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 $\frac{t}{\tau }$, where $\tau$ is a constant characteristic time. In the transformed time frame $\theta$ this process simulates one with TITR and the relaxation function is $\exp (-\frac{\theta }{{\tau }_s})$, where ${\tau }_s$ is a constant. It is shown that the only time-scale transformation that has the property $\theta \left(\tau \right)={\tau }_s$ and which converts a TDTR into a TITR is a monomial $\theta \mathrm{}\left(t\right)=a{t}^b$. Equivalently the TDTR is $t^{b-1\mathrm{}}$. This class of processes has a relaxation function of the form $\exp (-a{t}^b)$. This model is widely applicable in the description of measured relaxation behavior and leads naturally to Ngai's renormalization relation for activation energies.