One-dimensional diffusion with stochastic random barriers

The dynamics of a one-dimensional Brownian particle are treated theoretically over a wide time scale in the presence of random reflecting barriers. Green functions for particle motion with two kinds of barriers are formulated, one being diffusion with fixed random barriers and the other with stochastic random barriers. The stochastic barriers are expected to appear randomly on the time axis as well as with positional disorder and then, after a while, to disappear. To treat these problems, we first develop a mean-field Green-function MFG theory. The effects of many barriers, regarded as a multiple perturbation to the free diffusion of the particle, are decomposed into many elements. Their average effects (the first cumulant of the random perturbation elements) are sequentially incorporated into the unperturbed Green function up to the number of the elements. The first-cumulant MFG is thus obtained and covers the whole range of the perturbation intensity of the mean element. To improve the approximation, the second cumulants of the perturbation elements are incorporated one by one into the first-cumulant MFG, yielding the second-cumulant MFG. Taking into account a higher-order correlation among the perturbation elements yields a higher-order cumulant MFG. By applying the first-cumulant and the second-cumulant MFG theories to the particle diffusion with fixed and with stochastic barriers, we calculated the mean-square displacement as a function of time. It was found that the first-cumulant MFG describes qualitatively the dynamic behavior of the particle on all the time scales, whereas the second-cumulant MFG gives a more accurate numerical coefficient in the estimation of the time-dependent mean-square displacement.