Emergent Hydrodynamics in an Exclusion Process with 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 $\partial_t ρ+\partial_x j[ρ]=0$ with $j[ρ]=(1/π)\sin(πρ(x,t))\sinh(π\mathcal{H}ρ(x,t))$, 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.

Figures (3)

• Figure 1: Evolution of the density profile from a single block initial condition (left) and from a double-block initial condition (right); the dashed gray line shows the initial density profile. We compare the result of the stochastic simulation of the SDEP for $N=100$ particles on $L=300$ sites with periodic boundary conditions, to the hydrodynamic prediction (black continuous lines) obtained by solving equation (\ref{['PP']}) numerically. The profiles for the stochastic evolution are averaged over 1000 independent realizations.
• Figure 2: Emergence of a limit-shape phenomenon as the system size grows: the particle trajectories are plotted in spacetime, starting from an initial configuration consisting of a single block of $N$ particles on the infinite line. The initial configuration is shown below the first figure on the left. The vertical axis denotes time. Alternating colours are used solely to distinguish neighbouring trajectories. From left to right: $N=$12, 60, and 600 particles, respectively. We clearly see the emergence, as $N$ increases, of a sharp boundary between 'frozen' regions where the density is $0$ or $1$, and 'fluctuating' regions where the density is strictly between $0$ and $1$.
• Figure 3: The Arctic curve delineates the frozen regions, where the density is one inside and zero outside. Numerical simulation of particle trajectories from an initial configuration of a single block of $N=$ 400 particles centered at the origin (left) and two blocks each containing $N/2=$ 200 particles (right)