Fluctuation-dissipation relations and universal behavior for relaxation processes in systems with static disorder and in the theory of mortality

A unified description for the parallel relaxation in systems with static disorder and for the competitive risk mortality theory in population biology is suggested by combining the physical and biological approaches presented in the literature. A multichannel parallel decay process is investigated by assuming that each channel is characterized by a state vector x and by a probability of decaying $p(\mathrm{x};t)$. A general fluctuation-dissipation relation is derived which relates the effective decay rate of the process to the fluctuations of the density of channels characterized by different state vectors. A limit of the thermodynamic type in x space is introduced for which both the volume available and the average number of channels tend to infinity, but the average volume density of channels remains constant. By using scaling arguments combined with a stochastic renormalization group approach, two types of universal laws are identified in the thermodynamic limit for the relaxation (survival) function corresponding to nonintermittent and intermittent fluctuations of the density of channels, respectively. For nonintermittent fluctuations the general relaxation equation of Huber is recovered, which includes the stretched exponential equation as a particular case, whereas for intermittent fluctuations a more complicated universal relaxation equation is obtained which includes Huber's equation, the stretched exponential, and the inverse power law relaxation equations as particular cases.