stationary product Bernoulli measures
For a conservative interacting particle system like the exclusion process, the natural question of equilibrium is: as the system runs forever, what is the statistical law of the configuration? The remarkable answer for the exclusion process is the product Bernoulli measures — the equilibrium states in which each site is independently occupied with the same probability rho, and rho can be any value in [0, 1]. So instead of a single equilibrium, a conservative system has a whole one-parameter family of stationary states indexed by the conserved density. This is a strong structural statement: in equilibrium the exclusion constraint, despite coupling neighbouring sites dynamically, leaves no spatial correlations at all.
Concretely, let nu_rho be the product measure on {0,1}^(Z^d) under which the eta(x) are independent Bernoulli(rho). The theorem is that for the simple exclusion process every nu_rho is invariant: if you start from nu_rho the law is preserved for all time, integral L f dnu_rho = 0 for all local f. For the symmetric process (SSEP) the nu_rho are also reversible (they satisfy detailed balance), because the symmetric exclusion dynamics is reversible — one checks that the rate of any transition equals the rate of its reverse under nu_rho. For the asymmetric process (ASEP/TASEP) the nu_rho are still invariant but NOT reversible: there is a nonzero stationary current j(rho) = (p - q) rho (1 - rho), reflecting the net drift even in equilibrium. The translation-invariant extremal invariant measures of the (irreducible) exclusion process are exactly the nu_rho, a complete classification. The verification is a clean local computation: under a product measure the gain and loss terms in the master equation for any single bond cancel because the two-site marginal of nu_rho factorises.
These measures matter because they are the equilibrium reference points for everything else: the hydrodynamic limit is the statement that locally the system relaxes to nu_{rho(t,x)} with a slowly varying density, and the fluctuation theory is built by perturbing around a fixed nu_rho. An honest caveat: product (uncorrelated) stationarity is special to the exclusion process and a handful of related systems; it is NOT a general feature of interacting particle systems — most have correlated, non-product, hard-to-describe stationary measures (the contact process's upper invariant measure is not product). For ASEP, the nonreversibility of the product measures is the source of the nontrivial KPZ-class current fluctuations; reversibility would have forced the simpler diffusive picture. And on a finite lattice with open boundaries the stationary measure is generally NOT product — it is the celebrated matrix-product-ansatz state, which carries long-range correlations.
For TASEP on Z, start from the product Bernoulli measure nu_rho with each site occupied independently with probability rho. The law is then stationary, and the average current of particles across any bond equals j(rho) = rho(1 - rho): a particle must be present at the source (probability rho) and absent at the target (probability 1 - rho). The current is maximised at rho = 1/2, the densest sustainable flow.
Each density rho gives a product-Bernoulli equilibrium; for TASEP it carries a stationary current rho(1 - rho), peaking at half-filling.
Product (uncorrelated) stationary measures are special to exclusion and a few related models, not a general feature of IPS. For ASEP they are invariant but not reversible (there is a stationary current); with open boundaries the stationary state is not product at all but a matrix-product-ansatz state with long-range correlations.