Anomalous diffusion approach to non-exponential relaxation in complex physical systems

• Considered the general stochastic scenario for the description of relaxation.
• Based on subordination by inverse infinity divisible random processes.
• Derived the kinetic equation under memory effects.
• Obtained memory functions related to empirical relaxation laws.
• Clusterization in complex systems is an origin of power-law memory effects.

We derive the relaxation function from the simple model of two-state systems under memory effects caused by the subordination. The non-exponential relaxation is shown to result from subordination by inverse infinity divisible random processes. The wide class of such random processes includes ordinary α-stable, tempered α-stable, exponential, gamma processes and many others as particular cases. This approach generalizes the Cole–Cole, Cole–Davidson and Havriliak–Negami laws well known in experimental physics of relaxation. The presented considerations discover a direct (one-to-one) relationship between the method of random relaxation rates and the anomalous diffusion approach based on subordination of random processes that are applied for the theory of relaxation phenomena. Moreover, it is found that the space and time clusterizations are responsible on equal foots for power-law memory effects in relaxation of complex physical systems.