Diffusion with restrictions

Abstract

A non-linear diffusion equation is derived by taking into account hopping rates depending on the occupation of next neighbouring sites. There appears additonal repulsive and attractive forces leading to a changed local mobiltiy. The stationary and the time dependent behaviour of the system are studied based upon the master equation approach. Different to conventional diffusion it results a time dependent bump the position of which increases with time described by an anomalous diffusion exponent. The fractal dimension of this random walk is exclusively determined by the space dimension. The applicabilty of the model to descibe glasses is discussed. 05.60.+w, 05.70.Ln, 82.20.Mj Typeset using REVT E X 1

Glassy dynamics of kinetically constrained models

“…As for the 1, 1-SFM, the spin autocorrelation function is then for short times just the return probability of a random walker in d dimensions with diffusion constant D eff . This is the d-th power of the result (116) for d = 1, giving for short times C(t) = (1−ct) d = 1−dct+O((ct) 2 ) consistent with the early time scaling found in [214].…”

“…A model similar to one-vacancy assisted hopping, but on a d-dimensional hypercubic lattice, was studied in [116]. There, the hopping rate was taken to be proportional to the number of vacancies on the n.n.…”

“…sites surrouding the hop path; hops are therefore allowed only if at least one such vacancy is present. Unfortunately the analysis of this model given in [116] was flawed since it involved an approximation which violates conservation of particle number.…”

Competing Glauber and Kawasaki Dynamics

“…The evolution equation for the higher order terms appearing in eq. ( 13) could be calculated in a straightforward manner, see 18 . However, in the large N-limit those terms has not to be evaluated provided the system is an ergodic one.…”

“…The method has been applied to study a spin-facilitated model 17 where the dynamics is restricted by specific constraints. It can be also used to find the above mentioned transition probability 18 which is not unique. It should be emphasized that the method is determined by the algebraic properties of the operators where an extension to a p-state model 19 and to a Q-statistics is straightforward 20 .…”