Effective diffusion in one-dimensional rough potential-energy landscapes

Abstract

Diffusion in spatially rough, confining, one-dimensional continuous energy landscapes is treated using Zwanzig's proposal, which is based on the Smoluchowski equation. We show that Zwanzig's conjecture agrees with Brownian dynamics simulations only in the regime of small roughness. Our correction of Zwanzig's framework corroborates well with numerical results. A numerical simulation scheme based on our coarse-grained Langevin dynamics offers significant reductions in computational time. The mean first-passage time problem in the case of random roughness is treated. Finally, we address the validity of the separation of length scales assumption for the case of polynomial backgrounds and cosine-based roughness. Our results are applicable to hierarchical energy landscapes such as that of a protein's folding and transport processes in disordered media, where there is clear separation of length scale between smooth underlying potential and its rough perturbation.

An effective one-dimensional approach to calculating mean first passage time in multi-dimensional potentials

“…Finally, previous work demonstrated that the effects of constant-amplitude roughness upon the mean first-passage time in one dimension can be accounted for easily, provided that there is a separation in length-scales between the roughness and the underlying potential. 32 We believe that this result continues to hold for this two-dimensional, constant-curvature potential. See the supplementary material for details.…”

“…In the supplementary material, we calculated the mean first passage time for a special class of potential where there is a clear separation of length-scales between the (short-scale) roughness of the potential and the underlying potential (long-scale variation). The analysis relies heavily on a recent paper, 32 which is an extension of Zwanzig's work. 36 In this case, the short-scale roughness adds a multiplicative factor that can be accounted for in an elegant manner.…”

Dynamics, statistics, and task allocation of foraging ants

“…The growth in the order parameters is due to the discovery of the food source by the first ant and the subsequent formation of a pheromone trail. The time at which the order parameter changes nontrivially is commonly known as the first-passage time (FPT), or first hitting time, in stochastic processes [22,79,80], and indicates when a stochastic event occurs for the first time. Due to the random nature of this time, it is more meaningful to look at the mean first-passage time (MFPT) instead, which in our case, is the average time taken for the first ant from the colony to find the food source.…”