Hydrodynamic Limit Theorem

Convergence of the empirical density of an interacting particle system (Markov dynamics) to a deterministic PDE under macroscopic scaling, exemplified by exclusion processes.

Hydrodynamic Limit Theorem

Prerequisites

Empirical density field

Let $(\eta_t)_{t\ge 0}$ be a conservative interacting particle system on a discrete torus $\mathbb{T}_N^d=(\mathbb{Z}/N\mathbb{Z})^d$ with Markov generator $L_N$ (described by a master equation / Markov semigroup ).

A standard macroscopic observable is the empirical measure (density field)

\pi_t^N(du) ={} \frac{1}{N^d}\sum_{x\in\mathbb{T}_N^d}\eta_t(x)\,\delta_{x/N}(du),

viewed as a random measure on the continuum torus $\mathbb{T}^d$ . Equivalently, for a smooth test function $G$ ,

\langle \pi_t^N, G\rangle ={} \frac{1}{N^d}\sum_{x\in\mathbb{T}_N^d}\eta_t(x)\,G(x/N).

Theorem (hydrodynamic limit, typical form)

Fix a macroscopic time scale and a macroscopic initial profile $\rho_0:\mathbb{T}^d\to[0,1]$ . Assume the initial distributions of $\eta_0$ are “associated” to $\rho_0$ in the sense that $\pi_0^N$ converges (in probability) to $\rho_0(u)\,du$ .

Then, under an appropriate scaling of time (diffusive or hyperbolic, depending on the model), the path $(\pi_t^N)_{t\in[0,T]}$ converges in probability to a deterministic trajectory $\rho(t,u)\,du$ , where $\rho$ solves a macroscopic PDE of conservation/diffusion type.

Example: symmetric simple exclusion process (SSEP)

For SSEP, the natural scaling is diffusive: observe the process at times $tN^2$ . The hydrodynamic limit states that $\rho(t,u)$ solves the heat equation on $\mathbb{T}^d$ :

\partial_t \rho(t,u) = \Delta \rho(t,u), \qquad \rho(0,u)=\rho_0(u).

Example: asymmetric exclusion (ASEP, 1D)

For ASEP with nonzero drift, the natural scaling is often hyperbolic: observe at times $tN$ . In one dimension, the limiting density typically solves a scalar conservation law (inviscid Burgers type):

\partial_t \rho(t,u) + \partial_u\big(\rho(t,u)(1-\rho(t,u))\big)=0, \qquad \rho(0,u)=\rho_0(u),

under appropriate entropy-solution interpretation.

What “hydrodynamic limit” encodes

Law of large numbers at the macroscopic scale: fluctuations vanish after scaling, yielding deterministic PDE behavior.
Local equilibrium principle: microscopic configurations near a macroscopic point behave approximately like equilibrium Gibbs measures at the local density.
Connection to dynamical large deviations: hydrodynamic limits are often paired with path-space LDPs; the time-averaged DV theory (Donsker–Varadhan ) is one cornerstone in this direction.

Typical proof inputs (conceptual)

Many rigorous proofs rely on:

martingale formulations of $\langle \pi_t^N, G\rangle$ derived from the generator,
replacement lemmas (local functions replaced by functions of the empirical density),
and entropy/relative-entropy methods comparing the law of $\eta_t$ to local equilibrium references.