Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: Extracting information from transients

A Langevin process describing diffusion in a periodic potential landscape has a time-dependent diffusion constant, which means that its average mean-squared displacement (MSD) only becomes linear at late times. The long-time, or effective diffusion, constant can be estimated from the slope of a linear fit of the MSD at late times. Due to the crossover between a short time microscopic diffusion constant, which is independent of the potential, to the effective late-time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a nonzero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.

Figure 1

$\kappa \left(t\right)t$ estimated from the MSD in a numerical simulation of ${10}^5$ Langevin particles in the potential in Eq. (67) given by $V\left(x\right)=cos\left(x\right)$, $\beta =3$, and $l=4.0$ (continuous) black line. Shown as thick dashed line is the linear fit of $\kappa \left(t\right)t$ for $t>1$ yielding the estimate ${\kappa }_e=0.04258$ and $C=0.15432$. The analytical predictions are ${\kappa }_e=0.04198$ and $C=0.15778$.

Figure 2

A homogeneous potential $\varphi \left(x\right)=0$ perturbed by a narrow barrier with height $V_0>0$ or a well of depth $V_0<0$.

Figure 3

The scaled relaxation amplitude $12C/{\xi }^2{(1-\xi )}^2{l}^2$ in (73) versus $\beta V_0$ for $\xi =0.9$.