the hydrodynamic limit
The hydrodynamic limit is the bridge from the microscopic, random, particle-level dynamics of an interacting particle system to a macroscopic, deterministic partial differential equation for a density. It answers the question that physics has always assumed but probability must prove: how does the smooth diffusion or transport equation of a fluid emerge from the chaotic hopping of countless individual particles? It is, in essence, a law of large numbers, not for a single sum of random variables, but for an entire space-time density field, with space and time rescaled together so that the macroscopic picture comes into focus.
The set-up rescales the lattice by a small parameter 1/N (so microscopic sites x become macroscopic positions x/N in a continuum) and speeds up time appropriately. One studies the empirical measure pi^N_t = (1/N^d) sum_x eta_t(x) delta_{x/N}, which records the spatial profile of particle density. The hydrodynamic limit theorem states that if at time 0 the empirical measure is close to a profile rho_0(u) du (local equilibrium), then at the rescaled later time it concentrates, in probability, on a deterministic profile rho(t, u) du, where rho solves a PDE determined by the microscopic rates. The choice of time scale and the resulting PDE depend on the symmetry of the dynamics: for the symmetric exclusion process, time is sped up by N^2 (diffusive scaling) and rho solves the linear heat equation partial_t rho = (1/2) Laplacian rho; for the asymmetric exclusion process, time is sped up by only N (hyperbolic / Euler scaling) and rho solves the inviscid Burgers equation partial_t rho + partial_u [ rho(1 - rho) ] = 0. The standard proof techniques are the entropy method (Guo-Papanicolaou-Varadhan), controlling relative entropy with respect to local-equilibrium product measures, and the relative entropy method (Yau); both rest on a 'replacement lemma' / local equilibrium that lets one replace a microscopic average by a function of the local density.
Hydrodynamic limits matter because they put the derivation of macroscopic transport laws (Fick's law, Fourier's law of heat conduction, the Euler and Navier-Stokes equations) on a rigorous probabilistic footing, starting from particles obeying simple stochastic rules. Several honesty points are essential. First, the limit is deterministic — the randomness washes out at leading order — so the next, far harder, question is the fluctuations around it (the fluctuation field, a central limit theorem for the density). Second, the emergence of a closed PDE relies on a local equilibrium assumption: that on the microscopic scale the system relaxes fast to the stationary product measure of the appropriate local density before the macroscopic profile changes. Third, the asymmetric (Burgers) case is genuinely harder: the inviscid Burgers equation develops shocks, its solutions are non-unique without an entropy condition, and proving the particle system selects the correct entropy solution is a deep result. Hydrodynamics is the macroscopic law of large numbers; it does not by itself tell you the fluctuations or the rare-event large deviations, which are separate theories.
For SSEP on Z under diffusive scaling, prepare particles so the initial density profile is a smooth function rho_0(u). Speed time by N^2. Then the empirical density converges in probability to rho(t, u) solving the heat equation partial_t rho = (1/2) partial_u^2 rho with initial data rho_0 — a rigorous microscopic derivation of Fick's law of diffusion. For TASEP under hyperbolic scaling (time by N), the same kind of theorem yields the inviscid Burgers equation instead.
Same particles, different time scale: diffusive N^2 gives the heat equation (SSEP); hyperbolic N gives Burgers (ASEP).
The limit is a law of large numbers — deterministic, with the randomness averaged out — and rests on a local-equilibrium hypothesis (fast microscopic relaxation to the product measure of the local density). The asymmetric (Burgers) case develops shocks, so the limiting PDE needs an entropy condition for uniqueness, and proving the system selects it is a deep result.