Relaxation processes and time-scale transformations

Stochastic processes with a special class of time-dependent transition rates (TDTR) that can be related to time-independent transition rates (TITR) by time-scale transformations are considered. A proper subclass of TDTR processes are those which have their associated relaxation function depending only on the ratio tτ, where τ is a constant characteristic time. In the transformed time frame θ this process simulates one with TITR and the relaxation function is exp(-θτₛ), where τₛ is a constant. It is shown that the only time-scale transformation that has the property θ(τ)=τₛ and which converts a TDTR into a TITR is a monomial θ(t)=atᵇ. Equivalently the TDTR is t^b-1. This class of processes has a relaxation function of the form exp(-atᵇ). This model is widely applicable in the description of measured relaxation behavior and leads naturally to Ngai's renormalization relation for activation energies.