Green-Kubo relations for odd transport phenomena in chiral active matter

Non-equilibrium statistical mechanics connects random thermal fluctuations in equilibrium with macroscopic transport phenomena, as famously encapsulated in Onsager's regression hypothesis. Transport phenomena usually involve down-gradient fluxes; for example particles tend to diffuse from high to low concentration, and heat tends to flow from high to low temperature. In this talk, we examine "odd" transport phenomena, in which fluxes can be orthogonal to gradients, and thus need not affect the relaxation of the gradients. We particularly emphasize how odd transport phenomena such as odd diffusion, odd thermal conduction, and odd viscosity may arise in chiral active matter. By applying Onsager's regression hypothesis in the context of such steady states, and by reformulating this hypothesis at the level of the constitutive relations rather than that of the relaxation equations, we show that Green-Kubo relations of the standard form hold in general for odd transport coefficients. These relations reveal the connection between time-reversal symmetry breaking and odd transport, as well as yielding Onsager-Casimir reciprocal relations. Importantly, these Green-Kubo relations hold even in contexts where linear response (i.e. fluctuation-dissipation) relations break down. We conclude by demonstrating the applicability of these Green-Kubo relations through simulations of odd diffusion in concentrated solutions of chiral active particles.