Isothermal Langevin dynamics in systems with power-law spatially dependent friction

We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially dependent diffusion coefficient of the form $D\left(x\right)\sim {\left|x\right|}^c$, at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At long times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for $c<0$ and superdiffusion for $0<c<1$. For $c>1$, the diffusion equation predicts that the particles accelerate. Here we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit ($〈x^2〉\sim t^2$). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin and is correctly captured by Langevin's equation.

Figure 1

(a) The PDF at $t=1000$ obtained from the numerical integration (open circles) vs. the PDF given by Eq. (5) (solid line), which solves Fick's diffusion equation for $c=-0.5$. (b) Same as (a) but for $c=0.5$. (c) The MSD $\langle x^2\rangle$ of the particle as a function of time for $c=-0.5,0.25,0.5,0.75$ (markers) vs. the expected $\langle x^2\rangle$ for $c=-0.5,0.25,0.5,0.75$ according to Eq. (6) (lines).

Figure 2

The PDF at (a) $t=10$ and (b) $t=1000$ of a Brownian particle starting at the origin and moving in a medium where $\alpha \left(x\right)=1/(1+x^2)$.

Figure 3

(a) Circles show the bimodal VPDF at $t=1000$. The dashed line depicts the initial Gaussian equilibrium VPDF. (b) Computed MSD of the particle (circles) vs. the asymptotic power-law form $\langle x^2\rangle =2.41t^2$ (solid line).

Figure 4

(a) Ensemble averages of ${\left(m\mathrm{\Delta }v\right)}^2$ [first term on the lhs of Eq. (17), thin solid line], $2m\mathrm{\Delta }v\mathrm{\Delta }A$ (second term, thick solid line), and ${\left(\mathrm{\Delta }A\right)}^2$ (third term, dashed line), as a function of time $t$. (b) Sum of the three averages shown in (a) (dashed line) vs. the average of $2\alpha k_{B}T$ integrated over time [rhs of Eq. (17), solid line], as a function of $t$.