Hydrodynamic limit

The hydrodynamic limit can be seen as a dynamic version of the law of large numbers. A microscopic stochastic system — typically an interacting particle system or a continuous-spin lattice model — converges, on the right space-time scale, to a deterministic macroscopic evolution governed by a partial differential equation.

Concretely: under the Kawasaki dynamics, the empirical spin profile on a lattice converges to the solution of a non-linear heat equation; under the simple exclusion process, it converges to a Burgers-type equation. Many such correspondences are known qualitatively, and a long literature establishes them rigorously.

My own work focuses on quantifying this convergence. With Deniz Dizdar, Felix Otto, and Tianqi Wu we derived the first explicit error estimates for the hydrodynamic limit of the Kawasaki dynamics — see The quantitative hydrodynamic limit for an informal account, and arXiv:1807.09850 for the paper. Quantitative statements give error estimates, confidence intervals, and computer-time bounds that purely qualitative results cannot.