Odd transport phenomena from broken symmetries*

Linear constitutive relations connect fluxes of quantities such as mass, heat, and momentum to corresponding thermodynamic driving forces. Such relations reveal macroscopic consequences of microscopic fluctuations. When time-reversal and parity symmetries are broken at the microscale, as is often the case in chiral active matter, new "odd" transport coefficients can emerge at the continuum scale. We discuss two odd transport coefficients in particular: the odd diffusivity, which generates diffusive fluxes perpendicular to concentration gradients, and the odd viscosity, which generates normal stresses perpendicular to shear flow. In both cases, we show that the behavior under time reversal and parity inversion can be understood through deriving Green-Kubo relations, which additionally provide an avenue for direct numerical verification in model active matter systems using molecular dynamics simulations. Our work provides steps towards extending the framework of non-equilibrium thermodynamics to systems like active matter, which are inherently out of equilibrium, or for which no equilibrium reference state exists.