the exclusion process (SEP and TASEP)
/ TASEP = TAH-sep; ASEP = AY-sep /
The exclusion process is the standard interacting particle system with a conservation law: indistinguishable particles hop on the lattice Z^d, but with one rule — at most one particle per site, the exclusion constraint. A particle at x carries an independent Poisson clock; when it rings the particle attempts to jump to a neighbour y at a prescribed rate p(x, y), and the jump is executed if y is empty and suppressed (rejected) if y is occupied. The number of particles is conserved, which makes the exclusion process the natural microscopic model of a driven or diffusing fluid of interacting particles, and the cleanest setting for deriving macroscopic transport equations from microscopic dynamics.
Two regimes are fundamental. In the symmetric simple exclusion process (SSEP or SEP), p(x, y) is symmetric (a particle is equally likely to try to go left or right), there is no net drift, and the conserved density diffuses; the process is self-dual and reversible with respect to its product Bernoulli stationary measures, and its hydrodynamic limit is the heat (diffusion) equation. In the asymmetric simple exclusion process (ASEP) the rates are biased — rate p to the right, rate q < p to the left — so there is a net current; its totally asymmetric special case (TASEP) allows jumps in one direction only (rate 1 to the right, blocked if occupied). The asymmetry is not a small perturbation: it changes the macroscopic equation from the linear heat equation to the nonlinear Burgers equation, whose characteristics can collide and form shocks. The generator of the exclusion process is (L f)(eta) = sum_{x, y} p(x, y) eta(x) (1 - eta(y)) [ f(eta^{x,y}) - f(eta) ], where the factor eta(x)(1 - eta(y)) enforces 'occupied source, empty target' and eta^{x,y} moves the particle from x to y.
The exclusion process matters because it is simultaneously simple enough to be analysed in remarkable detail and rich enough to display deep universal phenomena. SSEP gives a fully rigorous derivation of Fick's law / the heat equation; ASEP and TASEP are exactly solvable enough (via the Bethe ansatz, the matrix product ansatz, and connections to random matrices and the RSK correspondence) to be the canonical lattice model in the KPZ universality class for interface growth and current fluctuations. An honest caveat: 'simple' refers only to the nearest-neighbour, single-particle-jump structure — the analysis is anything but simple, and the symmetric and asymmetric cases are qualitatively different worlds (diffusive scaling t^(1/2) and Gaussian fluctuations for SSEP versus t^(1/3) scaling and Tracy-Widom fluctuations for TASEP). The exclusion constraint, though it looks like a minor bookkeeping rule, is exactly what creates the interactions and the nonlinearity.
TASEP step initial condition: fill every nonpositive site with a particle and leave every positive site empty. Particles can only jump right and only into empty sites. The packed front melts and spreads; rescaling space by t and density appropriately, the macroscopic density profile solves the Burgers equation with a rarefaction fan, while the fluctuation of the integrated current across the origin is of order t^(1/3) and, suitably normalised, converges to the Tracy-Widom (GUE) distribution.
TASEP from a step: a deterministic Burgers profile at the fluid scale, with current fluctuations of order t^(1/3) governed by Tracy-Widom — the KPZ signature.
The exclusion constraint (one particle per site) is what creates the interaction; without it the particles would be independent random walks. Symmetric and asymmetric exclusion are qualitatively different: SSEP is diffusive (heat equation, Gaussian, t^(1/2)) while ASEP/TASEP is hyperbolic and nonlinear (Burgers equation, Tracy-Widom, t^(1/3)).