[Paper Review] The stochastic heat equation as the limit of a stirring dynamics perturbed by a voter model

Milton Jara, Cláudio Landim|arXiv (Cornell University)|Aug 7, 2020

Stochastic processes and statistical mechanics15 references4 citations

TL;DR

This paper establishes that the density fluctuations of a symmetric exclusion process on a d-dimensional torus, perturbed by a slower-scale voter model, converge to the solution of the stochastic heat equation in dimensions d ≤ 3. The key innovation lies in the non-conservative noise in the limiting SPDE, arising from the voter model's non-equilibrium dynamics, and the proof relies on a refined Boltzmann-Gibbs principle using entropy bounds from recent advances in interacting particle systems.

ABSTRACT

We prove that in dimension $d\le 3$ a modified density field of a stirring dynamics perturbed by a voter model converges to the stochastic heat equation.

Motivation & Objective

• To understand nonequilibrium fluctuations in interacting particle systems, a long-standing open problem in statistical mechanics.
• To establish the hydrodynamic limit of a modified exclusion process perturbed by a voter model in dimensions d ≤ 3.
• To derive the stochastic heat equation as the scaling limit of the density fluctuation field, even when the limiting noise is non-conservative.
• To develop a new approach to the Boltzmann-Gibbs principle in non-equilibrium settings using refined entropy bounds.

Proposed method

• Introduce a coupled dynamics combining symmetric exclusion (fast scale) and voter model (slow scale), both on a d-dimensional discrete torus.
• Define a modified density field that is not normalized by the square root of the system size, enabling convergence to a non-conservative noise SPDE.
• Apply a refined version of the Boltzmann-Gibbs principle to replace space-time averages of cylinder functions by averages of the local density.
• Use entropy bounds with respect to Bernoulli product measures, leveraging recent improvements in entropy production estimates from Jara and Menezes (2016).
• Establish tightness of the fluctuation field and prove convergence to the solution of the stochastic heat equation via martingale problem techniques and moment bounds.
• Derive a uniform moment bound on the martingale component of the density field using Young’s inequality and uniform integrability estimates.

Experimental results

Research questions

• RQ1 Can the density fluctuations of a perturbed exclusion process converge to the stochastic heat equation in dimensions d ≤ 3?
• RQ2 What happens to the noise structure in the limiting SPDE when the perturbation is a non-conservative voter model?
• RQ3 Can the Boltzmann-Gibbs principle be extended to non-equilibrium settings using entropy bounds with respect to Bernoulli measures?
• RQ4 Is the fluctuation field tight and convergent under a modified normalization that avoids square-root scaling?

Key findings

• The density fluctuation field of the perturbed exclusion process converges weakly to the solution of the stochastic heat equation in dimensions d ≤ 3.
• The limiting SPDE features a non-conservative noise term, a novel feature compared to most prior results in this class.
• The proof establishes a uniform moment bound on the martingale component of the density field: E^n_η[M^n_t(F)^4] ≤ C₀T²(‖F‖⁴_∞ + ‖∇F‖⁴_∞) for all t ≤ T and n ≥ 1.
• The Boltzmann-Gibbs principle is extended to non-equilibrium via entropy bounds, relying on the improved entropy production estimate from Jara and Menezes (2016).
• The decomposition of cylinder functions into degree-1 and higher-degree terms in L²(ν_ρ) enables precise control of fluctuations via the operator Π_ρ.
• The convergence holds under a modified normalization of the density field, which avoids the standard square-root scaling and allows for non-conservative noise in the limit.