Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

TL;DR

This paper develops an age-structured hydrodynamic theory for large ensembles of anomalously diffusing particles with renewal resetting, revealing steady-state behaviors and densities under various resetting protocols.

Contribution

It introduces a novel age-structured hydrodynamic framework for anomalous diffusion with resetting, accounting for inter-particle correlations and analyzing different resetting models.

Findings

Steady states are reached in all models analyzed.

Steady-state densities differ significantly between models, with some having compact support.

The theory applies to various anomalous diffusion processes with resetting.

Abstract

We develop an age-structured hydrodynamic (HD) theory which describes the collective behavior of $N\gg 1$ anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle -- the time since its last reset -- as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. We apply this theory to three different resetting protocols: independent resetting to the origin (model~A), resetting to the origin of the particle farthest from it (model~B), and a scaled-diffusion extension of the ``Brownian bees" model of Berestycki et al, Ann. Probab. \textbf{50}, 2133 (2022). In all these models non-equilibrium steady states are reached at long times, and we determine the steady-state densities. For model A the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all $H>0$. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols which introduce global inter-particle correlations.