A stochastic lattice gas driven out of equilibrium by unequal reservoirs at its two ends has a stationary state that is not Boltzmann-Gibbs and must be computed directly from the master equation. The matrix ansatz writes the unnormalised steady-state weight of a configuration as a product of operators - one operator per site, chosen by that site's occupancy - sandwiched between two boundary vectors, so that the normalisation is a single matrix element and observables such as the particle current and density profile follow from operator algebra rather than from summing exponentially many configurations. The ansatz works exactly when the operators satisfy a bulk quadratic relation forced by the Markov generator plus two boundary relations forced by the injection and extraction rates; these are the Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations of integrable scattering theory, evaluated at the point where the R-matrix and K-matrices degenerate to the stochastic model's rates. Integrability therefore supplies the algebra rather than merely accompanying it: a solvable model is one whose R-matrix satisfies Yang-Baxter and whose K-matrices satisfy the reflection equation, and the matrix ansatz is the steady-state shadow of that structure. Covers the construction for the totally asymmetric simple exclusion process, where the algebra DE = D + E admits infinite-dimensional representations whose matrix elements are counted by Catalan numbers, and its extension to multi-species exclusion and to models such as DiSSEP where the same machinery yields exact currents and correlation functions in the thermodynamic limit.
A conference note (ISQS24, Prague 2016) summarising work of Ragoucy with Crampe, Evans, Finn, Mallick and Vanicat on deriving the matrix product ansatz from integrable-systems structure. A driven lat…