Universal asymptotic solution of the Fokker-Planck equation with time-dependent periodic potentials

Abstract

Brownian motion, as one of the most fundamental concepts in statistical physics, has everlasting interests in interdisciplinary fields in the past century. Although this motion with static potentials have been widely explored, its physics in time-dependent periodic potentials are far less well understood. Here we generalize this motion to the realm of time-dependent periodic potentials, showing that the asymptotic solution of the probability distribution function (PDF) can have a universal form, that is, a Boltzmann weight multiplied by a Gaussian envelope function. We derive a partial equation for this Boltzmann weight and demonstrate that many different potentials can give the same Boltzmann weight. We first present an exact solvable model to illustrate the validity of our solution. For the periodic potential with a time-dependent tilt potential, we can determine the Boltzmann weight by numerical solving the partial equation. These results are confirmed by solving the Langevin equation numerically. With this idea, we can determine the asymptotic solution of the Fokker-Planck equation, in which the entropy satisfies the thermodynamic law. Our results can have wide applications, including quasi-periodic potentials, two-dimensional potentials and even with models with inertia, which should greatly broaden our perspective of Brownian motion.

Figures (4)

• Figure 1: The Brownian motion and PDF with $w = B + A \cos(\omega t) \cos(x) + C \cos(x)$ and $f(t)=0$. (a) The variance $\sigma(t)= \langle x^2\rangle - \langle x\rangle^2$ versus $t$, from which the slope yields the effective diffusion constant $D^*$. (b) PDF at $t = 2000$. (c) $\varphi(x, t) = \ln Bp/\mathcal{G}$ versus $x$ at $t = 2000$ from theory and numerical simulation, without fitting parameters involved. (d) $Bp/e^{\varphi}$, yielding the Gaussian profile. (e) Demonstration of the Gaussian profile in the large $x$ limit. (f) $\varphi = \ln Bp/\mathcal{G}$ for various $t$.
• Figure 2: The Brownian motion and PDF with $w = B + A \cos(\omega t) \cos(x) + C \cos(x)$ and $f(t) = f_0+f_1 \cos(\sqrt{2}t)$. (a) The mean position $x_c = \langle x \rangle$ versus $t$, where the Gaussian envelope is given by Eq. \ref{['eq-gauss']}. (b) The variance $\sigma(t)= \langle x^2 \rangle-\langle x \rangle^2$, from which the slope yields the effective diffusion constant $D^*$. (c) PDF at $t = 200$. (d) $\varphi(x, t) = \ln Bp/\mathcal{G}$ versus $x$ at $t = 200$ from theory and numerical simulation, without fitting parameters involved. (e) $Bp/e^{\varphi}$, yielding the Gaussian profile. (f) Demonstration of the Gaussian envelope in the large $x$ limit.
• Figure 3: The same as Fig. \ref{['fig-fig2']} with $\phi = U \cos(x) - a \sin(\omega t) x$. The PDF in (c)-(f) consider a Gaussian envelope $\mathcal{G}(x-x_c,t)$, where $x_c = \langle x \rangle$ at $t = 500$ as shown in (a). The theoretical $\varphi(x,t)$ in (d) is computed by $\varphi = \ln w$, where $w$ is the solution to Eq. \ref{['eq-WJphix']} obtained using finite difference method.
• Figure 4: The potential $\varphi$ and effective diffusion constant $D$ in the high frequency limit. (a) The theoretical and numerical result of $\varphi(x,t)$ at $t= 200$, with $\omega =10$. (b) $D^*$ and variance of position (inset) as a function of $\omega$. The black dashed line represents $D^*$ calculated using LJ formula.

Theorems & Definitions (1)

• Theorem 1