Equilibrium forces on nonreciprocal materials

We discuss and analyze the properties of Casimir forces acting between nonreciprocal objects in thermal equilibrium. By starting from fluctuation-dissipation theorem splitting force into those arising individual sources, we show that if all temperatures are equal, resulting is reciprocal derivable as gradient a (free) energy. While expression for free energy identical to one objects, there several distinct features: To leading order reflections, can be decomposed sum two terms, first corresponding second antireciprocal objects. The term negative typically yields attraction, while have either sign. For case each other's mirror images, positive repulsion. terms lead overall repulsive forces, agreement with previous observations. Stable configurations, ruled out cases, appear possible three three-body exists, indicating previously found persistent heat currents situations cannot used produce torques.