Subordination and memory dependent kinetics in diffusion and relaxation phenomena

Abstract

AbstractThe concept of subordination, originally introduced in the probability and stochastic processes theories, has also appeared in analysis of evolution equations. So it is not surprising that we meet it in physics of complex systems, in particular when study equations describing diffusion and dielectric relaxation phenomena. Grace to intuitively understood decomposition of complex processes into their simpler and better known components, called parent and leading processes, subordination formalism enables us to attribute physical interpretation to integral decompositions representing plethora of solutions to anomalous diffusion and relaxation problems. Moreover, it makes investigation of properties obeyed by these solutions far easier and more effective. Using the Laplace-Fourier transform method to solve memory-dependent evolution equations we show that subordination can be naturally implemented in their solutions. The key to achieve this goal is the use of operational calculus merged with the application of the Efros theorem [1]. Adopting exclusively methods of classical mathematical analysis we are able to derive the memory-stemmed origin of subordination and build a bridge connecting functional analysis/operator calculus based methods of solving the evolution equations with well established stochastic and probabilistic approaches. With such a developed general formalism in hands we apply it to several models of anomalous diffusion and relaxation phenomena.