Subdiffusive master equation with space dependent anomalous exponent: `Black Swan' effects

TL;DR

This paper derives a space-dependent fractional master equation for subdiffusion, revealing that non-uniform anomalous exponents lead to unstable stationary states and dominate long-term behavior through rare events.

Contribution

It introduces a fractional master equation with spatially varying anomalous exponents and analyzes its asymptotic behavior both analytically and via simulations.

Findings

Constant anomalous exponent equations are not structurally stable under spatial variations.

The Gibbs-Boltzmann distribution is not the stationary solution in non-homogeneous cases.

Long-term behavior is dominated by rare, unlikely events in the distribution of the anomalous exponent.

Abstract

We derive the fractional master equation with space dependent anomalous exponent. We analyze the asymptotic behavior of corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent $\mu $ in a bounded domain $[ 0,L]$ are not structurally stable with respect to the non-homogeneous variations of parameter $\mu $. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of $% \mu$ in space and find that in the long time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events.