The Paley-Wiener Criterion for Relaxation Functions.

It is shown that a rigorous mathematical theorem in the theory of Fourier transforms due to Paley and Wiener provides the bound for physically acceptable relaxation functions for long times. The exponential decay function, exp-ttau, with a constant relaxation time tau, and hence also a superposition of exponential decay functions corresponding to a distribution of relaxation times, does not provide an acceptable description of relaxation phenomena. On the other hand, the assumable bound of the Paley-Wiener theorem does. This bound turns out to have exactly the same form as a class of relaxation functions that have been successfully applied in the description of many relaxation phenomena in condensed matter. An important consequence of the Paley-Wiener theorem is the necessity for time-dependent relaxation rates which provides insight into the reason for deviation from exponential behavior for long times. Author