Two extensions of exact nonequilibrium steady states of a boundary-driven cellular automaton

Recently Prosen and Mejía-Monasterio (2016 J. Phys. A: Math. Theor. 49 185003) obtained exact nonequilibrium steady states of an integrable and reversible cellular automaton driven by some stochastic boundary conditions. In this paper, we explore the possible extensions of their method by generalizing the boundary conditions. As the result, we find two cases where such an extension is possible. One is obtained by generalizing probabilities for values of virtual cells while imposing a special condition on emission and absorption rates. Although this actually coincides with the conditional boundary driving discussed in Prosen and Buča (2017 J. Phys. A: Math. Theor. 50 395002), we present a solution in a different form. The other is obtained by considering a conserved quantity as energy, and by considering boundaries as heat reservoirs. The latter contains the original solution and the former one as the special cases. Properties of both solutions are discussed. (paper)