Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. 19, 1 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multiwell potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov) equation, the minimization of the free-energy functional, and a continuation algorithm for the stationary solutions.

Figure 1

A two-dimensional (2D) multiscale potential.

Figure 2

A polynomial potential of the form (12), where $M=3$.

Figure 3

A realization of a potential of the form (13) with 10 local minima located at the integers between $-5$ and 5 ($c_{\ell }=1$) and separated by arbitrary heights $[{\delta }_{\ell }\sim U\left([0,1]\right)]$.

Figure 4

Free-energy surface (a) and the corresponding bifurcations of the steady states of the Fokker-Planck equation (b) in a simple bistable potential $V\left(x\right)=\frac{x^4}{4}-\frac{x^2}{2}$ (c). The control parameter is $\beta$, and the solution norm is given by the first moment $m$, with $\theta =0.5$. In (a) and (b), solid and dotted lines correspond to stable and unstable steady states of the Fokker-Planck equation, respectively.

Figure 5

Phase diagram in the $\beta -m$ space, for the potential $V_8\left(x\right)$, at $\theta =1.5$ (a) and $\theta =2.5$ (b). Solid and dotted branches are, respectively, stable and unstable steady states of the Fokker-Planck equation.

Figure 6

Critical temperature ${\beta }_C$ as a function of $\theta$ for the potentials $V_6\left(x\right)$ with $h=0.1$ (left panel) and $V_8\left(x\right)$ with $h=0.001$ (right panel) given by Eqs. (21) and (22), respectively.

Figure 7

Bifurcation diagrams of $m$ as a function of $\beta$ for tilted bistable potentials given by Eq. (25) with $a_0=0.249998581434761$, and $\kappa =0,0.01,0.1,1$ (see legend). Here, we used $\theta =2.5$. The symmetric pitchfork bifurcation is broken at any $\kappa >0$. We note that the locus of the critical points forms a distinct critical line.

Figure 8

Evolution of the density and first moment in a tilted bistable potential given in Eq. (25), with $\kappa =0.1$ and an initial condition distributed according to a $N(0,{\beta }^{-1})$ distribution. (a) Mean position against time for the evolution of the Fokker-Planck equation (blue line) and for the interacting particles system (red line). (b) Fokker-Planck distributions and corresponding MC histograms for selected times, designated in (a) by vertical dashed lines of respective colors (see also the Supplemental Material [57] movies MovM1.avi, showing simultaneously the first moment on the bifurcation diagram and the distribution, and MovFig8.avi, showing good agreement between Fokker-Planck and MC simulations).

Figure 9

Evolution of the density and first moment in a tilted bistable potential given in Eq. (25), with $\kappa =0.1$ and an initial condition distributed according to a $N(-0.1,{\beta }^{-1})$ distribution. (a) Mean position against time for the evolution of the Fokker-Planck equation (blue line) and for the interacting particles system (red line). (b) Fokker-Planck distributions and corresponding Monte Carlo histograms for selected times, designated in (a) by vertical dashed lines of respective colors (see also movie MovFig9.avi in Supplemental Material [57] showing good agreement between Fokker-Planck and MC simulations).

Figure 10

Free-energy surface (a) and the corresponding bifurcation diagram in the $\beta -m$ parameter space (b) for the potential in Eq. (27) with $\theta =5$ and $h=1$. Stable and unstable branches are plotted with solid and dotted lines, respectively.

Figure 11

(a) A potential from Eq. (26), with the minima positioned at integers between $-10$ and 10, separated by local maxima of arbitrary heights. (b) Corresponding bifurcation diagram of the steady states of the Fokker-Planck equation (4), obtained from the computed free-energy surface. Stable and unstable states are designated by solid and dashed curves, respectively.

Figure 12

Numerical solution of the Fokker-Planck equation (4), for the potential given in Fig. 11 (see also MovFig12.avi in the Supplemental Material [57]). (a) $p(x,t)$ (solid curves) and $V\left(x\right)$ (dashed curve). The first moments for each $p(x,t)$ are designated by dashed verticals, and the respective values of $t$ are provided. (b) States from (a) on the bifurcation diagram. We note that the basins of attraction of the stable states are effectively demarcated by the metastable branches of the bifurcation diagram.

Figure 13

Same as in Fig. 12, against the corresponding MC simulation data. (a) Mean position against time for the evolution of the Fokker-Planck equation (full blue line) and the interacting particles system for two realizations of the noise (full and dashed red lines). (b) Fokker-Planck distributions and corresponding MC histograms for selected times, designated in (a) by vertical dashed lines of respective colors. We can see an overall good agreement between the simulations and computations.

Figure 14

Phase diagram for the case of the piecewise linear potential (28) with six wells for $\theta =5$. (a) Shows the solution of $R\left(m\right)-m=0$ for $\beta =1$ (black) and $\beta =10$ (gray). A horizontal dotted line is drawn at $R\left(m\right)-m=0$. (b) Shows the bifurcation diagram in the $\beta -m$ space. (c) Shows the critical inverse temperature ${\beta }_C$ at which a pitchfork bifurcation occurs from the mean-zero solution as a function of $\theta$.

Figure 15

Phase diagram for the case of the piecewise linear potential (28) with four wells at random heights and depths for $\theta =5$. (a) Shows the solution of $R\left(m\right)-m=0$ for $\beta =1$ (black) and $\beta =10$ (gray). A horizontal dotted line is drawn at $R\left(m\right)-m=0$. (b) Shows the bifurcation diagram in the $\beta -m$ space. (c) Shows the critical inverse temperature ${\beta }_C$ at which a pitchfork bifurcation occurs from the mean-zero solution as a function of $\theta$.