Exclusion processes with two or more species are relevant for a variety of non-equilibrium phenomena including laning and motility induced phase separation. Their hydrodynamic equations are complicated in that they include coefficients that can be characterised variationally but are not explicit. I will describe a systematic method for obtaining the coefficients [1] and a recent application of these methods in an active matter model [2]. If time permits, I will also discuss some results for fluctuations around the hydrodynamic limit [3].

[1] J. Mason, R. L. Jack, and M. Bruna, J. Stat. Phys. 190, 47 (2023) [2] J. Mason, C. Erignoux, R. L. Jack, and M. Bruna, arXiv:2307.11673. [3] T. Agranov, M. E. Cates and R. L. Jack, Phys. Rev. Lett. 131, 017102 (2023).