Limit properties of periodic one dimensional hopping model

TL;DR

This paper investigates the limiting behaviors of velocity and diffusion in a periodic one-dimensional hopping model, establishing theoretical equivalences with Langevin and Fokker-Planck descriptions, and providing insights into microscopic particle motion in thermal environments.

Contribution

It derives the limits of velocity and diffusion constants as the number of states increases and demonstrates their equivalence with Langevin and Fokker-Planck formulations.

Findings

Limits of velocity and diffusion constants as states tend to infinity.

Theoretical equivalence of hopping model and Langevin/Fokker-Planck formulations.

Numerical results show diffusion coefficients are nearly identical across models.

Abstract

Periodic one dimensional hopping model is useful to study the motion of microscopic particles, which lie in thermal noise environment. The mean velocity $V_N$ and diffusion constant $D_N$ of this model have been obtained by Bernard Derrida [J. Stat. Phys. 31 (1983) 433]. In this research, we will give the limits $V_D$ and $D_D$ of $V_N$ and $D_N$ as the number $N$ of mechanochemical sates in one period tends to infinity by formal calculation. It is well known that the stochastic motion of microscopic particles also can be described by overdamped Langevin dynamics and Fokker-Planck equation. Up to now, the corresponding formulations of mean velocity and effective diffusion coefficient, $V_L$ and $D_L$ in the framework of Langevin dynamics and $V_P, D_P$ in the framework of Fokker-Planck equation, have also been known. In this research, we will find that the formulations $V_D$ and $V_L, V_P$ are theoretically equivalent, and numerical comparison indicates that $D_D, D_L$, and $D_P$ are almost the same. Through the discussion in this research, we also can know more about the relationship between the one dimensional hopping model and Fokker-Planck equation.