Some phenomenological relaxation rate equations based on Bose–Einstein similar kinetics

Abstract

The possibility of describing transient phenomena associated with flow and consolidation of solids, such as stress relaxation or physical aging, in terms of a kinetic mechanism comprising spontaneous and induced events is discussed. The starting point is the differential equation dṅ/dt=−aṅ[1−(b/a)ṅ], with n denoting the number of relaxed entities and ṅ=dn/dt (a,b are constants, t is time), yielding an ṅ(t) function reminiscent of a Bose–Einstein distribution. The corresponding n(t) relation describes the linear variation of n with log t, and the exponential dependence of ṅ on n, as often found experimentally. Replacing ṅ in the starting equation by the relative rate ṅ/n yields a power-law-type ṅ(n) dependence. A further modification, where the induction term ṅ/n is not linear but raised to a power ≳1, finally produces a generalized version of the stretched exponential. When interpreted formally in terms of a spectrum of relaxation times τ, all three equations produce response functions with discrete τ distributions, provided a≠0.