Emergent Hydrodynamics in an Exclusion Process with Long-Range Interactions

TL;DR

This paper investigates a long-range interacting exclusion process, revealing non-local hydrodynamics and providing exact solutions for certain initial states, bridging stochastic processes and quantum spin models.

Contribution

It introduces a new long-range exclusion process with Coulomb interactions, maps it to a quantum spin chain, and derives non-local hydrodynamic equations with exact solutions.

Findings

Hydrodynamics governed by a non-local PDE involving the Hilbert transform.

Exact solutions for melting block initial states and limit shapes.

Model demonstrates emergent non-local hydrodynamics from long-range interactions.

Abstract

We study the symmetric Dyson exclusion process (SDEP) - a lattice gas with exclusion and long-range, Coulomb-type interactions that emerge both as the maximal-activity limit of the symmetric exclusion process and as a discrete version of Dyson's Brownian motion on the unitary group. Exploiting an exact ground-state (Doob) transform, we map the stochastic generator of the SDEP onto the spin-$1/2$ XX quantum chain, which in turn admits a free-fermion representation. At macroscopic scales we conjecture that the SDEP displays ballistic (Eulerian) scaling and non-local hydrodynamics governed by the equation $\partial_t \rho+\partial_x j[\rho]=0$ with $j[\rho]=(1/\pi)\sin(\pi\rho(x,t))\sinh(\pi\mathcal{H}\rho(x,t))$, where $\mathcal{H}$ is the Hilbert transform, making the current a genuinely non-local functional of the density. This non-local one-field description is equivalent to a local two-field "complex Hopf" system for finite particle density. Closed evolution formulas allow us to solve the melting of single and double block initial states, producing limit shapes and arctic curves that agree with large-scale Monte Carlo simulations. The model thus offers a tractable example of emergent non-local hydrodynamics driven by long-range interactions.