Overdamped limit and inverse-friction expansion for Brownian motion in an inhomogeneous medium

We revisit the problem of the overdamped (large-friction) limit of the Brownian dynamics in an inhomogeneous medium characterized by a position-dependent friction coefficient and a multiplicative noise (local temperature) in one-dimensional space. Starting from the Kramers equation and analyzing it through the expansion in terms of eigenfunctions of a quantum harmonic oscillator, we derive analytically the corresponding Fokker-Planck equation in the overdamped limit. The result is fully consistent with the previous finding by Sancho, San Miguel, and Dürr [J. Stat. Phys. 28, 291 (1982)]. Our method allows us to generalize the Brinkman's hierarchy, and thus it would be straightforward to obtain higher-order corrections in a systematic inverse-friction expansion without any assumption. Our results are confirmed by numerical simulations for simple examples.

Figure 1

Probability distribution function $\widehat{P}(x,t)$ at $t=5$. We take $\gamma \left(x\right)={\gamma }_0(1+e^{-x^2/2}),T\left(x\right)=2/\left[(1+e^{-x^2/2}){(1+2x^2)}^2\right]$, and $f\left(x\right)=0$ with a large value of ${\gamma }_0=10$. Circles (without approximation) and crosses (correct limit) represent the data obtained from Eqs. (30) and (31), respectively, which overlap each other very well. Squares (Ito) and triangles (anti-Ito) represent the data obtained from Eq. (32) with $\alpha =0,1$, respectively.

Figure 2

Probability distribution function $\widehat{P}(x,t)$ at $t=10$. We take $\gamma \left(x\right)={\gamma }_0[1+\frac{4.4x}{3(x^2+1)}],T\left(x\right)={(3+\frac{4x}{x^2+1})}^2/\left[4(3+\frac{4.4x}{x^2+1})\right]$, and $f\left(x\right)=-2x$ with ${\gamma }_0=30$. The same symbols are used as described in the legend of Fig. 1.

Figure 3

Probability distribution function $\widehat{P}(x,t)$ at $t=0.2$. We take $\gamma \left(x\right)={\gamma }_0[1+\frac{4.4x}{3(x^2+1)}],T\left(x\right)={(3+\frac{4x}{x^2+1})}^2/\left[4(3+\frac{4.4x}{x^2+1})\right]$, and $f\left(x\right)=-2x$ with ${\gamma }_0=10$. For initial conditions, we take the Gaussian distribution with variance 4 centered on $x=1$ for the position and the Maxwell velocity distribution with local temperature $T\left(x\right)$. Data obtained from Eqs. (30) and (27) (with exponential correction) overlap each other very well, while data obtained from Eq. (20) show a noticeable difference.

Figure 4

Probability distribution function $\widehat{P}(x,t)$ at $t=1$. We take $\gamma \left(x\right)=1+e^{-x^2/2},T\left(x\right)=1/\left[2(1+e^{-x^2/2}){(x^2+1)}^2\right]$, and $f\left(x\right)=-x/5$ with a small value of $m=0.01$. The same symbols are used as described in the legend of Fig. 1.