From the nonlinear Fokker-Planck equation to the Vlasov description and back: Confined interacting particles with drag

Nonlinear Fokker-Planck equations endowed with power-law diffusion terms have proven to be valuable tools for the study of diverse complex systems in physics, biology, and other fields. The nonlinearity appearing in these evolution equations can be interpreted as providing an effective description of a system of particles interacting via short-range forces while performing overdamped motion under the effect of an external confining potential. This point of view has been recently applied to the study of thermodynamical features of interacting vortices in type II superconductors. In the present work we explore an embedding of the nonlinear Fokker-Planck equation within a Vlasov equation, thus incorporating inertial effects to the concomitant particle dynamics. Exact time-dependent solutions of the $q$-Gaussian form (with compact support) are obtained for the Vlasov equation in the case of quadratic confining potentials.

Figure 1

The quotient $g\left(t\right)/g\left(0\right)$, where $g\left(t\right)={\beta }_{vv}exp(-2at)$ [see Eq. (46)] and $g\left(0\right)$ stands for the initial value of $g\left(t\right)$, is plotted against time, for $a=1$, $B=1$, $m=1$, $C=1$, ${\rho }_{0,i}=1$, and ${\mathrm{\Delta }}_0={(\pi /3)}^2$. The quantity $g\left(t\right)/g\left(0\right)$ is dimensionless, and time is measured in units of $a^{-1}$.

Figure 2

Plots of ${\beta }_{xx}\left(t\right)$ against time (a) and ${\beta }_{vv}\left(t\right)$ against time (b). The values of the physical parameters characterizing the system are the same as those detailed in the caption of Fig. 1. The quantity ${\beta }_{xx}$ is measured in units of ${\left[\frac{m{a}^2}{B}\right]}^{2/3}$, ${\beta }_{vv}$ in units of ${\left[\frac{m}{Ba}\right]}^{2/3}$, and time in units of $a^{-1}$.

Figure 3

Plot of the scaled spatial density $\sqrt{g\left(t\right)}\tilde{\rho }\left(x\right)$ as a function of the scaled and centered spatial variable $[x-x_0\left(t\right)]/\sqrt{g\left(t\right)}$. The scaled density $\sqrt{g}\tilde{\rho }$ is measured in units of ${\left[\frac{m^{2}a}{B^2}\right]}^{1/3}$ and the spatial coordinate $[x-x_0\left(t\right)]/\sqrt{g}$ in units of ${\left[\frac{B^2}{m^{2}a}\right]}^{1/3}$.

Figure 4

Plot of ${\beta }_{xx}$ against ${\beta }_{vv}$ (a) and ${\beta }_{xv}$ against ${\beta }_{vv}$ (b) along one orbit of the conservative system ($a=0$), with the other physical parameters characterizing the system being the same as in Fig. 1. The quantity ${\beta }_{vv}$ is measured in units of ${\left[\frac{m^3}{B^{2}C}\right]}^{1/3}$, ${\beta }_{xx}$ is measured in units of ${\left[\frac{C}{B}\right]}^{2/3}$, and ${\beta }_{xv}$ in units of $\sqrt{m/C}{\left[C/B\right]}^{2/3}$.

Figure 5

Plots of ${\beta }_{xx}$ (a), ${\beta }_{vv}$ (b), and ${\beta }_{xv}$ (c), against time, for the conservative case. The quantity ${\beta }_{xx}$ is measured in units of ${\left[\frac{C}{B}\right]}^{2/3}$, ${\beta }_{vv}$ in units of ${\left[\frac{m^3}{B^{2}C}\right]}^{1/3}$, ${\beta }_{xv}$ in units of $\sqrt{m/C}{\left[C/B\right]}^{2/3}$, and time in units of $\sqrt{m/C}$.

Figure 6

Plot of the effective potential $\mathrm{\Phi }\left(z\right)$ given by Eq. (54). The quantities $z$ and $\mathrm{\Phi }$ are respectively measured in units of ${\left[\frac{m^3}{B^{2}C}\right]}^{1/6}$ and ${\left[\frac{C}{B}\right]}^{2/3}$.