  The statistical mechanics of equilibrium interfaces has been well-established
for over a half century. In the last decade, a wealth of observations have made
increasingly clear that a new perspective is required to describe interfaces
arbitrarily far from equilibrium. In this work, beginning from microscopic
particle dynamics that break time-reversal symmetry, we systematically derive
the interfacial dynamics of coexisting motility-induced phases. Doing so allows
us to identify the athermal energy scale that excites interfacial fluctuations
and the nonequilibrium surface tension that resists these excitations. Our
theory identifies that, in contrast to equilibrium fluids, this active surface
tension contains contributions arising from nonconservative forces which act to
suppress interfacial fluctuations and, crucially, is distinct from the
mechanical surface tension of Kirkwood and Buff. We find that the interfacial
stiffness scales linearly with the intrinsic persistence length of the
constituent active particle trajectories, in agreement with simulation data. We
demonstrate that at wavelengths much larger than the persistence length, the
interface obeys surface area minimizing Boltzmann statistics with our derived
nonequilibrium interfacial stiffness playing an identical role to that of
equilibrium systems.
