  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.
