Memory effects and macroscopic manifestation of randomness

Abstract

It is shown that due to memory effects the complex behavior of components in a stochastic system can be transmitted to macroscopic evolution of the system as a whole. Within the Markov approximation widely used in ordinary statistical mechanics, memory effects are neglected. As a result, a time-scale separation between the macroscopic and the microscopic level of description exists, the macroscopic differential picture is not a consequence of microscopic nondifferentiable dynamics. On the other hand, the presence of complete memory in a system means that all its components have the same behavior. If the memory function has no characteristic time scales, the correct description of the macroscopic evolution of such systems has to be in terms of the fractional calculus.