On the asymmetric simple exclusion process with multiple species

In the asymmetric simple exclusion process on the integers each particle waits exponential time, then with probability p it moves one step to the right if the site is unoccupied, otherwise it stays put; and with probability q=1-p it moves one step to the left if the site is unoccupied, otherwise it stays put. In previous work the authors, using the Bethe Ansatz, found for N-particle ASEP a formula --- a sum of multiple integrals --- for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations are satisfied.