# Limit properties of periodic one dimensional hopping model - Condensed Matter > Statistical Mechanics

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Abstract: Periodic one dimensional hopping model is useful to study the motion ofmicroscopic particles, which lie in thermal noise environment. The meanvelocity $V N$ and diffusion constant $D N$ of this model have been obtained byBernard Derrida J. Stat. Phys. 31 1983 433. In this research, we will givethe limits $V D$ and $D D$ of $V N$ and $D N$ as the number $N$ ofmechanochemical sates in one period tends to infinity by formal calculation. Itis well known that the stochastic motion of microscopic particles also can bedescribed by overdamped Langevin dynamics and Fokker-Planck equation. Up tonow, the corresponding formulations of mean velocity and effective diffusioncoefficient, $V L$ 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 thisresearch, we will find that the formulations $V D$ and $V L, V P$ aretheoretically equivalent, and numerical comparison indicates that $D D, D L$,and $D P$ are almost the same. Through the discussion in this research, we alsocan know more about the relationship between the one dimensional hopping modeland Fokker-Planck equation.

Author: ** Yunxin Zhang**

Source: https://arxiv.org/