Non-Markovian Brownian motion in a periodic potential is investigated with special attention to diffusion properties. The velocity autocorrelation function (VAF) and diffusion constant are calculated for three types of random forces whose correlations are given by delta function (the white noise), exponential and damped-oscillatory functions. Some comments are also given for the relation between the diffusion constants calculated from the VAF and the hopping rate.
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Igarashi et al. (1988) studied this question.
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