Abstract
A new, time-local (TL) reduced equation of motion for the probability distribution of excitations in a disordered system is developed. ToO(k2) the TL equation results in a Gaussian spatial probability distribution, i.e, 〈P(r, t)〉 = [(2πξ)1/2]−dexp(-r2/2ξ2), where ξ = ξ(t) is a correlation length, andr = ¦r¦. The corresponding distribution derived from the Hahn-Zwanzig (HZ) equation is more complicated and assumes the asymptotic (r→ ∞) form: 〈P(r, s)〉(sξ d)−1exp(−r/ξ) · (r/ξ)(1-d)/2 where ξ = ξ(s),d is the space dimensionality, ands is the Laplace transform variable conjugate tot. The HZ distribution generalizes the scaling form suggested by Alexanderet al. ford= 1. In the Markov limit ξ(t)√t, ξ(s)1/√s, and the two distributions are identical (ordinary diffusion).
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Mukamel, S. On the dynamics of excitations in disordered systems. J Stat Phys 30, 179–184 (1983). https://doi.org/10.1007/BF01010873
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DOI: https://doi.org/10.1007/BF01010873