Fluctuation-induced self-force and violation of action-reaction in a nonequilibrium steady state fluid

We show that the fluctuations of a fluid driven out of equilibrium can induce a net force on a single asymmetric object immersed in it. The force originates in the restriction of the fluid's fluctuations at the object's boundaries, as in the Casimir effect. In contrast to the equilibrium situation, its emergence on a single obstacle is not ruled out by the second law of thermodynamics since the fluid is in a nonequilibrium state. We explicitly calculate this self-force on a deformed circle embedded in a fluid whose density fluctuations obey a stochastic reaction-diffusion equation. When two objects are considered, the presence of self-forces can violate the action-reaction principle. We illustrate this by calculating the internal Casimir-type forces between a circle and a plate. Their sum, instead of vanishing, provides the self-force exerting on the circle-plate assembly.