Algebraic depletion interactions in two-temperature mixtures

The phase separation that occurs in two-temperature mixtures, which are driven out of equilibrium at the local scale, has been thoroughly characterized, but much less is known about the depletion interactions that drive it. Using numerical simulations in dimension 2, we show that the depletion interactions extend beyond two particle diameters in dilute systems, as expected at equilibrium, and decay algebraically with an exponent $-4$. Solving for the $N$-particle distribution function in the stationary state, perturbatively in the interaction potential, we show that algebraic correlations with an exponent $-2d$ arise from triplets of particles at different temperatures in spatial dimension $d$. Finally, simulations allow us to extend our results beyond the perturbative limit.