Phase space volume scaling of generalized entropies and anomalous diffusion scaling governed by corresponding non-linear Fokker-Planck equations

TL;DR

This paper establishes a direct relationship between the phase space volume scaling of generalized entropies and the anomalous diffusion scaling governed by non-linear Fokker-Planck equations, linking transient and stationary process characteristics.

Contribution

It introduces a simple algebraic relation connecting the scaling exponents of anomalous diffusion and generalized entropy, unifying their descriptions of complex stochastic processes.

Findings

The scaling exponents are bijectively related.

Stationary and transient behaviors are characterized by the same information.

The classification of processes is consistent across different measures.

Abstract

Many physical, biological or social systems are governed by history-dependent dynamics or are composed of strongly interacting units, showing an extreme diversity of microscopic behaviour. Macroscopically, however, they can be efficiently modeled by generalizing concepts of the theory of Markovian, ergodic and weakly interacting stochastic processes. In this paper, we model stochastic processes by a family of generalized Fokker-Planck equations whose stationary solutions are equivalent to the maximum entropy distributions according to generalized entropies. We show that at asymptotically large times and volumes, the scaling exponent of the anomalous diffusion process described by the generalized Fokker-Planck equation and the phase space volume scaling exponent of the generalized entropy bijectively determine each other via a simple algebraic relation. This implies that these basic measures characterizing the transient and the stationary behaviour of the processes provide the same information regarding the asymptotic regime, and consequently, the classification of the processes given by these two exponents coincide.