem and strict monotonicity of $G_{Z,x}$ there exists a unique ${m = m_Z(x) > 0}$ such that $G_{Z,x}(m_Z(x)) = 0$. Hence the first result follows. In order to show the regularity requirement that ${m_Z \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)}$, we need to show\n    \\begin{enumerate}\n        \\item $m_Z \\in C^2\\left(\\Omega\\right)$.\n        \\item For any $x \\in \\partial \\Omega$, $\\lim_{y \\to x, \\, y \\in \\Omega} m_Z(y)$ exists.\n        \\item For any $x \\in \\partial \\Omega$, $\\lim_{y \\to x, \\, y \\in \\Omega} \\nabla m_Z(y)$ exists.\n    \\end{enumerate}\n    The assertion that $m_Z \\in C^2(\\Omega)$ follows from the implicit function theorem. For the implicit function theorem to hold we require that $G_{Z,x}(m)$ is a $C^2$ function with respect to $x$ and $m$ at $(x,m_Z(x))$ and that $G_{Z,x}'(m_Z(x)) \\neq 0$. The first requirement is true from our assumption that $h \\in C^2\\left(\\Omega \\times (0,\\infty)\\right)$, the second requirement is true since\n    \\[ G_{Z,x}'(m) = - \\frac{2}{\\sigma^2 Z} \\partial_m h(x,m) e^{- \\frac{2}{\\sigma^2} h(x,m)} - 1 \\leq -1 < 0 \\, . \\]\n    To prove that $\\lim_{y \\to x, \\, y \\in \\Omega} m(y)$ exists for every $x \\in \\partial \\Omega$ it is enough to show $m_Z$ is uniformly Lipschitz in $\\Omega$. Since $m_Z \\in C^2(\\Omega)$ it is therefore enough to show $\\| \\nabla m_Z\\|_{\\infty} < \\infty$. Note that $C, \\epsilon$ defined above are independent of $x$, so we must have $ \\epsilon \\leq \\|m_Z\\|_{\\infty} \\leq \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2}C} + \\epsilon < \\infty$. Then by differentiating the implicit formula for $m_Z$ we get\n    \\begin{equation} \\label{eq:xu_brs_implicit_grad}\n        \\nabla m_Z = \\frac{- 2 m_Z \\nabla_x h(x,m_Z)}{\\sigma^2 + 2 m_Z \\partial_m h(x,m_Z)} \\, .\n    \\end{equation}\n    But $\\partial_m h \\geq 0$ since $h$ is increasing, also $m_Z$ is uniformly bounded as seen above. Similarly, using $f$ from assumption~\\ref{a:brs2} we find $\\|\\nabla_x h(\\cdot,m_Z(\\cdot))\\|_{\\infty} < \\infty$, hence $\\|\\nabla m_Z\\|_{\\infty} < \\infty$.\n    \n    To prove the final assertion that $\\lim_{y \\to x, \\, y \\in \\Omega} \\nabla m_Z(y)$ exists for any $x \\in \\partial \\Omega$, we note the formula for $\\nabla m_Z$ is given by~\\eqref{eq:xu_brs_implicit_grad}. Since $m_Z \\in C^0\\left(\\bar{\\Omega}\\right)$, $\\|m_Z\\|_{\\infty} < \\infty$, $\\nabla_x h, \\partial_m h \\in C^0\\left(\\bar{\\Omega} \\times (0,\\infty)\\right)$ and $\\sigma^2 + 2 m_Z \\partial_m h(x,m_Z) \\geq \\sigma^2$, then the right hand side of~\\eqref{eq:xu_brs_implicit_grad} has limit as $y \\to x$ for every $x \\in \\partial \\Omega$. Hence $\\nabla m_Z$ does as well.\n\\end{proof}\n\n\\begin{definition}\nLet $m_Z$ be given by~\\eqref{eq:xu_brs_mz}. Then we define the following function $\\Phi:(0, \\infty) \\to (0,\\infty)$\n    \\[ \\Phi(Z) = \\int_{\\Omega} m_Z~dx \\, . \\]\n\\end{definition}\n\n\\begin{lemma} \\label{lm:xu_brs_2}\n    There exists $\\bar{Z}, \\underaccent{\\bar}{Z}$ such that $\\Phi\\left(\\bar{Z}\\right) \\geq 1$ and $\\Phi\\left(\\underaccent{\\bar}{Z}\\right) \\leq 1$.\n\\end{lemma}\n\\begin{proof}\n    Take $C_1 = \\inf_{x \\in \\Omega} h \\left( x,\\frac{1}{|\\Omega|} \\right)$. So $C_1 \\in \\left[- g \\left( \\frac{1}{\\Omega} \\right),g \\left( \\frac{1}{\\Omega} \\right) \\right]$. Then, with $\\underaccent{\\bar}{Z} = |\\Omega| e^{- \\frac{2}{\\sigma^2} C_1} \\in (0,\\infty)$, we have\n    \\[ G_{\\underaccent{\\bar}{Z},x} \\left( \\frac{1}{|\\Omega|} \\right) \\leq 0 \\, \\text{ for every} \\, x \\in \\Omega \\, . \\]\n    Hence, $m_{\\underaccent{\\bar}{Z}}(x) \\leq \\frac{1}{|\\Omega|}$ because $G_{Z,x}$ is a strictly decreasing function. So \n    \\[ \\Phi\\left(\\underaccent{\\bar}{Z}\\right) \\leq \\|m_{\\underaccent{\\bar}{Z}}\\|_{\\infty} |\\Omega| \\leq 1 \\, . \\]\n    We can similarly find $\\bar{Z}$ by taking $C_2 = \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{|\\Omega|} \\right)$. So $C_2 \\in \\left[- g \\left( \\frac{1}{\\Omega} \\right),g \\left( \\frac{1}{\\Omega} \\right) \\right]$. Then, with $\\bar{Z} = |\\Omega| e^{- \\frac{2}{\\sigma^2} C_2} \\in (0,\\infty)$, we have\n    \\[ G_{\\bar{Z},x} \\left( \\frac{1}{|\\Omega|} \\right) \\geq 0 \\, \\text{ for every} \\, x \\in \\Omega \\, . \\]\n    Hence, $m_{\\bar{Z}}(x) \\geq \\frac{1}{|\\Omega|}$ because $G_{Z,x}$ is a strictly decreasing function. So \n    \\[ \\Phi\\left(\\bar{Z}\\right) \\geq \\|m_{\\bar{Z}}\\|_{\\infty} |\\Omega| \\geq 1 \\, . \\]\n\\end{proof}\n\n\\begin{lemma} \\label{lm:xu_brs_3}\n    There exists a unique $Z^* \\in (0,\\infty)$ such that $\\Phi\\left(Z^*\\right) = 1$.\n\\end{lemma}\n\n\\begin{proof}\n    If $\\Phi$ is continuous and strictly decreasing the intermediate value theorem and the Lemma~\\ref{lm:xu_brs_2} give the result. We start by proving that $\\Phi$ is strictly decreasing. First note that if $m_Z(x)$ is strictly decreasing in $Z$ for every $x$ then $\\Phi$ must be strictly decreasing because $m_Z$ is continuous with respect to $x$. Take $Z_1 < Z_2$. Then $m_{Z_1}(x)$ satisfies\n    \\[ \\frac{1}{Z_1} e^{-\\frac{2}{\\sigma^2}h(x,m_{Z_1}(x))} - m_{Z_1}(x) = 0 \\, . \\]\n    So\n    \\[ \\frac{1}{Z_2} e^{-\\frac{2}{\\sigma^2}h(x,m_{Z_1}(x))} - m_{Z_1}(x) < \\frac{1}{Z_1} e^{-\\frac{2}{\\sigma^2}h(x,m_{Z_1}(x))} - m_{Z_1}(x) = 0 \\, . \\]\nHence $G_{Z_2,x}(m_{Z_1}(x)) < 0$. Then $m_{Z_2}(x) < m_{Z_1}(x)$ for all $x \\in \\Omega$ since $G_{Z,x}$ is a strictly decreasing function. \nTo show that $\\Phi$ is continuous at $Z \\in (0,\\infty)$, take $\\epsilon < Z' < Z$. Then\n    \\[ \\begin{aligned}\n        |\\Phi(Z) - \\Phi(Z')| & = \\Phi(Z') - \\Phi(Z) = \\int_{\\Omega} m_{Z'}(x)~dx - \\Phi(Z) \\\\\n        & = \\frac{Z}{Z'} \\int_{\\Omega} \\frac{1}{Z} e^{- h(x,m_{Z'}(x))}~dx - \\Phi(Z) \\\\\n        & \\leq \\frac{Z}{Z'} \\int_{\\Omega} \\frac{1}{Z} e^{- h(x,m_Z(x))}~dx - \\Phi(Z) \\leq \\frac{\\Phi(Z)}{\\epsilon} (Z - Z') \\leq \\frac{\\Phi(\\epsilon)}{\\epsilon} (Z - Z') \\, .\n    \\end{aligned} \\]\nBy exchanging $Z$ and $Z'$ we can similarly show the analogous result for $\\epsilon < Z < Z'$, therefore $\\Phi$ is locally Lipschitz and hence continuous.\n\\end{proof}\n\\begin{theorem}\n    There exists a unique solution $m: \\Omega \\to (0,\\infty)$ to the stationary BRS~\\eqref{eq:brssystem}.\n\\end{theorem}\n\\begin{proof}\n    Take $m(x) = m_{Z^*}(x)$, with $Z^*$ defined as in Lemma~\\ref{lm:xu_brs_3}. Then from Lemmas~\\ref{lm:xu_brs_1} and~\\ref{lm:xu_brs_3} we have shown there exists a unique $m: \\Omega \\to (0,\\infty)$ satisfying\n    \\begin{subequations} \\label{eq:xu_brs_sys1}\n        \\begin{align}\n            & m \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right) \\\\\n            & \\text{There exists } Z \\in (0,\\infty) \\text{ such that } m = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}h(x,m)} \\, , \\quad x \\in \\Omega \\\\\n            & \\int_{\\Omega} m~dx = 1 \\, .\n        \\end{align}\n    \\end{subequations}\n    Now for any $m:\\Omega \\to (0,\\infty)$ we can define $\\phi(m)$ by $\\phi(m) = e^{- \\frac{2}{\\sigma^2} h(x,m)}$. Suppose $m$ is a solution to~\\eqref{eq:brssystem}, then $\\frac{m}{\\phi(m)} = m e^{\\frac{2}{\\sigma^2} h(x,m)}$ and so $\\frac{m}{\\phi(m)} \\in H^1\\left(\\Omega\\right)$ because $m \\in C^1\\left(\\bar{\\Omega}\\right)$, $h \\in C^1\\left(\\bar{\\Omega} \\times (0,\\infty)\\right)$ and $h$ is increasing in $m$. Therefore a solution to~\\eqref{eq:brssystem} is equivalent to a solution of\n    \\begin{subequations} \\label{eq:xu_brs_sys2}\n        \\begin{align}\n            m \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)& \\label{eq:xu_brs_sys2_reg1} \\\\\n            \\frac{m}{\\phi(m)} \\in H^1(\\Omega)& \\label{eq:xu_brs_sys2_reg2}\\\\\n            \\nabla \\cdot \\left(\\phi(m) \\nabla \\left( \\frac{m}{\\phi(m)} \\right)\\right) &= 0 \\, , \\quad x \\in \\Omega \\label{eq:xu_brs_sys2_pde}\\\\\n            \\phi(m) \\nabla \\left( \\frac{m}{\\phi(m)} \\right) \\cdot \\nu &= 0 \\, , \\quad x \\in \\partial \\Omega \\label{eq:xu_brs_sys2_bc}\\\\\n            \\int_{\\Omega} m~dx &= 1 \\, .\n        \\end{align}\n    \\end{subequations}\n    Now if we multiply~\\eqref{eq:xu_brs_sys2_pde} by $\\frac{m}{\\phi(m)}$, and integrate over $\\Omega$, then using Green's formula and the boundary condition~\\eqref{eq:xu_brs_sys2_bc} we get\n    \\[ 0 = \\int_{\\Omega} \\frac{m}{\\phi(m)} \\nabla \\cdot \\left(\\phi(m) \\nabla \\left(\\frac{m}{\\phi(m)} \\right)\\right)~dx = - \\int_{\\Omega} \\phi(m) \\left| \\nabla \\left(\\frac{m}{\\phi(m)} \\right) \\right|^2~dx \\, . \\]\n    But $\\phi(m) > 0$ and $\\left| \\nabla \\left(\\frac{m}{\\phi(m)} \\right) \\right|^2 \\geq 0$ for every $x \\in \\Omega$. Hence this is only true if $\\nabla \\left(\\frac{m}{\\phi(m)} \\right) = 0$ for every $x \\in \\Omega$, i.e. if there exists $Z \\in (0,\\infty)$ such that $m = \\frac{1}{Z}\\phi(m)$. Conversely, if $m \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$ and there exists $Z \\in (0,\\infty)$ such that $m = \\frac{1}{Z}\\phi(m)$, then $m$ satisfies~\\eqref{eq:xu_brs_sys2_reg1}--\\eqref{eq:xu_brs_sys2_bc}. So a solution of~\\eqref{eq:xu_brs_sys2} is equivalent to a solution of~\\eqref{eq:xu_brs_sys1}. Therefore the systems~\\eqref{eq:brssystem} and~\\eqref{eq:xu_brs_sys1} are equivalent. Hence we have shown existence and uniqueness of solutions to~\\eqref{eq:xu_brs_sys1} by proving existence and uniqueness of solutions to~\\eqref{eq:brssystem}.\n\\end{proof}\n\n\\subsection{Mean Field Games}\nNext we discuss existence and uniqueness of classical solutions to~\\eqref{eq:statmfg}, which is defined as follows.\n\\begin{definition}\n    The stationary MFG boundary value problem is to find $m:\\Omega \\to (0,\\infty)$, $u:\\Omega \\to \\R$ and $\\lambda \\in \\R$ satisfying the following PDE system\n    \\begin{subequations} \\label{eq:xu_mfg}\n    \\begin{align}\n        m \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)& \\label{eq:xu_mfg_mreg} \\\\\n        u \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)& \\label{eq:xu_mfg_ureg}\\\\\n        - \\frac{\\sigma^2}{2} \\nabla^2 m - \\nabla \\cdot (m \\nabla u) &= 0 \\, , \\quad x \\in \\Omega \\label{eq:xu_mfg_pde1} \\\\\n        - \\frac{\\sigma^2}{2} \\nabla^2 u + \\frac{| \\nabla u |^2}{2} - h(x,m) + \\lambda &= 0 \\, , \\quad x \\in \\Omega \\label{eq:xu_mfg_pde2}\\\\\n            - \\frac{\\sigma^2}{2} \\nabla m \\cdot \\nu &= 0 \\, , \\quad x \\in \\partial \\Omega \\label{eq:xu_mfg_bc1} \\\\\n            - \\nabla u \\cdot \\nu &= 0 \\, , \\quad x \\in \\partial \\Omega \\label{eq:xu_mfg_bc2} \\\\\n            \\int_{\\Omega} m~dx &= 1, \\label{eq:xu_mfg_ic1}\\\\\n            \\int_{\\Omega} u~dx &= 0 \\, . \\label{eq:xu_mfg_ic2}\n    \\end{align}\n\\end{subequations}\n\\end{definition}\n\n\\begin{remark} \\label{remark:xu_mfg_msol}\n    Here, following the method of Section~\\ref{section:xu_brs}, we note that for any $u \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$, a solution $m$ of~\\eqref{eq:xu_mfg_mreg},~\\eqref{eq:xu_mfg_pde1},~\\eqref{eq:xu_mfg_bc1},~\\eqref{eq:xu_mfg_ic1} is equivalent to a solution of\n    \\begin{subequations} \\label{eq:xu_mfg_mvar}\n        \\begin{align}\n            &m \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)\\\\\n            m &= \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u} \\, , \\quad x \\in \\Omega \\\\\n            Z &= \\int_{\\Omega} e^{- \\frac{2}{\\sigma^2} u}~dx \\, . \\label{eq:xu_mfg_mvar_ic}\n        \\end{align}\n    \\end{subequations}\n    Then by the arguments in Section~\\ref{section:xu_brs}, a unique $m$ satisfying~\\eqref{eq:xu_mfg_mvar} exists and is the unique solution to~\\eqref{eq:xu_mfg_mreg},~\\eqref{eq:xu_mfg_pde1},~\\eqref{eq:xu_mfg_bc1},~\\eqref{eq:xu_mfg_ic1}. So from now we only consider that solution.\n\\end{remark}\n\n\n\\begin{proposition} \\label{proposition:xu_mfg_transform}\n    There exists a unique solution $(m,u,\\lambda) \\in \\left[ C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right) \\right] \\times \\left[ C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right) \\right] \\times \\R$ to the stationary MFG boundary value problem if and only if there exists a unique solution $(u,\\lambda,Z) \\in \\left[ C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right) \\right] \\times \\R \\times (0,\\infty)$ to\n    \\begin{subequations}\\label{eq:xu_mfg_sys}\n    \\begin{align}\n        u \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right) \\label{eq:xu_mfg_bvp1}&\\\\\n        - \\frac{\\sigma^2}{2} \\nabla^2 u + \\frac{| \\nabla u |^2}{2} - h\\left( x,\\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u} \\right) + \\lambda &= 0 \\, , \\quad x \\in \\Omega \\label{eq:xu_mfg_bvp2} \\\\\n        - \\nabla u \\cdot \\nu &= 0 \\, , \\quad x \\in \\partial \\Omega \\label{eq:xu_mfg_bvp3} \\\\ \n        \\int_{\\Omega} \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u}~dx &= 1, \\label{eq:xu_mfg_bvp4} \\\\\n        \\int_{\\Omega} u~dx &= 0 \\label{eq:xu_mfg_bvp5} \\, .\n    \\end{align}\n    \\end{subequations}\n\\end{proposition}\n\n\\begin{proof}\n    First assume a unique solution $(m,u,\\lambda)$ to~\\eqref{eq:xu_mfg} exists, then thanks to remark~\\ref{remark:xu_mfg_msol}, we have $m = \\frac{1}{Z} e^{-\\frac{2}{\\sigma^2}u}$, for $Z$ satisfying~\\eqref{eq:xu_mfg_mvar_ic}. Then the triple $(u,\\lambda,Z)$ is clearly a solution to~\\eqref{eq:xu_mfg_sys}. Furthermore, suppose another solution $(u',\\lambda',Z')$ to~\\eqref{eq:xu_mfg_sys} exists. Then $(m',u',\\lambda')$, with $m' = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u'}$, is a solution to~\\eqref{eq:xu_mfg}. But since we assumed such solutions are unique, we have $(m',u',\\lambda') = (m,u,\\lambda)$ and hence $(u',\\lambda',Z') = (u,\\lambda,Z)$, so the solution to~\\eqref{eq:xu_mfg_sys} is unique.\\\\\n    Next we assume that a unique solution $(u,\\lambda,Z)$ to~\\eqref{eq:xu_mfg_sys} exists. Then, defining $m = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u}$, $(m,u,\\lambda)$ is a solution to~\\eqref{eq:xu_mfg}. Now suppose $(m',u',\\lambda')$ is another solution then (again using remark~\\ref{remark:xu_mfg_msol}) $m' = \\frac{1}{Z'} e^{- \\frac{2}{\\sigma^2} u'}$, where $Z'$ satisfies~\\eqref{eq:xu_mfg_mvar_ic}. So $(u',\\lambda',Z')$ satisfies~\\eqref{eq:xu_mfg_sys}. By uniqueness $(u',\\lambda',Z') = (u,\\lambda,Z)$ and so $(m,u,\\lambda)$ is also unique.\n\\end{proof}\n\n\\begin{theorem} \\label{thm:xu_mfg1}\n    There exists a unique solution $(m,u,\\lambda)$ of the MFG system~\\eqref{eq:xu_mfg}.\n\\end{theorem}\n\n\\begin{proof}[Proof (outline)]\n    First note, as a result of Proposition~\\ref{proposition:xu_mfg_transform}, we only need to prove existence and uniqueness of a solution to~\\eqref{eq:xu_mfg_sys} and existence and uniqueness for the MFG system~\\eqref{eq:xu_mfg} will follow. The proof is split into the following steps:\n    \\begin{enumerate}\n        \\item Show that for any pair of constants $(\\lambda,Z)$ there exists a unique solution, denoted by $u_{\\lambda,Z}$, to~\\eqref{eq:xu_mfg_bvp1},~\\eqref{eq:xu_mfg_bvp2}  (see Proposition \\ref{prop:xu_mfg_cont})\n        \\item Show that for any constant $Z$ there exists a unique $\\lambda = \\lambda(Z)$ such that $u_{\\lambda(Z),Z}$ satisfies~\\eqref{eq:xu_mfg_bvp4} (see Proposition \\ref{prop:xu_mfg_lambda})\n        \\item Show that there exists a unique $Z = Z^*$ such that $u_{\\lambda(Z^*),Z^*}$ satisfies~\\eqref{eq:xu_mfg_bvp5}.\n    \\end{enumerate}\n    \n    Then $\\left(u_{\\lambda\\left(Z^*\\right),Z^*},\\lambda\\left(Z^*\\right),Z^*\\right)$ is a solution to~\\eqref{eq:xu_mfg_sys}. Uniqueness follows from uniqueness obtained at each step of the proof outlined. We prove step 1 using a variant of the method of upper and lower solutions in the spirit of~\\cite{Schmitt1978}, so that it applies to our case of Neumann boundary conditions. We prove steps 2 and 3 by iteratively using the intermediate value theorem - in a similar manner to the proof of Lemma~\\ref{lm:xu_brs_3}. Note that we will first do this for $h$ which is strictly increasing in $m$. Then by considering ${h_{\\epsilon}(x,m) = h(x,m) + \\epsilon \\log\\left(|\\Omega| m\\right)}$, and taking the limit as $\\epsilon \\to 0$ we will prove it in the more general setting when $h$ is increasing.\n\\end{proof}\n\n\\begin{lemma} \\label{lm:xu_mfg_constbound}\n    There exists $\\Lambda_1, \\Lambda_2 \\in [- \\infty, \\infty]$ with $\\Lambda_1 < \\Lambda_2$ such that for every $\\lambda \\in \\left( \\Lambda_1, \\Lambda_2 \\right)$ and $Z > 0$, there exist two constants $\\underaccent{\\bar}{u}_{\\lambda,Z} \\leq 0 \\leq \\bar{u}_{\\lambda,Z}$ satisfying \n    \\begin{equation} \\label{eq:xu_mfg_constbound}\n        - h \\left(x, \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} \\underaccent{\\bar}{u}_{\\lambda,Z}} \\right) + \\lambda \\leq 0 \\leq - h \\left(x, \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} \\bar{u}_{\\lambda,Z}} \\right) + \\lambda \\, .\n    \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n     Take $\\Lambda_1 = \\lim_{m \\to 0} \\sup_{x \\in \\Omega} h(x,m)$ and $\\Lambda_2 = \\lim_{m \\to \\infty} \\inf_{x \\in \\Omega} h(x,m)$. First $\\Lambda_1 < \\Lambda_2$ by combining assumptions~\\ref{a:mfg_mto0} and~\\ref{a:mfg_mtoinf}. Then, since $h$ is continuous and increasing in $m$, for any $\\lambda \\in \\left( \\Lambda_1, \\Lambda_2 \\right)$ there exists $M_{\\lambda}^1, M_{\\lambda}^2 \\in (0,\\infty)$ such that $h(x,m) \\leq \\lambda$ for all $(x,m) \\in \\Omega \\times \\left(0,M_{\\lambda}^1\\right]$, and similarly $h(x,m) \\geq \\lambda$ for all $(x,m) \\in \\Omega \\times \\left[M_{\\lambda}^2,\\infty\\right)$. We define the upper and lower constants for $\\lambda \\in \\left( \\Lambda_1, \\Lambda_2 \\right)$ as \n     \\[ \\begin{aligned}\n            \\bar{u}_{\\lambda,Z} &= \\max \\left( - \\frac{\\sigma^2}{2} \\log Z M_{\\lambda}^1, 0 \\right) \\\\\n            \\underaccent{\\bar}{u}_{\\lambda,Z} &= \\min \\left( - \\frac{\\sigma^2}{2} \\log Z M_{\\lambda}^2, 0 \\right) \\, .\n        \\end{aligned} \\]\n    Then clearly \n    \\[ - h \\left( x, \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}\\bar{u}_{\\lambda,Z}} \\right) + \\lambda = - h \\left( x,\\min \\left(M_{\\lambda}^1, 1 \\right) \\right) + \\lambda \\geq - h \\left( x,M_{\\lambda}^1 \\right) + \\lambda \\geq 0 \\, , \\]\n    while the reverse inequality is true for $\\underaccent{\\bar}{u}_{\\lambda,Z}$. Hence $\\bar{u}_{\\lambda,Z}, \\underaccent{\\bar}{u}_{\\lambda,Z}$ are the required upper and lower constants.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:xu_mfg_1}\n    Define $C^{2,\\tau}\\left(\\bar{\\Omega}\\right)$ as the set of functions $u \\in C^2\\left(\\bar{\\Omega}\\right)$ whose second partial derivatives are all H\\\"older continuous with exponent $\\tau$ on $\\bar{\\Omega}$. Assume $h$ is strictly increasing with respect to $m$. Then, for every $\\lambda \\in (\\Lambda_1, \\Lambda_2)$ and every $Z \\in (0,\\infty)$ there exists a unique function, $u_{\\lambda,Z} \\in C^{2,\\tau}\\left(\\bar{\\Omega}\\right) \\subset C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$ for some $\\tau \\in (0,1)$, which satisfies~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3}. Furthermore, $\\underaccent{\\bar}{u}_{\\lambda,Z} \\leq u_{\\lambda,Z} \\leq \\bar{u}_{\\lambda,Z}$.\n\\end{proposition}\n\n\\begin{proof}\n    Existence is an application of Corollary 2.9 in~\\cite{Schmitt1978}, which states that a solution ${u_{\\lambda,Z} \\in C^{2,\\tau}\\left(\\bar{\\Omega}\\right)}$ to~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3} exists provided the following properties hold:\n    \n    \\begin{enumerate}\n        \\item There exist constants $\\underaccent{\\bar}{u}_{\\lambda,Z} \\leq 0 \\leq \\bar{u}_{\\lambda,Z}$ satisfying~\\eqref{eq:xu_mfg_constbound} for every $x \\in \\bar{\\Omega}$.\n        \\item There exists a continuous function $f:[0,\\infty) \\to [0,\\infty)$ such that the following inequality holds for every $(x,u,p) \\in \\bar{\\Omega} \\times \\R \\times \\R^d$\n        \\[ \\left| \\frac{|p|^2}{2} - h \\left( x,\\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}u} \\right) + \\lambda \\right| \\leq f(|u|) \\left( 1 + |p|^2 \\right) \\, . \\]\n    \\end{enumerate}\n    \n    Property 1 is true from Lemma~\\ref{lm:xu_mfg_constbound}. Property 2 can be shown to be true by taking\n    \\[ f(u) = \\max \\left( \\frac{1}{2}, |\\lambda| + \\max \\left[ g \\left( \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}u} \\right), g \\left( \\frac{1}{Z} e^{\\frac{2}{\\sigma^2}u} \\right) \\right] \\right) \\, , \\]\n    where $g$ is defined in assumption~\\ref{a:hbound}.\n    \n    We prove uniqueness using the strong maximum principle and Hopf's Lemma as stated in~\\cite{Evans1998}~(Section~6.4.2.~pp.~330--333). Suppose there are two solutions, $u_1,u_2 \\in C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$ to~\\eqref{eq:xu_mfg_bvp1},~\\eqref{eq:xu_mfg_bvp2}. Define $a = \\nabla (u_1 + u_2)$. Then $a \\in L^{\\infty}\\left(\\bar{\\Omega}\\right)$. Now suppose $u_1 \\neq u_2$ and define $v = u_1 - u_2$. Then $v$ must attain its maximum at some point $\\bar{x} \\in \\bar{\\Omega}$. First suppose $\\bar{x} \\in \\Omega$. Then there exists an open bounded and connected region $V$ such that $V \\subset \\Omega$, $\\bar{x} \\in V$ and $v > 0$ for all $x \\in V$. Hence, since $h(x,\\cdot)$ is increasing, we have\n    \\[ - \\frac{\\sigma^2}{2} \\nabla^2 v + \\frac{1}{2} a \\cdot \\nabla v \\leq 0 \\, , \\quad \\text{for every} \\, x \\in V \\, . \\]\n    So by the strong maximum principle $v$ is constant in $V$. Therefore we must have\n    \\[ h \\left( x,\\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}u_1(x)} \\right) = h \\left( x,\\frac{1}{Z} e^{- \\frac{2}{\\sigma^2}u_2(x)} \\right)  \\quad \\text{for every} \\, x \\in V \\, . \\]\n    So $u_1 = u_2$ in $V$ because $h$ is strictly increasing, which leads to a contradiction. Therefore the only other option is that $\\bar{x} \\in \\partial \\Omega$ and $v(x) < v(\\bar{x})$ for every $x \\in \\Omega$. Hence by Hopf's Lemma (which we can use because $\\partial \\Omega$ is $C^2$) $\\left. \\frac{\\partial v}{\\partial \\nu} \\right|_{\\bar{x}} > 0$, but by the boundary condition~\\eqref{eq:xu_mfg_bvp3}, $\\frac{\\partial v}{\\partial \\nu} = \\frac{\\partial u_1}{\\partial \\nu} - \\frac{\\partial u_2}{\\partial \\nu} = 0$. This again leads to a contradiction. Therefore $u_1 = u_2$, and therefore the solution is unique.\n\\end{proof}\n\n\\begin{remark} \\label{rmk:XU_mfg_r2}\n    It should be noted that the same method to prove uniqueness can be used to prove that $u_{\\lambda_1,Z} \\geq u_{\\lambda_2,Z}$ for all $\\lambda_1 \\leq \\lambda_2$\n\\end{remark}\n\n\\begin{lemma} \\label{lm:xu_mfg_monotone}\n    Assume $h$ is strictly increasing with respect to $m$. Then for every $x \\in \\Omega$, $u_{\\lambda,Z}(x)$ is decreasing with respect to $\\lambda$ and $Z$.\n\\end{lemma}\n\n\\begin{proof}\n    In Light of remark~\\ref{rmk:XU_mfg_r2} we need only to prove $u_{\\lambda,Z_1} \\geq u_{\\lambda,Z_2}$ for all $Z_1 \\leq Z_2$. However, by substitution we find that $u = u_{\\lambda,Z_1} - \\frac{\\sigma^2}{2} \\log \\frac{Z_2}{Z_1}$ satisfies~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3} with $Z = Z_2$. So by uniqueness of solutions to this PDE proved in Proposition~\\ref{prop:xu_mfg_1} we see that\n    \\[ u_{\\lambda,Z_2} = u \\leq u_{\\lambda,Z_1} \\, . \\]\n\\end{proof}\n\n\\begin{proposition} \\label{prop:xu_mfg_cont}\n    Define $\\Phi: (\\Lambda_1, \\Lambda_2) \\times (0,\\infty) \\to L^{\\infty}(\\Omega)$ by\n    \\[ \\Phi(\\lambda,Z) = u_{\\lambda,Z} \\, , \\]\n    where $u_{\\lambda,Z}$ is the unique solution to~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3} as found in the Proposition~\\ref{prop:xu_mfg_1}. Assume $h$ is strictly increasing with respect to $m$. Then $\\Phi$ is continuous (with respect to $L^{\\infty}$ norm).\n\\end{proposition}\n\\begin{proof}\n\n    We will prove $\\Phi$ is sequentially continuous. Let $(\\lambda_n,Z_n)$ be a sequence in $(\\Lambda_1,\\Lambda_2) \\times (0,\\infty)$ that converges to $(\\lambda,Z) \\in (\\Lambda_1,\\Lambda_2) \\times (0,\\infty)$. We consider two sequences: $(\\lambda_{n}^{(i)}, Z_{n}^{(i)})$ for $i = 1,2$, which we use to sandwich our original sequence. We set these sequences with the following conditions \n    \\begin{enumerate}\n        \\item $\\lambda_n^{(1)} = \\inf_{j \\geq n} \\lambda_j$\n        \\item $\\lambda_n^{(2)} = \\sup_{j \\geq n} \\lambda_j$\n        \\item $Z_n^{(1)} = \\inf_{j \\geq n} Z_j$\n        \\item $Z_n^{(2)} = \\sup_{j \\geq n} Z_j$\n    \\end{enumerate}\n    In the first part of this proof we show that for each $i = 1,2$, there exists a subsequence $(\\lambda_{n_k}^{(i)}, Z_{n_k}^{(i)})$ such that $u_{\\lambda_{n_k}^{(i)}, Z_{n_k}^{(i)}} \\to u_{\\lambda, Z}$. We will only show this for $i = 1$ as the case $i = 2$ is identical. Clearly the sequence $(\\lambda_{n}^{(1)}, Z_{n}^{(1)})$ also converges to $(\\lambda,Z)$. So there exists a subsequence $n_k$ such that $u_{\\lambda_{n_k}^{(1)},Z_{n_k}^{(1)}} \\to u_*$ in $C^2(\\Omega) \\cap C^1(\\bar{\\Omega})$ because $u_{\\lambda_{n_k}^{(1)},Z_{n_k}^{(1)}} \\in C^{2,\\tau}(\\bar{\\Omega})$ (by Proposition~\\ref{prop:xu_mfg_1}), which is compactly embedded in $C^2(\\bar{\\Omega}) \\subset C^2(\\Omega) \\cap C^1(\\bar{\\Omega})$. Therefore we also get the following pointwise convergence\n    \\begin{align*}\n            0 & = \\lim_{k \\to \\infty} \\left[-\\frac{\\sigma^2}{2} \\nabla^2 u_{\\lambda_{n_k},Z_{n_k}} + \\frac{1}{2}\\left| \\nabla u_{\\lambda_{n_k},Z_{n_k}}\\right|^2 - h \\left( x, \\frac{1}{Z_{n_k}} e^{- \\frac{2}{\\sigma^2} u_{\\lambda_{n_k},Z_{n_k}}} \\right) + \\lambda_{n_k} \\right]\\\\\n            & = - \\frac{\\sigma^2}{2} \\nabla^2 u_* + \\frac{\\left| \\nabla u_* \\right|^2}{2} - h \\left( x, \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u_*} \\right) + \\lambda\\\\\n         0 &= \\lim_{k \\to \\infty} \\nabla u_{\\lambda_{n_k},Z_{n_k}} \\cdot \\nu |_{x \\in \\partial \\Omega} = \\nabla u_* \\cdot \\nu |_{x \\in \\partial \\Omega} \\, .\n    \\end{align*}\n    So $u_* = u_{\\lambda,Z}$, by uniqueness proved in Proposition~\\ref{prop:xu_mfg_1}. Now by design we have $\\lambda_{n_k}^{(2)} \\geq \\lambda_n \\geq \\lambda_{n_k}^{(1)}$ for all $n \\geq n_k$ and similarly for $Z_n$, hence $u_{\\lambda_{n_k}^{(1)},Z_{n_k}^{(1)}} \\geq u_{\\lambda_n,Z_n} \\geq u_{\\lambda_{n_k}^{(2)},Z_{n_k}^{(2)}}$ by Lemma~\\ref{lm:xu_mfg_monotone}. So $u_{\\lambda_n,Z_n} \\to u_{\\lambda,Z}$ in $L^{\\infty}(\\Omega)$.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:xu_mfg_lambda}\n    For each $Z \\in (0,\\infty)$ define $I_1(\\cdot;Z): (\\Lambda_1,\\Lambda_2) \\to \\R$ by\n    \\[ I_1(\\lambda;Z) = \\int_{\\Omega} u_{\\lambda,Z}~dx = \\int_{\\Omega} \\Phi(\\lambda,Z)~dx \\, . \\]\n    Assume $h$ is strictly increasing with respect to $m$. Then for every $Z \\in (0,\\infty)$ there exists a unique $\\lambda = \\lambda(Z)$ such that $I_1(\\lambda(Z);Z) = 0$, furthermore \n    \\begin{equation} \\label{eq:xu_mfg_lambdabound}\n        \\inf_{x \\in \\Omega} h \\left( x,\\frac{1}{Z} \\right) \\leq \\lambda(Z) \\leq \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z} \\right) \\, .\n    \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n    We use the intermediate value theorem to prove this proposition. There are three parts we have to prove\n    \\begin{enumerate}\n        \\item For every $Z \\in (0,\\infty)$ there exists $\\lambda_1 \\leq \\lambda_2 \\in (\\Lambda_1, \\Lambda_2)$ such that $I_1(\\lambda_1;Z) \\leq 0$ and $I_1(\\lambda_2;Z) \\geq 0$.\n        \\item $I_1(\\lambda;Z)$ is continuous with respect to $\\lambda$ in $[\\lambda_1,\\lambda_2]$.\n        \\item $I_1(\\lambda;Z)$ is strictly decreasing with respect to $\\lambda$.\n    \\end{enumerate}\n    \n    Part (1) and part (2) allow us to use the intermediate value theorem to show that for every $Z \\in (0,\\infty)$ there exists some $\\lambda$ such that $I_1(\\lambda;Z) = 0$. Part (3) shows that this $\\lambda$ is unique, so the function $Z \\mapsto \\lambda(Z)$ is well defined.\n    \n    Part (1): Take $\\lambda_1 = \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z} \\right) > \\Lambda_1$. Then recall that $\\bar{u}_{\\lambda_1, Z} = \\max \\left( - \\frac{\\sigma^2}{2} \\log(Z M_{\\lambda_1}^1), 0 \\right)$, where $M_{\\lambda_1}^1$ satisfies $h \\left( x,M_{\\lambda_1}^1 \\right) \\leq \\lambda_1$. But we can take $M_{\\lambda_1}^1 = \\frac{1}{Z}$ by our choice of $\\lambda_1$. So $u_{\\lambda_1,Z} \\leq \\bar{u}_{\\lambda_1, Z} = 0$, and therefore $I_1(\\lambda_1;Z) \\leq 0$. The choice for $\\lambda_2$ is $\\lambda_2 = \\inf_{x \\in \\Omega} h \\left( x,\\frac{1}{Z} \\right)$ and the proof is similar to the above. \n    \n    Part (2): Take $\\lambda_1,\\lambda_2$ as above. By Propositions~\\ref{prop:xu_mfg_1} and~\\ref{prop:xu_mfg_cont} and Lemma~\\ref{lm:xu_mfg_monotone}, $u_{\\lambda,Z}$ is continuous with respect to $\\lambda$ in $L^{\\infty}(\\Omega)$ and $\\underaccent{\\bar}{u}_{\\lambda_2,Z} \\leq u_{\\lambda,Z} \\leq \\bar{u}_{\\lambda_1,Z}$ for any $\\lambda \\in [\\lambda_1,\\lambda_2]$. So by the dominated convergence theorem $I_1$ is continuous in $\\lambda$.\n    \n    Part (3): Take $\\lambda_1 < \\lambda_2$, from Lemma~\\ref{lm:xu_mfg_monotone} we know $u_{\\lambda_1,Z} \\geq u_{\\lambda_2,Z}$. Clearly, since solutions to the PDE~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3} are unique, there exists $a \\in \\Omega$ such that $u_{\\lambda_1,Z}(a) \\neq u_{\\lambda_2,Z}(a)$. Hence, $u_{\\lambda_1,Z}(a) > u_{\\lambda_2,Z}(a)$ and so by continuity $I_1(\\lambda_1,Z) > I_1(\\lambda_2,Z)$.\n\\end{proof}\n    \n\\begin{remark}\n    This proposition ensures that for any $Z \\in (0,\\infty)$ we can find $\\lambda = \\lambda(Z)$ and $u = u_{\\lambda(Z),Z}$ satisfying~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3},~\\eqref{eq:xu_mfg_bvp5}, so we are left to find $Z^*$ such that~\\eqref{eq:xu_mfg_bvp4} holds.\n\\end{remark}\n\n\\begin{lemma} \\label{lm:xu_lambda_monotone}\n    Assume $h$ is strictly increasing with respect to $m$. Then the function $\\lambda(Z)$ is strictly decreasing.\n\\end{lemma}\n    \n\\begin{proof}\n    From Lemma~\\ref{lm:xu_mfg_monotone}, $u_{\\lambda,Z}$ is strictly decreasing with respect to $Z$. Now suppose $Z_1 < Z_2$ then\n    \\[ 0 = I_1(\\lambda(Z_2),Z_2) = I_1(\\lambda(Z_1),Z_1) > I_1(\\lambda(Z_1),Z_2) \\, . \\]\n    Therefore, since $I_1$ is strictly decreasing in $\\lambda$, $\\lambda(Z_2) < \\lambda(Z_1)$ so $\\lambda(Z)$ is strictly decreasing with respect to $Z$.\n\\end{proof}\n    \n\\begin{lemma}\n    Assume $h$ is strictly increasing with respect to $m$. Then the function $\\lambda(Z)$ is continuous.\n\\end{lemma}\n    \n\\begin{proof}\n    We will prove $\\lambda(Z)$ is sequentially continuous. Let $Z_n$ be a sequence in $(0,\\infty)$ that converges to $Z \\in (0,\\infty)$. We consider two sequences: $Z_{n}^{(i)}$ for $i = 1,2$, which we use to sandwich our original sequence. We choose these sequences as follows\n    \\begin{enumerate}\n        \\item $Z_n^{(1)} = \\inf_{j \\geq n} Z_j$\n        \\item $Z_n^{(2)} = \\sup_{j \\geq n} Z_j$\n    \\end{enumerate}\n    \n    Now, $Z_n^{(1)},Z_n^{(2)} \\to Z$ and are increasing and decreasing sequences respectively. Furthermore, there exists $\\underaccent{\\bar}{Z},\\bar{Z}$ such that $Z_n^{(1)},Z_n^{(2)} \\in [\\underaccent{\\bar}{Z},\\bar{Z}]$ for every $n \\in \\mathbb{N}$. So, since $\\lambda(Z)$ is decreasing, $\\lambda(Z_n^{(1)}),\\lambda(Z_n^{(2)}) \\in [\\lambda(\\bar{Z}),\\lambda(\\underaccent{\\bar}{Z})]$ and are decreasing and increasing respectively. Therefore, $\\lambda(Z_n^{(1)}) \\to \\lambda^{(1)}$ and $\\lambda(Z_n^{(2)}) \\to \\lambda^{(2)}$. Using continuity of $I_1$ we get for $i = 1,2$:\n    \\[ 0 = \\lim_{n \\to \\infty} I_1 \\left( \\lambda(Z_n^{(i)}),Z_n^{(i)} \\right) =  I_1 \\left( \\lambda^{(i)},Z \\right) \\, . \\]\n    Hence by definition $\\lambda^{(1)}  = \\lambda^{(2)} = \\lambda(Z)$. Since $Z_n^{(i)}$ bound $Z_n$, then $\\lambda(Z_n^{(i)})$ bound $\\lambda(Z_n)$. Hence $\\lambda(Z_n) \\to \\lambda(Z)$.\n\\end{proof}\n\n\\begin{proposition} \\label{prop:xu_mfg_Z}\n    Define $I_2:(0,\\infty) \\to (0,\\infty)$ by\n    \\begin{equation}\n        I_2(Z) = \\int_{\\Omega} \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} u_{\\lambda(Z),Z}}~dx \\, .\n    \\end{equation}\n    Assume $h$ is strictly increasing with respect to $m$. Then there exists a unique $Z^* \\in (0,\\infty)$ such that $I_2(Z^*) = 1$.\n\\end{proposition}\n    \n\\begin{proof}\n    Similar to the proof of Proposition~\\ref{prop:xu_mfg_lambda}, we prove this proposition using the intermediate value theorem. Again there are three parts we have to prove\n     \\begin{enumerate}\n        \\item There exists $Z_1 \\leq Z_2 \\in (0, \\infty)$ such that $I_2(Z_1) \\geq 1$ and $I_2(Z_2) \\leq 1$.\n        \\item $I_2(Z)$ is continuous with respect to $Z$ for all $Z \\in [Z_1,Z_2]$.\n        \\item $I_2(Z)$ is strictly decreasing with respect to $Z$.\n    \\end{enumerate}\n    Steps (1) and (2) prove existence via the intermediate value theorem, step (3) proves uniqueness.\n    \n    Step (1): From assumption~\\ref{a:mfg_mto0}, we can find $Z_2 \\geq |\\Omega|$ such that \n    \\begin{equation} \\label{eq:xu_mfg_Zbound}\n        \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z_2} \\right) \\leq \\inf_{x \\in \\Omega} h \\left(x, \\frac{1}{|\\Omega|} \\right) \\, .\n    \\end{equation}\n    Then, since $\\lambda(Z_2) \\leq \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z_2} \\right)$ (from~\\eqref{eq:xu_mfg_lambdabound}), it follows that\n    \\[ u_{\\lambda(Z_2),Z_2} \\geq u_{\\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z_2} \\right), Z_2} \\geq \\underaccent{\\bar}{u}_{\\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z_2} \\right), Z_2} = \\min \\left( - \\frac{\\sigma^2}{2} \\log Z_2 M,0 \\right) \\, , \\]\n    where $M$ satisfies $h(x,M) \\geq \\sup_{x \\in \\Omega} h \\left( x,\\frac{1}{Z_2} \\right)$ for all $x$ (from the proof of Lemma~\\ref{lm:xu_mfg_constbound}). But from~\\eqref{eq:xu_mfg_Zbound}, this is clearly satisfied by $M = \\frac{1}{|\\Omega|}$, and in this case $\\min \\left( - \\frac{\\sigma^2}{2} \\log Z_2 M,0 \\right) = - \\frac{\\sigma^2}{2} \\log \\frac{Z_2}{|\\Omega|}$. Thus\n    \\[ I_2(Z_2) \\leq \\int_{\\Omega} \\frac{1}{Z_2} e^{- \\frac{2}{\\sigma^2} \\left( - \\frac{\\sigma^2}{2} \\log \\frac{Z_2}{|\\Omega|} \\right)}~dx  = \\int_{\\Omega} \\frac{1}{|\\Omega|}~dx = 1 \\, . \\]\n    A similar procedure works to find $Z_1$, in which case $Z_1$ satisfies $Z_1 \\leq |\\Omega|$ and $\\inf_{x \\in \\Omega} h \\left( x, \\frac{1}{Z_1} \\right) \\geq \\sup_{x \\in \\Omega} h \\left( x, \\frac{1}{|\\Omega|} \\right)$.\n    \n    Step (2): Take $Z_1 \\leq Z_2$ as in Step (1). Then for every $Z \\in [Z_1,Z_2]$ there exists $C_1,C_2 \\in \\R$ such that ( by~\\eqref{eq:xu_mfg_lambdabound})\n    \\[ C_2 = \\inf_{x \\in \\Omega} h \\left(x, \\frac{1}{Z_2} \\right) \\leq \\lambda(Z_2) \\leq \\lambda(Z) \\leq \\lambda(Z_1) \\leq  \\sup_{x \\in \\Omega} h \\left(x, \\frac{1}{Z_1} \\right) = C_1 \\, . \\]\n    So $\\underaccent{\\bar}{u}_{C_1,Z_2} \\leq u_{\\lambda(Z),Z} \\leq \\bar{u}_{C_2,Z_1}$ for every $Z \\in [Z_1,Z_2]$. So we can use the dominated convergence theorem along with continuity of $u_{\\lambda,Z}$ with respect $(\\lambda,Z)$ and continuity of $\\lambda(Z)$ with respect to $Z$ to show $I_2(Z)$ is continuous.\n    \n    Step (3): Take $\\underaccent{\\bar}{Z} < \\bar{Z}$ then there exists $a \\in \\Omega$ such that\n    \\[ u_{\\lambda(\\underaccent{\\bar}{Z}),\\bar{Z}}(a) < u_{\\lambda(\\bar{Z}),\\bar{Z}}(a) \\, . \\]\n    Therefore, at $a \\in \\Omega$:\n    \\[ \\frac{1}{\\underaccent{\\bar}{Z}} e^{- \\frac{2}{\\sigma^2} u_{\\lambda(\\underaccent{\\bar}{Z}),\\underaccent{\\bar}{Z}}} = \\frac{1}{\\bar{Z}} e^{- \\frac{2}{\\sigma^2} u_{\\lambda(\\underaccent{\\bar}{Z}),\\bar{Z}}} > \\frac{1}{\\bar{Z}} e^{- \\frac{2}{\\sigma^2} u_{\\lambda(\\bar{Z}),\\bar{Z}}} \\, . \\]\n    So $I_2(\\underaccent{\\bar}{Z}) > I_2(\\bar{Z})$ because of the continuity of $u_{\\lambda,Z}$. This proves $I_2$ is strictly decreasing.\n\\end{proof}\n    \n\\begin{proof}[End of proof of Theorem~\\ref{thm:xu_mfg1}]\n    First let's assume $h$ is a strictly increasing function in $m$. Then we can choose the unique $Z^* \\in (0,\\infty)$ such that $I_2(Z^*) = 1$. Then clearly the triple ${\\left( u_{\\lambda(Z^*),Z^*}, \\lambda(Z^*), Z^* \\right)}$ is a solution to the system~\\eqref{eq:xu_mfg_sys}. Furthermore, suppose $(u',\\lambda',Z')$ is also a solution of~\\eqref{eq:xu_mfg_sys}. But this implies that $u'$ satisfies~\\eqref{eq:xu_mfg_bvp1}--\\eqref{eq:xu_mfg_bvp3}, so $u' = u_{\\lambda',Z'}$ from uniqueness proven in Proposition~\\ref{prop:xu_mfg_1}. Then $u_{\\lambda',Z'}$ also solves~\\eqref{eq:xu_mfg_bvp5}, so by uniqueness proven in Proposition~\\ref{prop:xu_mfg_lambda} we can show $\\lambda' = \\lambda(Z')$. Finally we now have $u' = u_{\\lambda(Z'),Z'}$ meets the integral constraint~\\eqref{eq:xu_mfg_bvp4}. So from uniqueness proven in Proposition~\\ref{prop:xu_mfg_Z} we have $Z' = Z^*$. Therefore ${(u',\\lambda',Z') = \\left( u_{\\lambda(Z^*),Z^*}, \\lambda(Z^*), Z^* \\right)}$. Hence the unique solution to the MFG problem is given by $\\left( m_{Z^*},u_{\\lambda(Z^*),Z^*},\\lambda(Z^*) \\right)$, where $m_{Z^*}$ is defined by $m_{Z^*} = \\frac{1}{Z^*} e^{- \\frac{2}{\\sigma^2}u_{z^*}}$.\n    \n    Now let's assume $h$ is an increasing function in $m$ and define $h_{\\epsilon}(x,m)$ by\n    \\[ h_{\\epsilon}(x,m) = h(x,m) + \\epsilon \\log \\left(|\\Omega| m \\right) \\, . \\]\n    Then for every $\\epsilon \\in (0,1]$, $h_{\\epsilon}$ is a strictly increasing function of $m$. Furthermore $h_{\\epsilon}$ still satisfies assumptions~\\ref{a:mfg_hreg}--\\ref{a:mfg_mtoinf}. Therefore there exists a unique solution $(u_{\\epsilon},\\lambda_{\\epsilon},Z_{\\epsilon})$ to the MFG system~\\eqref{eq:xu_mfg_sys}. From Proposition~\\ref{prop:xu_mfg_Z}, $Z_{\\epsilon} \\in [Z^1_{\\epsilon},Z^2_{\\epsilon}]$ for some $Z^1_{\\epsilon},Z^2_{\\epsilon} \\in (0,\\infty)$ such that\n    \\[ \\begin{aligned}\n        0 < Z^1_{\\epsilon} \\leq &|\\Omega| \\leq Z^2_{\\epsilon} < \\infty \\\\\n        \\sup_{x \\in \\Omega} h_{\\epsilon} \\left(x,\\frac{1}{Z^2_{\\epsilon}}\\right) &\\leq \\inf_{x \\in \\Omega} h_{\\epsilon} \\left(x,\\frac{1}{|\\Omega|}\\right) \\\\\n        \\inf_{x \\in \\Omega} h_{\\epsilon} \\left(x,\\frac{1}{Z^1_{\\epsilon}}\\right) &\\geq \\sup_{x \\in \\Omega} h_{\\epsilon} \\left(x,\\frac{1}{|\\Omega|}\\right) \\, .\n    \\end{aligned} \\]\n    But by the definition of $h_{\\epsilon}$ we have $h_{\\epsilon} \\left(x,\\frac{1}{Z^2_{\\epsilon}}\\right) \\leq h \\left(x,\\frac{1}{Z^2_{\\epsilon}}\\right)$ and $h_{\\epsilon} \\left(x,\\frac{1}{|\\Omega|}\\right) = h \\left(x,\\frac{1}{|\\Omega|}\\right)$, and a similar inequality holds for $Z^1_{\\epsilon}$ . So we can find $Z^1 \\in (0,|\\Omega|]$ and $Z^2 \\in [|\\Omega|,\\infty)$ independent of $\\epsilon$ such that $Z_{\\epsilon} \\in [Z^1,Z^2]$ for every $\\epsilon \\in (0,1]$. Now, from Lemma~\\ref{lm:xu_lambda_monotone} \n    \\[ \\lambda_{\\epsilon} = \\lambda(Z_{\\epsilon}) \\in \\left[\\lambda(Z^2),\\lambda(Z^1)\\right] \\, . \\]\n    So take a sequence $\\epsilon_n$ such that $\\lim_{n \\to \\infty} \\epsilon_n = 0$. Then, since $u_{\\epsilon} \\in C^{2,\\tau}\\left(\\bar{\\Omega}\\right)$, which is compactly embedded in $C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$, there exists a subsequence also denoted by $n$ such that $u_{\\epsilon_n} \\to u_0$ with convergence in $C^2\\left(\\Omega\\right) \\cap C^1\\left(\\bar{\\Omega}\\right)$, $Z_{\\epsilon_n} \\to Z_0 \\in [Z^1,Z^2]$, and $\\lambda_{\\epsilon_n} \\to \\lambda_0 \\in \\left[\\lambda(Z^2),\\lambda(Z^1)\\right]$. So we find, by taking limits\n    \\[ - \\frac{\\sigma^2}{2} \\nabla^2 u_0 + \\frac{|\\nabla u_0|^2}{2} - h\\left(x,\\frac{1}{Z_0}e^{- \\frac{2}{\\sigma^2}u_0}\\right) + \\lambda_0 = 0 \\, .\\]\n    Similarly we can show $(u_0,\\lambda_0,Z_0)$ satisfy~\\eqref{eq:xu_mfg_sys}. So we have proven existence of solutions for increasing $h$. \n    \n    For uniqueness, let's assume $(u_1,\\lambda_1,Z_1)$ and $(u_2,\\lambda_2,Z_2)$ are both solutions of~\\eqref{eq:xu_mfg_sys} and $\\lambda_1 \\leq \\lambda_2$. Define $u = u_2 - \\frac{\\sigma^2}{2} \\log\\left(\\frac{Z_1}{Z_2}\\right)$, then $u$ satisfies\n    \\[ - \\frac{\\sigma^2}{2} \\nabla^2 u + \\frac{|\\nabla u|^2}{2} - h\\left(x,\\frac{1}{Z_1}e^{- \\frac{2}{\\sigma^2}u}\\right) + \\lambda_1 = 0 \\, . \\] \n    Now define $v = u - u_2$ and suppose there exists $x \\in \\bar{\\Omega}$ such that $v(x) > 0$. Then $v$ has a maximum at $x^*$ and $v(x^*) > 0$. By Hopf's lemma $x^* \\in \\Omega$ and by the maximum principle $v$ is constant in the set $\\Omega_+ = \\{x \\in \\Omega : v(x) \\geq 0\\}$ (see the proof of Proposition~\\ref{prop:xu_mfg_1} for the details of such an argument). By assumption $x^* \\in \\Omega_+$ and $v(x^*) > 0$, so $\\Omega_+ = \\Omega$ by continuity of $v$. Hence $v$ is constant in $\\Omega$ and $v > 0$. However, from the integral constraint~\\eqref{eq:xu_mfg_bvp4} we obtain\n    \\begin{equation} \\label{eq:xu_mfg_proof1}\n        0 = \\int_{\\Omega} \\frac{1}{Z_1} e^{- \\frac{2}{\\sigma^2} u_1} \\left(1 - e^{- \\frac{2}{\\sigma^2} v}\\right)~dx = |\\Omega| \\left(1 - e^{- \\frac{2}{\\sigma^2} v}\\right) \\, ,\n    \\end{equation}\n    since $1 - e^{- \\frac{2}{\\sigma^2} v}$ is constant. So $v = 0$, contradicting the assumption $v(x^*) > 0$. Therefore $v(x) \\leq 0$ for all $x \\in \\Omega$. This implies that $1 - e^{- \\frac{2}{\\sigma^2} v} \\leq 0$, and subsequently that\n    \\[0 = \\int_{\\Omega} \\frac{1}{Z_1} e^{- \\frac{2}{\\sigma^2} u_1} \\left(1 - e^{- \\frac{2}{\\sigma^2} v}\\right)~dx \\leq 0 \\, ,\\]\n    with equality if and only if $v = 0$. Therefore $u_2 = u$, which implies (using the integral constraint~\\eqref{eq:xu_mfg_bvp5}) that $Z_1 = Z_2$, and subsequently that $u_1 = u_2$. Finally, by subtracting the PDE~\\eqref{eq:xu_mfg_bvp2} satisfied by $u_1$ from the one satisfied by $u_2$ we find $\\lambda_2 = \\lambda_1$. Therefore solutions are unique.\n\\end{proof}\n\n\\section{Quadratic Potential} \\label{sec:quad potential}\n\n\\noindent In this section we consider a specific example with quadratic potential and a logarithmic congestion term, $h(x,m) = \\beta x^2 + \\log m$ for some constant $\\beta \\geq 0$ on the real line. This problem has been studied extensively in~\\cite{Gomes2016a} and~\\cite{Gueant2009} and admits explicit solutions. This allows us to compare the solutions of the BRS and the MFG. Note that we do not impose any boundary conditions or integral constraints on $u$, since we consider the model on $\\R$ rather than on a bounded domain. Therefore it doesn't fit directly into the framework for existence and uniqueness proven in the previous section. It is however, one of the few illustrative examples, where explicit solutions are known. This allows us to make an analytical comparison of the two models and use the solution to validate the proposed numerical methods.\n\n\\subsection{The MFG}\n\nThe stationary MFG model studied in~\\cite{Gomes2016a} and~\\cite{Gueant2009}, with the integral constraints used in this paper, is given by:\n\\begin{subequations}\\label{eq:quad_stat_mfg1}\n\\begin{align} \n    \\frac{\\sigma^2}{2} \\partial_{xx}^2 m + \\partial_x \\left( m \\partial_x u \\right) &= 0 \\, , \\quad x \\in \\R \\, , \\\\ \\label{eq:quad_stat_mfg2}\n    - \\frac{|\\partial_x u|^2}{2} + \\log m + \\beta x^2 + \\frac{\\sigma^2}{2} \\partial_{xx}^2 u + \\lambda &= 0 \\, , \\quad x \\in \\R \\, , \\\\\n    \\int_{\\R} m~dx &= 1 \\, .\n\\end{align}\n\\end{subequations}\nwhere $\\lambda \\in (-\\infty, \\infty)$ is a constant to be found as part of the solution, and $\\sigma, \\beta \\geq 0$ are given parameters.\n\\begin{proposition}\n    A solution to the stationary MFG system~\\eqref{eq:quad_stat_mfg1} exists and has an explicit form \n    \\begin{subequations} \\label{eq:quad_stat_mfg_sol}\n        \\begin{align}\n            & m(x) = \\left( \\frac{a}{\\pi} \\right)^{1\/2} e^{- a x^2} \\\\\n            & u(x) = b x^2 \\\\\n            & \\lambda = \\log \\left( \\frac{\\pi}{a} \\right) - \\sigma^2 b\\, ,\n        \\end{align}\n    \\end{subequations}\n    where the constants $a,b,c \\geq 0$ are given by\n    \\[ a = \\beta,~ b = 0 \\, , \\]\n    if $\\sigma = 0$, or\n    \\[ a = \\frac{-1 + \\left( 1 + 2 \\sigma^4 \\beta \\right)^{1\/2}}{\\sigma^4},~b = \\frac{-1 + \\left( 1 + 2 \\sigma^4 \\beta \\right)^{1\/2}}{2 \\sigma^2} \\, , \\]\n    if $\\sigma > 0$.\n\\end{proposition}\nThe proof is straight-forward using substitution.\n\n\\subsection{The BRS}\nNext we consider the respective stationary BRS model. It is given by\n\\begin{subequations} \\label{eq:quad_stat_brs}\n    \\begin{align}\n        \\partial_x \\left( m \\partial_x (\\log m + \\beta x^2) \\right) + \\frac{\\sigma^2}{2} \\partial_{xx}^2 m &= 0 \\, , \\quad x \\in \\R \\\\\n        \\int_{\\R} m~dx &= 1 \\, .\n    \\end{align}\n\\end{subequations}\n\\begin{proposition}\n    The solution to the stationary BRS equation~\\eqref{eq:quad_stat_brs} is given by\n    \\[ m(x) = \\left( \\frac{2 \\beta}{(2 + \\sigma^2) \\pi} \\right)^{1\/2} e^{- \\frac{2 \\beta}{(2 + \\sigma^2)} x^2} \\, . \\]\n\\end{proposition}\nAgain the claim follows from substitution. \n\n\\begin{figure}[h!]\n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-00.jpg}\n\t    \\caption{$\\beta = 0.1$, $\\frac{\\sigma^2}{2} = 10$}\n    \\end{subfigure}\n    \\begin{subfigure}{0.5 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-01.jpg}\n\t    \\caption{$\\beta = 0.1$, $\\frac{\\sigma^2}{2} = 0.2$}\n    \\end{subfigure}\n    \n    \\bigskip\n    \n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-02.jpg}\n\t    \\caption{$\\beta = 1$, $\\frac{\\sigma^2}{2} = 10$}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-03.jpg}\n\t    \\caption{$\\beta = 1$, $\\frac{\\sigma^2}{2} = 0.2$}\n    \\end{subfigure}\n    \n     \\bigskip\n    \n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-04.jpg}\n\t    \\caption{$\\beta = 10$, $\\frac{\\sigma^2}{2} = 10$}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-05.jpg}\n\t    \\caption{$\\beta = 10$, $\\frac{\\sigma^2}{2} = 0.2$}\n    \\end{subfigure}\n    \\caption{Simulations of BRS and MFG with quadratic potential and logarithmic congestion}\n    \\label{fig:quad-log}\n\\end{figure}\n\n\\subsection{Comparison}\n\\begin{proposition}\n    For $\\sigma > 0$, the stationary distributions of the MFG system~\\eqref{eq:quad_stat_mfg1} and the BRS~\\eqref{eq:quad_stat_brs} are given by normal distributions with mean $0$ and variances $a_1$ and $a_2$ respectively, where\n    \\[ \\begin{aligned}\n        a_1 & = \\frac{\\sigma^4}{-2 + 2(1 + 2 \\sigma^4 \\beta)^{1\/2}} \\\\\n        a_2 & = \\frac{2 + \\sigma^2}{4 \\beta} \\, .\n    \\end{aligned} \\]\n    Then for fixed $\\beta \\geq 0$\n    \\begin{subequations}\n        \\begin{align}\n            \\lim_{\\sigma^2 \\to 0} \\frac{a_2}{a_1} &= 1 \\label{eq:quad_lim1} \\\\\n            \\lim_{\\sigma^2 \\to \\infty} \\frac{a_2}{a_1} &= \\frac{1}{(2 \\beta)^{1\/2}} \\, . \\label{eq:quad_lim2}\n        \\end{align}\n    \\end{subequations}\n    While for fixed $\\sigma > 0$\n    \\begin{subequations}\n        \\begin{align}\n            \\lim_{\\beta \\to 0} \\frac{a_2}{a_1} &= 1 + \\frac{\\sigma^2}{2} \\label{eq:quad_lim3} \\\\\n            \\lim_{\\beta \\to \\infty} (2 \\beta)^{1\/2} \\frac{a_2}{a_1} &= \\frac{2 + \\sigma^2}{\\sigma^2} \\, . \\label{eq:quad_lim4}\n        \\end{align}\n    \\end{subequations}\n\\end{proposition}\n\n\\begin{proof}\n    The first part of this proof is trivial from the previous propositions. Now \n    \\[ \\frac{a_2}{a_1} = \\frac{(2 + \\sigma^2) \\left( (1 + 2 \\sigma^4 \\beta)^{1\/2} - 1 \\right)}{2 \\sigma^4 \\beta} \\, . \\]\n    Using a Taylor expansion of $(1 + x)^{1\/2}$ around $x = 0$ gives behaviour for small $\\sigma^2$ i.e.\n   \\[ \\frac{a_2}{a_1} = \\frac{(2 + \\sigma^2) (\\sigma^4 \\beta + o(\\sigma^4))}{2 \\sigma^4 \\beta} \\, . \\]\n    Hence $\\lim_{\\sigma^2 \\to 0} \\frac{a_2}{a_1} = \\lim_{\\sigma^2 \\to 0} \\frac{2 + \\sigma^2}{2} = 1$. The other limit can be simply calculated\n    \\[ \\begin{aligned}\n            \\lim_{\\sigma^2 \\to \\infty} \\frac{a_2}{a_1} & = \\lim_{\\sigma^2 \\to \\infty} \\frac{(2 + \\sigma^2) (1 + 2 \\sigma^4 \\beta)^{1\/2}}{2 \\sigma^4 \\beta} = \\lim_{\\sigma^2 \\to \\infty} \\frac{(2 + \\sigma^2) (2 \\beta)^{1\/2} \\sigma^2}{2 \\sigma^4 \\beta} = \\frac{1}{(2 \\beta)^{1\/2}} \\, .\n        \\end{aligned} \\]\n    The limits as $\\beta \\to 0,\\infty$ for fixed $\\sigma$, follows from straight forward calculations.\n\\end{proof}\n\nThis result is an important first glimpse at how the behaviour of the BRS and MFG may vary, as well as the importance certain parameters play in the difference. The limit in~\\eqref{eq:quad_lim1} shows that the existence of noise is vital to see any difference between the two models. However, as soon as there is noise, its effect on the relative difference plays a less important role than the strength of the quadratic potential, this can be seen in~\\eqref{eq:quad_lim2} and~\\eqref{eq:quad_lim4}. Specifically the limit~\\eqref{eq:quad_lim4} shows that the relative difference between the variances of the two distributions grows like $\\beta^{\\frac{1}{2}}$, which means the BRS distribution reacts much more rapidly with changes to the potential strength than the MFG. This suggests that the MFG is more affected by congestion or is a more congestion-averse model than the BRS one. \n\nAt a conceptual level this agrees with the formulation of the MFG and BRS systems. The agents in the BRS are acting myopically, only reacting to the situation as it currently exists, which isn't the case in the MFG. As a result the BRS agents don't `see' the future congestion that will result from their behaviour and hence they move towards the minimum of $\\beta x^2$ more rapidly than the MFG agents who do see the future cost of the congestion that results from their behaviour. Therefore, thinking of the stationary solutions as the long time, time-averaged behaviour of the models then the stationary BRS will result in a distribution that appears to take into account the congestion less than the MFG and hence one with a smaller variance. This expectation is confirmed by the result~\\eqref{eq:quad_lim4} and it in fact quantifies the extent to which the BRS ignores the congestion compared with the MFG.\n\nFor this model we have run a variety of simulations --- both to confirm our numerical methods (see Section~\\ref{sec:numerical_sim} for methods) and to visualise how the parameters affect the distributions. Figure~\\ref{fig:quad-log} shows the results of these simulations on a bounded domain for a variety of parameter choices. Although the formulation on a bounded domain is slightly different than the one in this section, the same behaviour can be seen. For small $\\sigma$ the difference between the models doesn't change much as $\\beta$ increases, while for large values of $\\sigma$ the BRS model is much more dramatically affected by changes to $\\beta$. In both cases the BRS and MFG are more closely aligned when $\\beta$ is small. \n\n\\section{Simulations}\\label{sec:numerical_sim}\n\\noindent We conclude with presenting various computational experiments, which illustrate the difference between solutions to the BRS and MFG for different choices running costs and potentials. \n\n\\subsection{Solving the stationary BRS and MFG}\nSolutions to the stationary BRS~\\eqref{eq:brssystem} can be computed by finding the zeros of the function $G_{Z,x}(m)$ at every discrete grid point $x$ on a grid given $Z$. To compute the roots of $G_{Z,x}$ at every grid point $x$ we use a Newton-Raphson method. Then, having found $m_Z(x)$ for a particular value of $Z$, we can differentiate the implicit formula $m_Z = \\frac{1}{z} e^{- \\frac{2}{\\sigma^2} h(x,m_Z)}$ with respect to $Z$ and use a Newton-Raphson method to find $Z$ such that $\\Phi(Z) = \\int_{\\Omega}m_Z~dx = 1$. In practice this means iterating between the two Newton-Raphson methods: first finding $m_{Z_n}$, then computing $Z_{n+1}$, then recomputing $m_{Z_{n+1}}$ and repeating until convergence.\\\\\nThe solution to the stationary MFG~\\eqref{eq:xu_mfg} are found using an iterative procedure. Given an admissible initial iterate $m^l$, $l=0$ we solve the HJB equation~\\eqref{eq:xu_mfg_pde2} to obtain $u^l$ and $\\lambda^l$. Note that we include the constraint~\\eqref{eq:xu_mfg_ic2} via a Lagrange multiplier. In the final step of the iteration we update the distribution of agents by solving the FPE~\\eqref{eq:xu_mfg_pde1} using $u^{l+1}$ to obtain $m^{l+1}$. This procedure is repeated until convergence. Note that we sometimes perform a damped update\n\\begin{align*}\nu^l = \\omega u^{l-1} + (1-\\omega) v^l   \\text{ and } m^l = m^{l-1} + (1-\\omega) q^l \n\\end{align*}\nwhere $v^l,q^l$ are the undamped solutions of each iteration process. This damping helps to ensure convergence. Solutions to the HJB and the FPE are obtained by using an $H^1$ conforming finite element discretisation.\n\n\\begin{figure}[b]\n     \\centering\n    \\includegraphics[width=0.7\\textwidth]{brs-mfg-06.jpg}\n    \\caption{Simulations of BRS and MFG with $h(x,m) = m^{10} + 10 x^2$}\n    \\label{fig:quad-power}\n\\end{figure}\n\nWe noticed that in many of the simulations performed the ``cost'' associated to the MFG was higher than the ``cost'' associated with the BRS. At first this sounds counter-intuitive as the BRS (in the dynamic case) is a sub-optimal approximation of the MFG. However, as explained in Section~\\ref{sec:stat_prob}, the ``cost'' function $u$ is actually the long-time average difference between the cost and the space-average cost, whereas the stationary BRS cost is the equilibrium of the competitive minimisation of \\eqref{eq:Estat}. Therefore the MFG cost will always be centred around 0 while the BRS cost could be above or below it. As a result, it was not clear that comparing the ``costs'' of the two models is especially useful and hence we have solely focussed on comparing the distribution of agents.\n\n\\begin{figure}[t]\n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-07.jpg}\n\t    \\caption{$m_{\\max} = 10$}\n\t    \\label{fig:quad-max-10}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49 \\textwidth}\n\t    \\centering\n\t    \\includegraphics[width=\\linewidth]{brs-mfg-08.jpg}\n\t    \\caption{$m_{\\max} = 1$}\n\t    \\label{fig:quad-max-1}\n    \\end{subfigure}\n    \\caption{Simulations of BRS and MFG with $h(x,m) = \\frac{1}{m_{\\max} - m} + x^2$}\n    \\label{fig:quad-max}\n\\end{figure}\n\n\\subsection{Single well potential}\n\nIn the first examples we investigate the behavior of solutions to both models for cost functionals of the form ${h(x,m) = F(m) + \\beta x^2}$, using different functions $F$ and parameters $\\beta$. We also analyse how the noise level $\\sigma$ affects the two stationary states. Note that the case $F(m) = \\log(m)$ was already discussed in Section~\\ref{sec:quad potential}. We are particularly interested how penalising congestion by considering functions $F(m)$ of the form $F(m) = m^{\\alpha}$ for some $\\alpha > 0$ or $F(m) = \\frac{1}{m_{\\max} - m}$ for some $m_{\\max} > \\frac{1}{\\Omega}$ affect solutions. The last choice introduces a `barrier' above which the density can not exceed. Using such a congestion term is more realistic from a modelling perspective than either the logarithmic or power--law term as it forces densities to stay below a certain physical reasonable limit. We will observe a similar dependence on the parameters $\\beta,\\sigma$ compared with the logarithmic congestion term --- and in fact the same can be said for all of our simulations. So for all values of $\\sigma$ the MFG model responded less to changes in the strength of the potential $\\beta$ compared with the BRS model, however the difference is most pronounced as $\\sigma$ increases and again it may be expected that as $\\sigma \\to 0$ that the two models align very closely.\n\nThe most notable difference between the use of a logarithmic congestion term and a power--law congestion term is a difference in the shape of the distribution, particularly the flatness of the peak of the distribution as shown in figure~\\ref{fig:quad-power}. Importantly this characteristic is shared by both the MFG and the BRS, suggesting the congestion terms $F(m)$ affect both models in similar ways. When looking at congestion terms of the form $F(m) = \\frac{1}{m_{\\max} - m}$, we can find regimes where the behaviour is similar to the logarithm, or more like a vastly exaggerated version of the power--law congestion. When $m_{\\max}$ is large, as in figure~\\ref{fig:quad-max-10}, the resulting distribution for both models looks like a normal distribution, similar to the case with logarithmic congestion. In fact when $m_{max}$ is very large, a formal asymptotic analysis using a Taylor expansion around $m_{\\max}$ can be made which shows \n\\[\\frac{1}{m_{\\max} - m} \\approx \\frac{1}{m_{\\max}} \\, .\\]\nTherefore the BRS  satisfies the equation\n\\[ m \\approx \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} \\frac{ \\beta m_{\\max} x^2 - 1}{m_{\\max}}} \\approx \\frac{1}{Z} e^{- \\frac{2 \\beta}{\\sigma^2} x^2} \\, , \\]\nwith a normalisation constant $Z = \\int_{\\Omega} e^{- \\frac{2\\beta}{\\sigma^2} x^2}$, which corresponds, on the whole space $\\R$, to a normal distribution with zero mean and variance $\\frac{\\sigma^2}{4 \\beta}$. The variance of the BRS solution found here differs from the logarithmic congestion case by $\\frac{2}{4\\beta}$. A similar analysis shows that when $m_{\\max} \\to \\infty$ the MFG approximately resembles a normal distribution with zero mean and variance \n$\\frac{\\sigma^2}{2 \\beta}$. In summary, solutions to the MFG and the BRS are both normal distributions with zero mean and with variances whose relative difference is $\\frac{1}{2}$.\n\n\nHowever, when $m_{\\max}$ is reduced, as in figure~\\ref{fig:quad-max-1}, the peak flattens out in a similar but exaggerated way compared to the power--law congestion. It is interesting to note that the BRS seems to respond more to the change in $m_{\\max}$ than the MFG, this is in contrast to the role that $\\sigma$ plays in the two models where the MFG responds more to changes in $\\sigma$ compared with the BRS.\n\n\\begin{figure}[t]\n    \\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{mfg-brs-pot-1.jpg}\n        \\caption{Double well potential for Figure~\\ref{fig:double well 1}}\n        \\label{fig:double pot 1}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{mfg-brs-pot-2.jpg}\n        \\caption{Double well potential for Figure~\\ref{fig:double well 2}}\n        \\label{fig:double pot 2}\n    \\end{subfigure}\n    \n    \\bigskip\n    \n    \\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{mfg-brs-pot-3.jpg}\n        \\caption{Double well potential for Figure~\\ref{fig:double well 3}}\n        \\label{fig:double pot 3}\n    \\end{subfigure}\n    \\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{mfg-brs-pot-4.jpg}\n        \\caption{Double well potential for Figure~\\ref{fig:double well 4}}\n        \\label{fig:double pot 4}\n    \\end{subfigure}\n    \n    \\bigskip\n    \\centering\n    \\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{mfg-brs-pot-5.jpg}\n        \\caption{Double well potential for Figure~\\ref{fig:double well 5}}\n        \\label{fig:double pot 5}\n    \\end{subfigure}\n    \\caption{Double well potentials for simulations}\n    \\label{fig:double pots}\n\\end{figure}\n\n\\subsection{Double well potential}\n\nThe previous subsection has given insight into how the form of the congestion term affects both models. In this section we explore how the potential term affects each model. For this section we will consider costs of the form $h(x,m) = F_1(x) + \\log(m)$, where $F_1(x)$ will be a double well potential (see figure~\\ref{fig:double pots}). Since the key insight of this section is to understand how varying $F_1$ affects the similarity of solutions to the BRS and MFG models, we have decided not to include results with different congestion terms other than the logarithm. From simulations it can be seen that the effect of changing the congestion term from $\\log(m)$ to another term is very similar whether we are considering a single well or a double well.\n\nOur simulations focus on five different double wells, which can be seen in figure~\\ref{fig:double pots}. We vary the potentials as follows\n\\begin{enumerate}[topsep=0pt,itemsep=0pt,partopsep=2pt,parsep=1pt]\n    \\item same depth, same width,\n    \\item different depth, same width,\n    \\item same depth, different width,\n    \\item approximately similar perimeter,\n    \\item approximately similar volume.\n\\end{enumerate}\n\nThe simulations with the first two potentials, see figures~\\ref{fig:double well 1} and~\\ref{fig:double well 2}, where the widths of the two wells are always the same, show that the two models display similar qualitative behaviour. As with the single well potential, as $\\sigma$ increases the discrepancy between the two models also increases, with the MFG model being more affected by the level of noise than the BRS. As expected, when the two wells are of equal depth (as in figure~\\ref{fig:double well 1}) then both the MFG and BRS attribute equal weight between the wells, while when one well is deeper than the other (as in figure~\\ref{fig:double well 2}) both models give more mass to the location of the deeper well. This is true for all values of $\\sigma$, although the effect reduces as $\\sigma$ increases, as can be seen by comparing figures~\\ref{fig:double well 1a} and~\\ref{fig:double well 2a} to figures~\\ref{fig:double well 1b} and~\\ref{fig:double well 2b} respectively.\n\nUp to this point we have seen that the qualitative behaviour of the two models tends to agree. However when we look at double well potentials where the width of the well is varying, we start to observe differences. In figure~\\ref{fig:double well 3}, where the wells have the same depth but different width, we see that the BRS still distributes density equally to the two wells. However the MFG model results in a higher density focussed in the wider well than in the narrower well. The reason the width has no effect on the BRS can be seen from studying the implicit equation. In each well we are solving $m = \\frac{1}{Z} e^{- \\frac{2}{\\sigma^2} h(x,m)}$. In our case $h(x,m) = G(x) + F(m)$. Since the potential $G(x)$ is at the same depth in each well then the relative height of the distribution $m$ will be the same in each well. The reason the MFG is affected by the width of the of the well is that in finding the MFG solution we are in fact solving an elliptic equation to find the function $u$, hence at each point $x$ this $u$ will be affected by factors that can't be described by just looking at the value of $h$ at that point. In other words, the BRS depends only on local properties of the cost $h$ whereas the MFG depends also on non-local properties. To understand why the MFG assigns greater density to the wider well we need to look at the underlying optimisation problems related to the MFG and the BRS.\n\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-09.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 0.2$}\n    \t\\label{fig:double well 1a}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-10.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 1$}\n    \t\\label{fig:double well 1b}\n\t\\end{subfigure}\n\t\\caption{Simulation of BRS and MFG with logarithmic congestion and potential given in figure~\\ref{fig:double pot 1}}\n\t\\label{fig:double well 1}\n\\end{figure}\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-11.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 0.2$}\n    \t\\label{fig:double well 2a}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-12.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 1$}\n    \t\\label{fig:double well 2b}\n\t\\end{subfigure}\n\t\\caption{Simulation of BRS and MFG with logarithmic congestion and potential given in figure~\\ref{fig:double pot 2}}\n\t\\label{fig:double well 2}\n\\end{figure}\n\nWhen considering the optimisation problems related to the dynamic MFG and BRS models and the long-term behaviour of these models, which results in the stationary models, we can see that the BRS model has no anticipation about the future system, while the MFG model does. Therefore agents in the MFG model are willing to incur higher congestion costs in the wider well as they can see that the cost to move out of the well to an area of lower congestion will be higher than the cost incurred for staying in the well (the cost functional being optimised has a quadratic running cost on the control). Since the cost for moving out of the well increases with the width of the well (as the wider the well either the longer an agent has to use their control, or the larger their control has to be), fewer agents are willing to move out of the wider well than the narrower in the long-run. Hence in the stationary case the wider well has a higher density associated to it than the narrower well. In contrast, we see that in going from the MFG to the BRS we renormalise the cost of the control by $\\Delta t$ and take $\\Delta t \\to 0$ so the BRS doesn't consider the cost of moving along the width of the well in order to find an area of lower density. Therefore the BRS agents will not consider the width of the well when deciding whether to remain in it or leave. This further explains why the width of the well has no effect on the relative size of the density in each well for the BRS. \n\nWe have seen that increasing well width affects only the MFG while increasing well depth affects both the MFG and BRS. Now we can balance these effects to create situations in which the two models give completely different results. Figure~\\ref{fig:double well 4} involves a double well where the width and depth of the wells differ but the area of the well is the same, while in figure~\\ref{fig:double well 5}  the perimeter of the wells was kept the same. In the case of a small noise term, then both the MFG and BRS favour the deeper wells. However with a larger noise term, figure~\\ref{fig:double well 4b} shows that there are cases where the wider shallower well is favoured by the MFG while the narrower, deeper well is favoured by the BRS.\n\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-13.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 0.2$}\n    \t\\label{fig:double well 3a}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-14.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 1$}\n    \t\\label{fig:double well 3b}\n\t\\end{subfigure}\n\t\\caption{Simulation of BRS and MFG with logarithmic congestion and potential given in figure~\\ref{fig:double pot 3}}\n\t\\label{fig:double well 3}\n\\end{figure}\n\t\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-15.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 0.2$}\n    \t\\label{fig:double well 4a}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-16.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 1$}\n    \t\\label{fig:double well 4b}\n\t\\end{subfigure}\n\t\\caption{Simulation of BRS and MFG with logarithmic congestion and potential given in figure~\\ref{fig:double pot 4}}\n\t\\label{fig:double well 4}\n\\end{figure}\n\n\\section{Conclusion and outlook} \\label{sec:conclusion}\n\n\\noindent In this paper we have systematically compared two models of interacting multi-agent systems in the stationary case. Through a proof of existence and uniqueness for each model we have seen that the BRS model can be reformulated as an implicit equation. This shows that the BRS model really only depends on local data of the cost function, while the MFG model, the solution of which is given by an elliptic equation, may have non-local dependenicies on the data. The existence and uniqueness proofs were based on the important assumption that the congestion term is increasing. However, the regularity requirements on the MFG data are less strict than those on the BRS data. Finally the proof gave an insight into the dependence of each model on the diffusion coefficient. We want to remark that the strategy of the proof is interesting on its own and that the only similar results presented in ~\\cite{Cirant2015}, are based on different assumptions.\\\\\nWe supported our analytic results by numerical simulations and investigated the similarities and differences of the MFG and BRS models systematically in various computational experiments. We are planning to extend the analysis and simulations to the dynamic case in the future, and consider cost functions other than linear-quadratic ones.\n\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-17.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 0.2$}\n    \t\\label{fig:double well 5a}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.49 \\textwidth}\n        \\centering\n\t\t\\includegraphics[width=\\textwidth]{brs-mfg-18.jpg}\n    \t\\subcaption{$\\frac{\\sigma^2}{2} = 1$}\n    \t\\label{fig:double well 5b}\n\t\\end{subfigure}\n\t\\caption{Simulation of BRS and MFG with logarithmic congestion and potential given in figure~\\ref{fig:double pot 5}}\n\t\\label{fig:double well 5}\n\\end{figure}\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nClusters of galaxies are the most massive, individual, bound objects in the Universe.\nIn their gravitational potential well, the gas, called the intra-cluster medium (ICM),\nis heated to temperatures of $10^{7-8}$\\,K and, therefore, strongly emits at X-ray energies.\nThe ICM is commonly thought to be in hydrostatic equilibrium, but there are several\nfactors that may affect the dynamical state of the gas. The feedback from active galactic nuclei (AGN) creates bubbles \nthat may drive turbulence up to about 500\\,km\\,s$^{-1}$ \n(see, e.g., \\citealt{Bruggen2005} and \\citealt{Fabian2005}). \nSloshing of gas within the gravitational potential may produce similar velocities,\nwhile galactic mergers can give rise to even higher velocities of about 1000\\,km\\,s$^{-1}$ \n(see, e.g., \\citealt{Lau2009}, \\citealt{Ascasibar2006})\n\n{The AGN feedback is thought to offset radiative losses and \nto suppress cooling in isolated giant elliptical galaxies \nand in larger systems up to the richest galaxy clusters (see, e.g., \\citealt{McNamara2007}\nand \\citealt{Fabian2012}).\nSimulations and observations have confirmed that AGN feedback \nmay prevent cooling through the production of turbulence \n(see, e.g., \\citealt{Ruszkowski2004}, \\citealt{Zhuravleva2014}, and \\citealt{Gaspari2014}). \nOther work suggests that turbulent mixing may also be an important mechanism \nthrough which AGN heat cluster cores (see, e.g., \\citealt{Banerjee2014}).}\n\n\\vspace{-0.05cm}\n\nIt is possible to measure velocity broadening on the order of few hundreds km\\,s$^{-1}$ directly in the X-ray emission lines\nproduced by the hot ICM. The Reflection Grating Spectrometers (RGS, \\citealt{denherder2001}) aboard \nXMM-\\textit{Newton} are currently the only X-ray instruments, which have enough collecting area and spectral resolution\nto enable this measurement.\nHowever, the spatial extent of clusters complicates the process due to the slitless\nnature of the RGS.\n\\citet{Sanders2010} made the first measurement of cluster velocity broadening\nusing the luminous cluster \\object{A\\,1835} at redshift 0.25. Due to the limited spatial extent \nof its bright core, an upper limit of 274\\,km\\,s$^{-1}$ was obtained.\n\\citet{Sanders2011} then constrained turbulent velocities for a large sample\nof 62 sources observed with XMM-\\textit{Newton}\/RGS, which included clusters, groups, and elliptical galaxies.\nHalf of them show velocity broadening below 700\\,km\\,s$^{-1}$. Recently, \n\\citet{Sanders2013} used continuum-subtracted emission line surface brightness\nprofiles to account for the spatial broadening. This technique is affected by systematic errors of\nup to 150\\,km\\,s$^{-1}$.\n\n\\citet{Werner2009} and \\citet{dePlaa2012} measured turbulent velocities through the ratio of the \n\\ion{Fe}{xvii} emission lines at 15 and 17\\,{\\AA}. When the velocity broadening is low, the gas is optically thick \nin the 15\\,{\\AA} line due to resonant scattering, while the 17\\,{\\AA} lines remain optically thin. The comparison\nof observed with simulated line ratios for different Mach numbers constrains the level of turbulence.\nThis method is very efficient for cool core clusters rich in \\ion{Fe}{xvii} emission lines, but it is partly\nlimited by the systematic uncertainty ($\\sim$20\\%) in the line ratio for an optically thin plasma.\n\nIn this work, we measure the velocity broadening\nfor the 44 sources of the CHEmical Enrichment RGS cluster Sample (CHEERS),\nwhich is connected to a Very Large Program accepted for XMM-\\textit{Newton} AO-12. \nWe model the line spatial broadening using CCD images.\nThis method has systematics due to the spatial profile \nof the continuum, which may overestimate the line spatial broadening,\nbut it is still a useful technique to measure the level of velocity broadening\nwhen deep, high-spatial resolution maps are lacking. \nWe also test an alternative method, \nwhich uses a variable spatial-broadening.\nThe paper is organized as follows. \nIn Sect.\\,\\ref{sec:cheers}, we give a brief description of the CHEERS project. \nIn Sect.\\,\\ref{sec:data}, we present the data reduction. \nOur method is described in Sect.\\,\\ref{sec:spectral_modeling}. \nWe discuss the results in Sect.\\,\\ref{sec:discussion} and \ngive our conclusions in Sect.\\,\\ref{sec:conclusion}.\nFurther useful material is reported in Appendix\\,\\ref{sec:appendix}\nto speed up the paper reading.\n\n\\vspace{-0.35cm}\n\n\\section{The CHEERS project}\n\\label{sec:cheers}\n\nThe current catalog includes 44 nearby, bright clusters, groups of galaxies, and elliptical galaxies\nwith a value of a $\\gtrsim5\\sigma$ detection for the \\ion{O}{viii} 1s--2p line at 19\\,{\\AA}\nand with a well-represented variety of strong, weak, and non cool-core objects. \nThis catalog also contains 19 new observations of 1.6\\,Ms in total, which are taken during AO-12, \nPI: J. de Plaa (see Table~\\ref{table:log}). \nMore detail on the sample choice is provided by another paper \n(de Plaa et al., in preparation). \nAmong the several goals of this large project, we mention the following ones:\n\\vspace{-0.2cm}\n\\begin{itemize}\n \\item To understand the ICM metal enrichment by different SN types, \\\n       (see, e.g., Mernier et al. accepted) \n \\item to study substructures, asymmetries and multiphaseness,\n \\item to study heating and cooling in cluster cores,\n \\item to measure turbulence (this paper),\n \\item to improve the cross-calibration between X-ray satellites.\n\\end{itemize}\n\n\\vspace{-0.5cm}\n\n\\section{Data}\n\\label{sec:data}\n\nThe data used in this paper are listed in Table~\\ref{table:log}. \nIn boldface, we show the new observations\ntaken during AO-12. A few archival exposures have not been used, \nsince they were too short. \n\nThe XMM-\\textit{Newton} satellite is equipped with two types of X-ray detectors: \nThe CCD-type European Photon Imaging Cameras (EPIC) and the Reflection Grating Spectrometers (RGS). \nThe European photon imaging cameras are\nMOS\\,1, MOS\\,2, and pn (\\citealt{Struder2001} and \\citealt{Turner2001}). \nThe RGS camera consists of two similar detectors, which have both high effective area and \nspectral resolution between 6 and 38\\,{\\AA} \\citep{denherder2001}.\nThe MOS cameras are aligned with the RGS detectors and have \nhigher spatial resolution than the pn camera. \nWe have used MOS\\,1 for imaging and RGS for spectral analysis.\n\n\n\\subsection{RGS and MOS 1 data reduction}\n\nThe data were reduced with the XMM-\\textit{Newton} Science Analysis System (SAS) v13.5.0. \nWe processed the RGS data \nwith the SAS task \\textit{rgsproc} and the MOS\\,1 data with \\textit{emproc} \n{to produce event files, spectra, and response matrices for RGS and MOS data}.\n\nTo correct for contamination from soft-proton flares, we {used the SAS task \\textit{evselect}\nto extract} light curves for MOS\\,1 in the 10--12 keV\nenergy band, while we used the data from CCD number 9 for RGS \nwhere hardly any emission from each source is expected. We\nbinned the light curves in 100\\,s intervals. A Poissonian distribution was fitted to the\ncount-rate histogram, and all time bins outside the $2\\sigma$ level were rejected.\nWe built the good time intervals (GTI) files with the accepted time events for the MOS and RGS files \n{through the SAS task \\textit{tabgtigen} and \nreprocessed the data again with \\textit{rgsproc} and \\textit{emproc}}. \nThe RGS\\,1 total clean exposure times are quoted in Table\\,\\ref{table:log}.\n\n\\begin{table*}\n\\caption{XMM-\\textit{Newton}\/RGS observations used in this paper.}  \n\\vspace{-0.25cm}\n\\label{table:log}     \n\\renewcommand{\\arraystretch}{1.1}\n \\small\\addtolength{\\tabcolsep}{+2pt}\n \n\\scalebox{1}{%\n\\begin{tabular}{c c c c c c c }     \n\\hline\\hline            \nSource                               &  ID $^{(a)}$          & Total clean time (ks) $^{(b)}$ & $kT$ (keV) $^{(c)}$  & $z\\,^{(c)}$  & $N_{\\rm H}$ ($10^{24}\\,{\\rm m}^{-2}$) $^{(d)}$\\\\  \n\\hline   \n\\multirow{1}{*}{\\object{2A0335+096}} & 0109870101\/0201 0147800201               &  120.5      &    3.0      &  0.0349    & 30.7  \\\\\n\\multirow{1}{*}{\\object{A 85}}       &  \\textbf{0723802101\/2201}                &  195.8      &    6.1      &  0.0556    & 3.10  \\\\\n\\multirow{1}{*}{\\object{A 133}}      &  0144310101 \\textbf{0723801301\/2001}     &  168.1      &    3.8      &  0.0569    & 1.67  \\\\\n\\multirow{1}{*}{\\object{A 189}}      &  0109860101                              &   34.7      &    1.3      &  0.0320    & 3.38  \\\\\n\\multirow{1}{*}{\\object{A 262}}      &  0109980101\/0601 0504780101\/0201         &  172.6      &    2.2      &  0.0161    & 7.15  \\\\\n\\multirow{1}{*}{\\object{A 496}}      &  0135120201\/0801 0506260301\/0401         &  141.2      &    4.1      &  0.0328    & 6.00  \\\\\n\\multirow{1}{*}{\\object{A 1795}}     & 0097820101                               &   37.8      &    6.0      &  0.0616    & 1.24  \\\\\n\\multirow{1}{*}{\\object{A 1991}}     & 0145020101                               &   41.6      &    2.7      &  0.0586    & 2.72  \\\\\n\\multirow{1}{*}{\\object{A 2029}}     & 0111270201 0551780201\/0301\/0401\/0501     &  155.0      &    8.7      &  0.0767    & 3.70  \\\\\n\\multirow{1}{*}{\\object{A 2052}}     & 0109920101 0401520301\/0501\/0601\/0801     &  104.3      &    3.0      &  0.0348    & 3.03  \\\\\n                                     & 0401520901\/1101\/1201\/1301\/1601\/1701      &             &             &            &   \\\\\n\\multirow{1}{*}{\\object{A 2199}}     & 0008030201\/0301\/0601 \\textbf{0723801101\/1201} &  129.7 &    4.1      &  0.0302    & 0.909  \\\\\n\\multirow{1}{*}{\\object{A 2597}}     & 0108460201 0147330101 \\textbf{0723801601\/1701} & 163.9 &    3.6      &  0.0852    & 2.75  \\\\\n\\multirow{1}{*}{\\object{A 2626}}     & 0083150201 0148310101                    &   56.4      &    3.1      &  0.0573    & 4.59  \\\\\n\\multirow{1}{*}{\\object{A 3112}}     & 0105660101 0603050101\/0201               &  173.2      &    4.7      &  0.0750    & 1.38  \\\\\n\\multirow{1}{*}{\\object{A 3526}}     &  0046340101 0406200101                   &  152.8      &    3.7      &  0.0103    & 12.2  \\\\\n\\multirow{1}{*}{\\object{A 3581}}     &  0205990101 0504780301\/0401              &  123.8      &    1.8      &  0.0214    & 5.32  \\\\\n\\multirow{1}{*}{\\object{A 4038}}     & 0204460101 \\textbf{0723800801}           &   82.7      &    3.2      &  0.0283    & 1.62  \\\\\n\\multirow{1}{*}{\\object{A 4059}}     & 0109950101\/0201 \\textbf{0723800901\/1001} &  208.2      &    4.1      &  0.0460    & 1.26  \\\\\n\\multirow{1}{*}{\\object{AS 1101}}    & 0147800101 0123900101                    &  131.2      &    3.0      &  0.0580    & 1.17  \\\\\n\\multirow{1}{*}{\\object{AWM 7}}      & 0135950301 0605540101                    &  158.7      &    3.3      &  0.0172    & 11.9  \\\\\n\\multirow{1}{*}{\\object{EXO 0422}}   & 0300210401                               &   41.1      &    3.0      &  0.0390    & 12.4  \\\\\n\\multirow{1}{*}{\\object{Fornax}}     & 0012830101 0400620101                    &  123.9      &    1.2      &  0.0046    & 1.56  \\\\\n\\multirow{1}{*}{\\object{HCG 62}}     & 0112270701 0504780501 0504780601         &  164.6      &    1.1      &  0.0140    & 3.76  \\\\\n\\multirow{1}{*}{\\object{Hydra-A}}    & 0109980301 0504260101                    &  110.4      &    3.8      &  0.0538    & 5.53  \\\\\n\\multirow{1}{*}{\\object{M 49}}       & 0200130101                               &   81.4      &    1.0      &  0.0044    & 1.63  \\\\\n\\multirow{1}{*}{\\object{M 86}}       & 0108260201                               &   63.5      &    0.7      &  -0.0009   & 2.97  \\\\\n\\multirow{1}{*}{\\object{M 87} (Virgo)} & 0114120101 0200920101                    &  129.0      &    1.7      &  0.0042    & 2.11  \\\\\n\\multirow{1}{*}{\\object{M 89}}       & 0141570101                               &   29.1      &    0.6      &  0.0009    & 2.96  \\\\\n\\multirow{1}{*}{\\object{MKW 3s}}     & 0109930101 \\textbf{0723801501}           &  145.6      &    3.5      &  0.0450    & 3.00  \\\\\n\\multirow{1}{*}{\\object{MKW 4}}      & 0093060101 \\textbf{0723800601\/0701}      &  110.3      &    1.7      &  0.0200    & 1.88  \\\\\n\\multirow{1}{*}{\\object{NGC 507}}    &  \\textbf{0723800301}                     &   94.5      &    1.3      &  0.0165    & 6.38  \\\\\n\\multirow{1}{*}{\\object{NGC 1316}}   & 0302780101 0502070201                    &  165.9      &    0.6      &  0.0059    & 2.56  \\\\\n\\multirow{1}{*}{\\object{NGC 1404}}   & 0304940101                               &   29.2      &    0.6      &  0.0065    & 1.57  \\\\\n\\multirow{1}{*}{\\object{NGC 1550}}   & 0152150101 \\textbf{0723800401\/0501}      &  173.4      &    1.4      &  0.0123    & 16.2  \\\\\n\\multirow{1}{*}{\\object{NGC 3411}}   & 0146510301                               &   27.1      &    0.8      &  0.0152    & 4.55  \\\\\n\\multirow{1}{*}{\\object{NGC 4261}}   & 0056340101 0502120101                    &  134.9      &    0.7      &  0.0073    & 1.86  \\\\\n\\multirow{1}{*}{\\object{NGC 4325}}   & 0108860101                               &   21.5      &    1.0      &  0.0259    & 2.54  \\\\\n\\multirow{1}{*}{\\object{NGC 4374}}   & 0673310101                               &   91.5      &    0.6      &  0.0034    & 3.38  \\\\\n\\multirow{1}{*}{\\object{NGC 4636}}   & 0111190101\/0201\/0501\/0701                &  102.5      &    0.8      &  0.0037    & 2.07  \\\\\n\\multirow{1}{*}{\\object{NGC 4649}}   & 0021540201 0502160101                    &  129.8      &    0.8      &  0.0037    & 2.23  \\\\\n\\multirow{1}{*}{\\object{NGC 5044}}   & 0037950101 0584680101                    &  127.1      &    1.1      &  0.0090    & 6.24  \\\\\n\\multirow{1}{*}{\\object{NGC 5813}}   & 0302460101 0554680201\/0301\/0401          &  146.8      &    0.5      &  0.0064    & 6.24  \\\\\n\\multirow{1}{*}{\\object{NGC 5846}}   & 0021540101\/0501 \\textbf{0723800101\/0201} &  194.9      &    0.8      &  0.0061    & 5.12  \\\\\n\\multirow{1}{*}{\\object{Perseus}}    & 0085110101\/0201 0305780101               &  162.8      &    6.8      &  0.0183    & 20.7  \\\\\n\\hline                \n\\end{tabular}}\n\n$^{(a)}$ Exposure ID number. $^{(b)}$ RGS net exposure time. \n$^{(c)}$ Redshifts and temperatures are adapted from \\cite{Chen2007} and \\cite{Snowden2008}. \n$^{(d)}$ Hydrogen column density (see http:\/\/www.swift.ac.uk\/analysis\/nhtot\/).\nNew observations from our proposal are shown in boldface.\\\\\n      \\vspace{-0.5cm}\n\\end{table*}\n\n\n\\subsection{RGS spectra extraction}\n\\label{sec:rgs_regions}\n\nWe extracted the RGS source spectra in two alternative regions centered on the emission peak:\na broader 3.4' region, which includes most of the RGS field of view and a narrower 0.8' region that provides\nthe cluster cores but with high statistics. {This was done by launching \\textit{rgsproc} twice\nby setting the \\textit{xpsfincl} mask to include 99\\% and 90\\% of point-source events \ninside the spatial source extraction mask, respectively.} We have used the model background spectrum \ncreated by the standard RGS \\textit{rgsproc} pipeline, which is a template background file,\nbased on the count rate in CCD\\,9. \nThe RGS spectral extraction regions and the MOS\\,1 image of M\\,87 are shown in Fig.~\\ref{fig:rgs_regions}. \nThe spectra were converted to SPEX\\footnote{www.sron.nl\/spex} format through the SPEX task \\textit{trafo}.\n{During the spectral conversion, we chose the option of \\textit{sectors} in the task \\textit{trafo}\nto create as many sectors as the different exposures of each source. This permits us to simultaneously\nfit the multiple RGS spectra of each source by choosing which parameters to either couple or unbind \nin the spectral models of different observations.}\n\n\\begin{figure}\n  \\begin{center}\n      \\subfigure{ \n      \\includegraphics[bb=15 15 515 348, width=7.5cm]{ds9_M87_2_edited.ps}}\n      \\vspace{-0.25cm}\n      \\caption{RGS extraction regions and MOS\\,1 stacked image of M\\,87.}\n          \\label{fig:rgs_regions}\n  \\end{center}\n      \\vspace{-0.5cm}\n\\end{figure}\n\n \\vspace{-0.5cm}\n\n\\subsection{MOS 1 spatial broadening profiles}\n\\label{sec:spatial_profile}\n\nThe RGS spectrometers are slitless, and, therefore,\nthe spectra are broadened because of the spatial extent of the source in the dispersion direction. \nThe effect of this spatial broadening is described by the following wavelength shift\n\\begin{equation}\n\\Delta\\lambda = \\frac{0.138}{m} \\, \\Delta\\theta \\, {\\mbox{\\AA}},\n\\end{equation}\nwhere $m$ is the spectral order and $\\theta$ is the offset angle of the source in arcmin\n(see the XMM-\\textit{Newton} Users Handbook).\n\nThe MOS\\,1 DET\\,Y direction is parallel to the RGS\\,1 dispersion direction \nand can be used to correct for the spatial broadening. \n{With \\textit{evselect}}, we extracted MOS\\,1 images for each exposure in the 0.5--1.8\\,keV\n(7--25\\,{\\AA}) energy band and their surface brightness profiles in the dispersion\ndirection.\nWe account for spatial broadening using the \\textit{lpro} multiplicative model in SPEX,\nwhich convolves the RGS response with our model of the spatial extent of the source. \nWe show some cumulative profiles of spatial broadening in Fig.~\\ref{fig:profiles}.\nWe have also produced stacked Fe-L band (10--14\\,{\\AA}) images for each source. \nThe central 10' region contains most of the cluster emission \n(see Fig.~\\ref{fig:mos1}). \n\n\\begin{figure}\n  \\begin{center}\n      \\subfigure{ \n      \\includegraphics[bb=66 66 536 707, width=6cm, angle=90]{xxx_IDL_MOS1_comparing_profiles_overplots_sources.ps}}\n      \\vspace{-0.3cm}\n      \\caption{MOS\\,1 average 7--25\\,{\\AA} cumulative spatial profiles.}\n          \\label{fig:profiles}\n  \\end{center}\n      \\vspace{-0.4cm}\n\\end{figure}\n\n\\vspace{-0.4cm}\n\n\n\\section{Spectral modeling}\n\\label{sec:spectral_modeling}\n\nOur analysis focuses on the $7-28$ {\\AA} ($0.44-1.77$ keV) first and second order RGS spectra.\nWe model the spectra with SPEX\nversion 2.03.03. \nWe scale elemental abundances to the proto-Solar values \nof \\citet{Lodders09}, which are the default in SPEX. \nWe adopt C-statistics and $1\\,\\sigma$ errors throughout the paper,\nunless otherwise stated, and the updated ionization balance calculations of \\citet{Bryans2009}.\n\n\nClusters of galaxies are not isothermal, and most of them have both hot and cool gas phases \n(see e.g. \\citealt{Frank2013}). Therefore, we have used a two-temperature thermal plasma model\nof collisional ionization emission (CIE). This model is able to fit all the spectra in our database. \nThe \\textit{cie} model in SPEX calculates the spectrum of a plasma \nin collisional ionization equilibrium. The basis for this\nmodel is given by the mekal model, but several updates have been included (see the SPEX manual).\nFree parameters in the fits are the emission measure $Y=n_{\\rm e}\\,n_{\\rm H}\\,dV$, the temperature $T$, \nthe abundances (N, O, Ne, Mg, and Fe), and the turbulent broadening $v$ of the two \\textit{cie} models.\n\nWe bind the parameter $v$ and the abundances of two \\textit{cie} components with each other\nand assume that the gas phases have the same turbulence and abundances. \nThis decreases the degree of degeneracy. \n{This assumption is certainly not true, but some clusters just need one CIE component\nand the spectra of several clusters do not have good enough statistics in both high- and low-ionization emission lines,\nwhich prohibits constraining the velocities and the abundances for both hot and cool phases.\nWe attempt to constrain the turbulence in the different phases in Sect.\\,\\ref{sec:temperature}.}\n\nThe \\textit{cie} models are multiplied by a \\textit{redshift} model \nand a model for Galactic absorption, which is provided by the \\textit{hot} model in SPEX \nwith $T=0.5$\\,eV and $N_{\\rm H}^{TOT}$, as estimated through the tool of \\citet{Willingale2013}. \nThis tool includes the\ncontribution to absorption from both atomic and molecular hydrogen. The redshifts and\ncolumn densities that have been adopted are shown in Table~\\ref{table:log}.\nTo correct for spatial broadening, we have multiplied the spectral model \nby the \\textit{lpro} component that receives as input the surface brightness profile\nextracted in the MOS\\,1 images (see Sect.\\,\\ref{sec:spatial_profile} and Fig.\\,\\ref{fig:profiles}).\n\nWe do not explicitly model the cosmic X-ray background in the RGS spectra \nbecause any diffuse emission feature \nwould be smeared out into a broad continuum-like component. \n\nFor a few sources, including the Perseus and \\object{Virgo} (M\\,87) clusters, \nwe have added a further power-law emission component\nof slope $\\sim2$ to take the emission from the central AGN into account.\nThis component is not convolved with the spatial profile because \nit is produced by a central point-like source.\n\nTo avoid the systematic effects due to the stacking of multiple observations\nwith different pointing,\nwe have simultaneously fitted the individual spectra\nof each source extracted in the two regions defined in Sect.~\\ref{sec:rgs_regions} \nand shown in Fig.\\,\\ref{fig:rgs_regions}. \nThe plasma model is coupled between the observations. \nThe only uncoupled parameters are the emission measures \nof the two collisional-ionized gas components. \nFor each observation we adopt the spatial profile extracted in the MOS\\,1 image taken during that exposure.\nFor those exposures, during which the MOS\\,1 detector had a closed filter, \nwe have adopted an exposure-weighted average profile as given by the other available observations, \nbut the $\\delta\\lambda$ parameter in the \\textit{lpro} component is left free. \nThis factor allows us to shift the model lines by the same amount (in {\\AA}) for each specific spectrum\nand strongly decreases the systematic effects.\nThe $\\delta\\lambda$ parameter is always free in our fits to account for any redshift variation,\nwhich would otherwise affect the line modeling (see e.g. \\citealt{Sanders2011}).\n\nThe simultaneous modeling of multiple observations has been done through the use\nof the \\textit{sectors} option in SPEX (see also Sect.\\,\\ref{sec:rgs_regions}).\nThe RGS\\,1 and 2 spectra of the same observation\nhave exactly the same model and provide a single sector, while RGS spectra of other observations\ncontribute additional sectors and have the \\textit{cie} normalizations uncoupled.\n\n\\subsection{{Results using a fixed spatial broadening}}\n\\label{sec:results}\n\nWe have successfully applied this multi-temperature model to both the 3.4' and 0.8' RGS spectra.\nWe show the spectral modeling for the 3.4' region of the 44 sources \nin Figs.~\\ref{fig:rgs_fits}, \\ref{fig:rgs_fits2}, and \\ref{fig:rgs_fits3} in Appendix\\,\\ref{sec:appendix}. \nWe display the first-order stacked spectra to underline the high quality of these observations\nand to show the goodness of the modeling.\n\nFor some sources like Fornax, M\\,49, M\\,86, NGC\\,4636, and NGC\\,5813, \nthe 15 and 17\\,{\\AA} \\ion{Fe}{xvii} emission lines are not well fitted. \nPrecisely, the model underestimates the line peaks and overestimates the broadening.\nThis may be due to the different spatial distribution of the gas responsible for the cool \\ion{Fe}{xvii} \nemission lines and for the one producing most of the high-ionization Fe-L and \\ion{O}{viii} lines. \nThe cool gas is indeed to be found predominantly in the center of the clusters \nshowing a profile more peaked than that one of the hotter gas. \nThe estimated spatial profiles depend on the emission of the hotter gas due to its higher emission measure,\nand, therefore, they overestimate the spatial broadening of the 15--17\\,{\\AA} lines.\nIt is hard to extract a spatial profile for these lines because MOS\\,1 has a limited spectral resolution,\nand the images extracted in such a short band lack the necessary statistics \n(see e.g. \\citealt{Sanders2013}). {In Sect.\\,\\ref{sec:temperature}, we attempt to constrain the turbulence\nfor lines of different ionization states.} The 15\\,{\\AA}\\,\/\\,17\\,{\\AA} line ratio is also affected \nby resonant scattering, which would require a different approach. \nWe refer to a forthcoming paper on the analysis of the resonant scattering in the CHEERS sources.\n\nWe skip the discussion of the abundances and the supernova yields \nbecause these will be treated by other papers \nof this series (de\\,Plaa et al. in preparation and Mernier et al. submitted).\n\nIn Fig.\\,\\ref{fig:turbulence2} (\\textit{\\textit{left panel}}), we show the upper limits on the velocity broadening obtained \nwith the simultaneous fits of the 0.8' 7--28\\,{\\AA} RGS spectra.\nWe obtain upper limits for most clusters, while \nNGC\\,507 shows high kinematics. {More detail on our results for the 3.4' and 0.8' regions \nand their comparison are reported in Table\\,\\ref{table:velocity_results} and Fig.\\,\\ref{fig:velocities_comparison} (\\textit{\\textit{left panel}}).\nThe 3.4' limits are more affected by the source continuum,\nas clearly seen for M\\,87, AWM\\,7, and A\\,4038,\nwhich makes them less reliable.\n\n\\begin{figure*}\n  \\begin{center}\n      \\subfigure{ \n      \\includegraphics[bb=110 77 535 723, width=9cm]{CHEERS_turbulence_reg0_ABC_bryans09_AVG_10am_sectors_1sigma_sectors_all_cut_edited.txt.ps} \\hspace{0cm}  \n      \\includegraphics[bb=110 77 535 723, width=9cm]{CHEERS_turbulence_reg0_ABC_bryans09_AVG_10am_sectors_1sigma_sectors_all_cut_edited_sfree.txt.ps} }\n      \\vspace{-0.2cm}\n      \\caption{\\textit{Left panel}: Velocity 68\\% (red) and 90\\% (green) limits for the 0.8' region\n               with the spatial broadening determined with MOS\\,1 images (see Sect.\\,\\ref{sec:results}). \n               \\textit{Right panel}: Velocity limits obtained using the best-fit spatial broadening\n               (see Sect\\,\\ref{sec:results_combined}).}\n          \\label{fig:turbulence2}\n  \\end{center}\n      \\vspace{-0.6cm}\n\\end{figure*}\n\n\\subsection{{Results using the best-fit spatial broadening}}\n\\label{sec:results_combined}\n\nIt is known that the spatial profile of the source continuum may be\nbroader than the spatial distribution of the lines. The MOS\\,1 \nimages are strongly affected by the profile of the source continuum\nand, therefore, may overestimate the spatial line broadening\nand underestimate the residual velocity broadening. \nFor instance, NGC\\,1316 and NGC\\,5846 \nshow $1\\sigma$ limits of 20\\,km\\,s$^{-1}$, which are not realistic (see Table\\,\\ref{table:velocity_results}).\n\nTo obtain more conservative limits, we have simultaneously modeled\nthe spatial and the velocity broadening. This was done by fitting the RGS 0.8' spectra\nwith a free \\textit{s} parameter in the \\textit{lpro} component.\nThis factor simply scales the width of the spatial broadening \nby a factor free to vary (see the SPEX manual).\nThe free \\textit{s} parameter increases the degeneracy in the model \nbut provides conservative upper limits on the residual velocity broadening,\nwhich is measured with the $v$ parameter of the \\textit{cie} component.\nThe new limits on the velocities are plotted in Fig.\\,\\ref{fig:turbulence2} (\\textit{right panel})\nand quoted in the last two columns of Table\\,\\ref{table:velocity_results}.\nIn Fig.\\,\\ref{fig:velocities_comparison} (\\textit{right panel}), we compare the velocity upper limits\nestimated with the standard method (MOS\\,1 spatial profile with $s\\equiv1$ in the \\textit{lpro} component)\nwith this new approach using a free $s$ parameter. They generally agree,\nbut the new upper limits on the hotter Abell clusters\nare systematically larger by an average factor $\\sim2$. \nThis confirms that some of the previous velocity limits were \nunderestimated due to the broader spatial profiles.\nTherefore, we believe the new upper limits to be the most conservative.\nInterestingly, the conservative velocity limits of the hot clusters \nare generally higher than the cool galaxy groups with the exception of NGC\\,507,\nwhich is expected since the sound speed scales as a power\nof the temperature (see Sect.\\,\\ref{sec:turbulence}).\n\n      \\vspace{-0.4cm}\n\n\\subsection{Further tests}\n\\label{sec:tests}\n\nTo estimate the contribution of the spatial broadening to the line widths,\nwe have temporarily removed the convolution of the spectral model \nfor the spatial profile and re-fitted the data. \nIn these fits the $v$ parameter of the \\textit{cie} component \naccounts for any contribution to the line broadening. \nThe total (spatial + velocity) widths are also quoted in Table\\,\\ref{table:velocity_results}.\n\nWe have also tested the continuum-subtracted line surface brightness profiles \nintroduced by \\cite{Sanders2013}. This new method consists of subtracting the surface brightness profiles\nof two regions that are clearly line-dominated (core) and continuum-dominated (outskirts).\nIt can be applied only to those objects with a narrow core \nwhere it is possible to distinguish between line-rich and line-poor regions.\nWe have locally fitted the \\ion{O}{viii} 19.0\\,{\\AA} emission line of \nA\\,2597, A\\,3112, Hydra-A, Fornax (\\object{NGC\\,1399}), and NGC\\,4636 \nand we have found a general agreement with the results of \\cite{Sanders2013}.\nHowever, our MOS\\,1 images have much lower spatial resolution than the \\textit{Chandra} maps\nused by them, which increases the uncertainties that are present in this method. \nA thorough, extensive, analysis would require deep \\textit{Chandra} maps \nthat are not yet available.\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this work we have analyzed the data of 44 clusters, groups of galaxies, and elliptical galaxies\nincluded in the CHEERS project, a Very Large Program that was accepted \nfor XMM-\\textit{Newton} AO-12 (see Sect.\\,\\ref{sec:cheers}) \ntogether with complementary archival data.\n\nWe have measured upper limits of velocity broadening for these objects\nwith a method similar to the previous one used by \\citet{Bulbul2012} and \\citet{Sanders2013}. \nThis consists of fitting high-quality grating spectra\nby removing the spatial broadening through surface brightness profiles of the sources as provided\nby CCD imaging detectors. \nThese profiles are unfortunately affected by the source continuum\nand tend to overestimate the line spatial broadening with a consequent\nshrinking of the residual velocity broadening. \n\\citet{Sanders2013} addressed this point in their Sect.\\,2.2 on A\\,3112\nwhere they decreased these systematic effects by using \\textit{Chandra} continuum-subtracted\nline spatial profiles. We have tested this method by using the MOS\\,1 observations\nthat were taken simultaneously with the RGS spectra (see Fig.\\,\\ref{fig:mos1}). \nXMM-\\textit{Newton} CCDs have a spatial resolution lower than \\textit{Chandra} CCDs, \nwhich increase the systematic effects in the creation of continuum-subtracted maps.\nDeep \\textit{Chandra} observations, enabling an accurate  \nsubtraction of different energy bands, are missing for most sources.\nWe have therefore tried to use the MOS\\,1 integral maps\nand to fit the contribution of the spatial broadening\nas an alternative method.\n\n\n\\subsection{Temperature dependence of the upper limits}\n\\label{sec:temperature}\n\n{So far we adopted the same velocity broadening for all the emission lines.\nFor most sources it is possible to measure the velocity broadening of the \n\\ion{O}{viii} and \\ion{Fe}{xx-to-xxiv} emission lines, which are mainly produced by hot gas.\nOnly a few sources have high-statistics \\ion{Fe}{xvii} lines produced by cool ($T<1$\\,keV) gas.\nSix objects exhibit both strong low- and high-ionization lines \nand allow to fit the velocity broadening of the two \\textit{cie} components, separately,\nin the full-band spectral fits.\n17 sources allow to measure 90\\% upper limits on turbulence\nfor the \\ion{O}{viii}, \\ion{Fe}{xvii}, and \\ion{Fe}{xx} lines, \nby fitting the $18.0-23.0$\\,{\\AA}, $14.0-18.0$\\,{\\AA}, and $10.0-14.3$\\,{\\AA} \nrest-frame wavelength ranges, respectively. \nFor each local fit we adopt an isothermal model and correct for spatial broadening\nby using additional surface-brightness profiles calculated through MOS\\,1 images\nextracted in the same rest-frame wavelength ranges using the same method\nshown in Sect.\\,\\ref{sec:spatial_profile}. These profiles are still affected by the \ncontinuum but provide a better description of the spatial broadening in each line.\nIn Fig.\\,\\ref{fig:turb_vs_band} we compare the \\ion{O}{viii} velocity limits\nwith those measured for the \\ion{Fe}{xvii} and \\ion{Fe}{xx} line systems.\nThe high-ionization Fe lines clearly show higher upper limits,\nwhich is confirmed by the results of the full-band fits: \nthe hotter (T1) CIE component allows for higher values of velocity broadening.\nThe hotter gas is distributed over a larger extent than that of the cold (T2) gas \nand has larger spatial broadening,\nwhich affects the T1-T2 results shown in this plot. The \\ion{O}{viii}, \n\\ion{Fe}{xvii}, and \\ion{Fe}{xx} lines were fitted by subtracting the spatial broadening \nextracted exactly in their energy band, which should partly correct this systematic effect,\nbut it is difficult to estimate the systematic uncertainties\ndue to the low spatial (and spectral) resolution of the CCD data.\nOn some extent, the hotter phase may still have larger turbulence. \nFor clarity, we also tabulate these line-band fits in Table\\,\\ref{table:physical_properties}.\nWe also note that the the velocity limits of low- and high-ionization iron lines\nfall at opposite sides of the Fe--\\ion{O}{viii} 1:1 line, which means that the metallicity \ndistribution in the sources should not affect our broad-band, multi-ion, limits \nshown in Fig.\\,\\ref{fig:turbulence2}.}\n\n\\begin{figure}\n      \\subfigure{ \n      \\includegraphics[bb=58 54 540 730, width=6.5cm, angle=+90]{IDL_line_spatial_comparison_bands.ps}}\n      \\caption{{90\\% upper limits on velocity broadening obtained in the 0.8' region \n               for the \\ion{O}{viii} lines compared with those measured for high-ionization \\ion{Fe}{xx} \n               (open red triangles) and for the low-ionization \\ion{Fe}{xvii} (filled black triangles) line systems.\n               For six sources we could also measure the limits for the hot (open green circles) \n               and the cool (open magenta boxes) CIE components {(see Sect.\\,\\ref{sec:temperature})}.\n                \\label{fig:turb_vs_band}}}\n\\end{figure}\n\n\n\\subsection{Turbulence}\n\\label{sec:turbulence}\n\nIn Fig.\\,\\ref{fig:turbulence2} we show the velocity broadening \nof the RGS spectra extracted in the 0.8' core region. \nWe find upper limits to the velocity broadening with the possible exception of NGC\\,507. \nThey generally range between 200 and 600\\,km\\,s$^{-1}$. \nFor several objects like A\\,85, A\\,133, M\\,49, and most NGC elliptical\nwe found velocity levels below 500\\,km\\,s$^{-1}$, which would\nsuggest low turbulence.\nThe broader 3.4' region is more affected by spatial broadening \nas shown by the higher upper limits, but non-detection, of\nA\\,4038, AWM\\,7, and M\\,87 (see Table\\,\\ref{table:velocity_results}\nand Fig.\\,\\ref{fig:velocities_comparison}). \nFor these sources it is difficult to constrain the velocity broadening \nbecause their large extent smears out the emission lines.\n\nTo understand how much energy can be stored in turbulence, we \ncompare our upper limits with the sound speeds and the temperatures of \ndominant \\textit{cie} component in these objects.\nThe sound speed is given by $c_S = \\sqrt{\\gamma \\, k \\, T \/ \\mu m_{\\rm p}}$,\nwhere $\\gamma$ is the adiabatic index, which is 5\/3 for ideal monoatomic gas,\n$T$ is the RGS temperature, $\\mu=0.6$ is the mean particle mass, and $m_p$ is proton mass.\nThe ratio between turbulent and thermal energy is \n$\\varepsilon_{\\rm turb}\/\\varepsilon_{\\rm therm}=\\gamma\/2\\,M^2$,\nwhere $M=v_{\\rm turb}\/c_S$ is the Mach number (see also \\citealt{Werner2009}).\nIn Fig.\\,\\ref{fig:velocities_comparison} we compare our $2\\sigma$ upper limits \non the velocities in the central 0.8' region\nwith the sound speed and some fractions of turbulent energy.\nWe also show the more conservative velocity upper limits\nthat were measured with a variable spatial broadening. \nAt least for half the sample, our 90\\% upper limits are below the sound speed in the system.\nIn about ten objects the turbulence contains less than the 40\\%\nof the thermal energy. This is similar to the previous results\nof \\cite{Sanders2013}. Apparently, the hotter objects allow for higher velocities.\n\n{We note that the spectral extraction region had a fixed angle, and, therefore,\nthe actual physical scale -- where we estimated the velocity broadening -- depends on the source distance.\nIn Fig.\\,\\ref{fig:turb_vs_temp_scaled} (\\textit{left panel}), we show the Mach numbers\nfor the 90\\% conservative upper limits as a function of the temperature, and we compare\nthe average upper limits on Mach number calculated within different ranges of physical scales.\nThere is no significant trend with the temperature,\nbut the average upper limit on the Mach number is lower for narrower physical scales.\nAssuming a Kolmogorov spectrum for the turbulence in these objects, \nthe root-mean-square velocity scale depends on the 1\/3rd power of the physical length.\nTherefore, we scaled the upper limits by ${(sc\/sc_{\\rm min})}^{1\/3}$, \nwhere $sc_{\\rm min}$ is the \nminimum physical scale per arcsec $\\sim0.07$\\,kpc\/1\" of NGC\\,4636,\nthe nearest object in our sample.\nIn other words, we divided our upper limits by the relative physical scale per arcsec \nrelative to NGC\\,4636, which is equivalent to normalizing by the ratio\nbetween the size of the spectral extraction region of each cluster\nand that one of NGC\\,4636.\nThe scaled upper limits on the Mach numbers are tabulated \nin Table\\,\\ref{table:physical_properties} and plotted\nin Fig.\\,\\ref{fig:turb_vs_temp_scaled} (\\textit{right panel}).\nThey are randomly distributed around $Ma\\sim0.8$ and \ndo not depend any more on the physical scale. \nWe coded the point-size and the colors with the values of $r_{500}$ and $K_0$ taken from \nthe literature. The $r_{500}$ is the radius within which the mean over-density \nof the cluster is 500 times the critical density at the cluster redshift,\nand $K_0$ is the value of the central entropy in the same cluster.\nAll the adopted values and their references are reported in Table\\,\\ref{table:physical_properties}.\nWe do not find any significant relation between the upper limits on the Mach number and \nthese physical properties, possibly due to the limited sample.}\n\n{To understand whether dissipation of turbulence may prevent cooling\nin our sample, we computed the Mach number that is required to balance \nthe heating and cooling, according to the following equation:}\n\\begin{equation}\\label{eq:mach}\nMa_{REQ} \\approx 0.15 \\left( \\frac{n_e}{10^{-2} \\, {\\rm cm}^{-3}} \\right)^{1\/3} \n                   \\, \\left( \\frac{c_s}{10^3 \\, {\\rm km\\,s}^{-1}} \\right)^{-1} \n                   \\, \\left( \\frac{l}{10 \\, {\\rm kpc}} \\right)^{1\/3}\n\\end{equation}\n{where $n_e$ is the density at the cavity location, $c_s$ the sound speed\nthat we have estimated through the RGS temperature, \nand $l$ the characteristic eddy size, which we take as the\naverage cavity size (see \\citealt{Zhuravleva2014}).\nThe Mach numbers required to balance cooling are tabulated \nin Table\\,\\ref{table:physical_properties}.\nMost cavity sizes were taken from \\citet{Panagoulia2014b}.\nFor clusters with multiple cavities, we used an average size.\nFor the 19 sources outside of their sample, we used their $r-T$ relation \nto determine the cavity size. Most densities were taken from the ACCEPT\ncatalog.\n}\n\n{In Fig.\\,\\ref{fig:Mach_vs_temp_scaled}, we compare the ratios between \nthe conservative upper limits of the scaled Mach numbers assuming Kolmogorov turbulence,\nand those that are required to balance cooling with the RGS temperatures.\nFor most sources, our upper limits are larger than the balanced Mach numbers,\nwhich means that dissipation of turbulence can provide enough heat\nto prevent the cooling of the gas in the cores.}\n\n{It is difficult to know which is the main mechanism that produces turbulence\nin these objects. Our scaled upper limits are mostly below 500\\,km\\,s$^{-1}$,\nwhich can be produced by bubbles inflated by past AGN activity (see, e.g., \\citealt{Bruggen2005}).\nFor some objects, our upper limits are consistent with velocities up to 1000\\,km\\,s$^{-1}$, \nwhich would correspond to Mach numbers larger than one.\nFor NGC\\,507, we detect transonic motions presumably due to merging\n(see, e.g., \\citealt{Ascasibar2006}).\nIn a forthcoming paper, we will analyze the resonant scattering of the \\ion{Fe}{xvii} lines\nexhibited by half of our sample to place lower limits on turbulent broadening \nand provide more insights on its origin and its role in preventing cooling.}\n\n\\begin{figure}\n      \\subfigure{ \n      \\includegraphics[bb=65 85 525 686, width=6.8cm, angle=+90]{IDL_sound_speed_combined_referee_Mach_needed.ps}}\n      \\vspace{-0.5cm}\n      \\caption{{Ratios between the 90\\% conservative upper limits on the Mach number (velocity \/ sound speed)\n               that are scaled by the 1\/3rd power of the spatial scalel assuming Kolmogorov turbulence\n               (see {Sect.\\,\\ref{sec:turbulence}}), and the Mach number, which is required\n               to make a heating--cooling balance (see Eq.\\,\\ref{eq:mach}).\n               The point size provides the $r_{500}$, and the color is coded according to the central entropy,\n               $K_0$, in units of keV cm$^{2}$.\n                \\label{fig:Mach_vs_temp_scaled}}}\n\\end{figure}\n\n\n\\subsection{Comparison with previous results}\n\\label{sec:comparison}\n\nOur velocity limits broadly agree with the previous results obtained by \\citet{Sanders2013} \nusing a similar method and by other authors, who use the measurements\nof resonant scattering (\\citealt{Werner2009} and \\citealt{dePlaa2012}).\nIn particular, our limits for M\\,49 (also known as \\object{NGC\\,4472}), \nNGC\\,4636, and NGC\\,5813 \nagree with the $100$\\,km\\,s$^{-1}$ upper limit obtained by \\citet{Werner2009}.\nWe also found upper limits of a few $100$s\\,km\\,s$^{-1}$ for A\\,3112, which is similar\nto the results of \\citet{Bulbul2012}.\nHowever, we measured higher limits with a variable spatial broadening\nthat agree with continuum-subtracted profiles method of \\citet{Sanders2013}.\n\nRecently, \\citet{Zhuravleva2014} used the surface brightness fluctuations \nin the \\textit{Chandra} images of the Perseus and Virgo clusters to derive turbulent \nvelocities in the range 70--210\\,km\\,s$^{-1}$ for Perseus and 43--140\\,km\\,s$^{-1}$ for Virgo,\n{where the smaller values refer to the central 1.5' region.}\n{Our upper limits in the cores of the clusters are consistent with their values,\nespecially when normalized by the physical scale factor $1.5'\/0.4'$.}\nThey show that these turbulent motions should dissipate enough energy to \noffset the cooling of the central ICM in these clusters.\n{For ten objects, the scaled Mach number can be transonic, and a major fraction of energy\ncan be stored in turbulence, which could significantly heat the gas through dissipation \n(see, e.g., \\citealt{Ruszkowski2004}).\nRecently, \\citet{Gaspari2014} noted that even if the turbulence \nin the hot gas is subsonic, it may be transonic in the cooler gas phases.\n\\citet{Zhuravleva2014} reported that  dissipation of turbulence may balance cooling\neven under subsonic regime.\nOur upper limits on Mach number are larger than the values necessary to balance cooling\nand are consistent with this scenario.\nHowever, it is possible that other processes are dominant, \nsuch as turbulent mixing (see, e.g., \\citealt{Banerjee2014}).}\n\nThe NGC\\,507 group exhibits velocities larger than $1000$\\,km\\,s$^{-1}$ in both the 0.8' and 3.4' regions,\n{corresponding to a scaled Mach number $Ma=4.2\\pm1.7$ (1$\\sigma$)}. \nThe 15\\,{\\AA} \\ion{Fe}{xvii} line is stronger than the one at 17\\,{\\AA}, which\nwould suggest low resonant scattering (see Fig.\\,\\ref{fig:rgs_fits3})\nand, therefore, high kinematics in the galaxy group.\nThis object is known to have a disturbed shape and to host radio lobes presumably\nin a transonic expansion\/inflation (\\citealt{Kraft2004}). However, our high values\nsuggest the presence of bulk motions. \n{In Fig.\\,\\ref{fig:NGC507}, we show the velocities of the galaxies in the NGC\\,507 group\nas taken from \\cite{Zhang2011}. They are not necessary\nlinked to that of the ICM, but there are high kinematics and hints \nof infalling clumps, which indicate a substructure extended toward the observer.\nIn this group, the galaxy velocities generally double those observed in NGC\\,4636,\nwhere we measure lower velocity broadening (see also the different line widths in Fig.\\,\\ref{fig:rgs_fits3}).}\n\n\\begin{figure}\n  \\begin{center}\n      \\subfigure{ \n      \\includegraphics[bb=50 115 535 590, width=7.5cm, angle=-90]{vlos_dist_leallr500.ps}}\n  \\vspace{-0.4cm}\n      \\caption{Line-of-sight velocity versus projected distance from the central cD galaxy \n               for the member galaxies of NGC\\,507 group.\n               Optical spectroscopic redshifts are taken from \\cite{Zhang2011}.}\n          \\label{fig:NGC507}\n  \\end{center}\n\\end{figure}\n\n\\subsection{Toward ASTRO-H}\n\\label{sec:simulations}\n\nThe RGS gratings aboard XMM-\\textit{Newton} are currently the only instruments that can\nmeasure $100$s\\,km\\,s$^{-1}$ velocities in X-ray spectra of extended sources like clusters of galaxies. \nHowever, they are slitless spectrometers and, therefore, affected by spatial broadening.\nWe have partly solved this issue by using line surface brightness profiles, \nbut there are still systematic uncertainties larger than 100\\,km\\,s$^{-1}$.\nOur models provide an important workbench once the new ASTRO-H\nX-ray satellite (\\citealt{Takahashi2010}) is launched. The spectra, as provided by its microcalorimeter (SXS),\ndo not suffer from spatial broadening as for the RGS and will revolutionize the method. \nMoreover, its constant spectral resolution\nin terms of energy increases the sensitivity at high energies, which allows us \nto use higher-ionization lines up to 6-7\\,keV (Fe-K line complex)  \nnecessary to constrain the turbulence in hotter gas phases. \nThe position of the lines unveil evidence of bulk motions.\n\nIn Fig.\\,\\ref{fig:simulations} (\\textit{\\textit{left panel}}), we compare the effective area of the ASTRO-H SXS\nwith that of the first order RGS 1 and 2. ASTRO-H provides clearly better results than the sum of RGS\\,1 and 2 below 14\\,{\\AA} \n(above 1\\,keV). The RGS has still a better spectral resolution than the SXS \nin the wavelength range that includes the \\ion{Fe}{xvii} lines of the cool gas,\nbut the absence of spurious line-broadening in the SXS makes it a great alternative tool.\nWe have simulated a 100\\,ks exposure with the ASTRO-H SXS for four interesting objects\nin our catalog: Perseus (500\\,km\\,s$^{-1}$), NGC\\,5846 (10\\,km\\,s$^{-1}$), NGC\\,4636 (100\\,km\\,s$^{-1}$), \nand NGC\\,507 (1000\\,km\\,s$^{-1}$, see Fig.\\,\\ref{fig:simulations} \\textit{right panel}). \nWe have used the model fitted for the full (-1.7',+1.7') RGS spectra as a template, which are shown\nin Fig.\\,\\ref{fig:rgs_fits3}, because this extraction region is \ncomparable to the 3.05'\\,$\\times$\\,3.05' field-of-view of the microcalorimeter.\nThe spatial broadening was excluded from the model. \nThe simulated SXS spectra are characterized by a richness of resolved emission lines, which provides\nvelocity measurements with an accuracy of 50\\,km\\,s$^{-1}$ or better.\nThe line widths clearly increase throughout NGC\\,5846, NGC\\,4636, and NGC\\,507.\nThe hotter gas present in the Perseus cluster produces strong higher-ionization lines\nabove 1\\,keV, which constrain the turbulence in different (Fe-L and Fe-K) gas phases. \n\nWe also note that the 1' spatial resolution of ASTRO-H provides,\nfor the first time the means for a spatially-resolved high-resolution spectral analysis\nand the measurements of turbulence in different regions of the clusters.\nThe ATHENA X-ray observatory that is to be launched by the late 2020s will further \nrevolutionize our measurements due to its combined high spectral (2.5\\,eV) and spatial ($<5$'') resolution.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nWe have presented a set of upper limits and measurements of the velocity widths \nfor the soft X-ray emitting gas of a sample of clusters, groups of galaxies, \nand elliptical galaxies included in the CHEERS project. \nWe have subtracted the instrumental spatial broadening \nthrough the use of surface brightness profiles\nextracted in the MOS\\,1 images. \n\nFor most sources, we obtain upper limits ranging within 200-600\\,km\\,s$^{-1}$, \n{where the turbulence may originate in AGN feedback or sloshing of the ICM.\nHowever, for some sources, such as NGC\\,507, we find upper limits of 1000\\,km\\,s$^{-1}$ \nor larger, suggesting other origins, such as mergers and bulk motions.\nThe measurements depend on the angular scale and the temperature.\nFor a small sample producing strong high- and low-ionization\nlines, we measured significantly broader upper limits for the hot gas phase,\nwhich may be partly due to its larger spatial extent as compared\nto the cool phase.\nWhen we normalize the Mach numbers for the physical scale, assuming \nKolmogorov turbulence, we constrain upper limits ranging within\n$0.3<Ma<1.5$. These values are above the Mach numbers necessary to balance\ncooling, which means that the dissipation of turbulence\nmay be the dominant mechanism to heat the gas and quench cooling flows.\nHowever, it is possible that additional processes are heating the ICM.}\nIn a forthcoming paper, we will use the resonant-scattering technique \nto place some lower limits on the velocities in half of our sample,\nwhich are characterized by strong \\ion{Fe}{xvii} emission lines, and \nmake an extensive use of higher-resolution \\textit{Chandra} maps.\nThis will provide alternative measurements and further insights\non the origin and the role of turbulence in clusters, groups of galaxies,\nand elliptical galaxies.\n\nThe current techniques are partly limited by systematics associated with the \nspatial line broadening,\nbut the future ASTRO-H and ATHENA missions \nwill dramatically improve the method due to their high broadband spectral resolution\nand to the absence of issues concerning the spurious spatial broadening.\nThey will also enable spatially-resolved turbulence measurements.\n\n\\begin{acknowledgements}\nThis work is based on observations obtained with XMM-\\textit{Newton}, an\nESA science mission funded by ESA Member States and the USA (NASA).\nYYZ acknowledges the BMWi DLR grant 50~OR~1304.{We thank Electra Panagoulia\nfor kindly providing the values of central entropy\nand the anonymous referee for very useful comments on the paper.}\n\\end{acknowledgements}\n\n\\begin{figure*}\n      \\subfigure{ \n      \\includegraphics[bb=118 78 533 689, width=5cm, angle=-90]{RGS_AstroH_effarea.ps}}\n      \\subfigure{ \n      \\includegraphics[bb=65 45 555 755, height=10.cm,width=6cm, angle=-90]{4clusters_AstroH_sim_Vedit.ps}}\n  \\vspace{-0.3cm}\n      \\caption{\\textit{Left panel}: RGS 1 and 2 first order and ASTRO-H SXS effective area.\n               \\textit{Right panel}: ASTRO-H SXS 100\\,ks simulations for the bright Perseus cluster and three groups of galaxies\n               zoomed in the RGS energy band (see Sect.\\,\\ref{sec:simulations}). \n               We adopted the 3.4' RGS models as a template (see Fig.\\,\\ref{fig:rgs_fits3}).}\n          \\label{fig:simulations}\n\\end{figure*}\n  \n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}