Competition between homogeneous and local processes in a diffusive many-body system

TL;DR

This paper analyzes a stochastic many-body system with particle injection and local trapping, revealing how the trap affects particle density profiles and steady states across different dimensions.

Contribution

It provides exact solutions for the particle concentration, density profiles, and steady states in a diffusive system with local traps, across various dimensions.

Findings

Density depletion depends on spatial dimension.

Exact steady-state concentrations are derived.

Long-time behavior of particle concentration is characterized.

Abstract

We consider a stochastic many-body system where a source refills uniformly the empty sites of a hypercubic lattice, on which each particle is allowed to jump (symmetrically) onto neighboring vacant sites. In addition, there is a local {\it trap}, in competition with the injection reaction, which perturbs the dynamics by removing particles from the system. In dimensions $d=1, 2$ and 3, for an ``imperfect'' and a ``perfect'' trap, the spatiotemporal effect of the local perturbation of the dynamics is investigated by computing the exact concentration of particles in the system and it is shown that the density profile exhibits a depletion (in one and two dimensions) which properties depend on the space dimension. The exact reactive (fluctuating) steady state and the long-time behavior of the concentration of particles are explicitly computed.