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The Casimir force and thermal Casimir force originating from quantumelectromagnetic fluctuations at zero and non-zero temperatures, respectively,are significant in nano- and microscale systems and are well-understood. Lessunderstood, however, are the Casimir and thermal Casimir forces in systemsbreaking Lorentz reciprocity. In this work, we derive a formalism for thermalCasimir forces between an arbitrary number of spheres based on fluctuationalelectrodynamics and scattering theory without the assumption of Lorentzreciprocity. We study the total Casimir force in systems of two and three Weylsemimetal spheres with time-reversal symmetry breaking for differentorientations of the momentum-space separation of Weyl nodes in both thermalequilibrium and nonequilibrium. In thermal nonequilibrium, we show that a netthermal Casimir force exists not only along the center-to-center displacementsof the spheres, but also in the transverse direction to it due to thermalemission with non-zero angular momentum. Different symmetries of the systemdrive a variety of dynamics such as global rotations, self-propulsion, andspinning of the spheres. We also show that the Casimir energy in thermalequilibrium depends on the orientations of the Weyl node directions in thespheres and that the lateral Casimir force will act between the spheres even inthermal equilibrium to relax the system into the minimum energy state withouttransferring net energy and momentum to the environment. The developedframework opens a way for investigating many-body dynamics by Casimir andthermal Casimir forces among arbitrary number of spheres with arbitrarydielectric function tensors in both thermal equilibrium and nonequilibrium.

Casimir force among spheres made of Weyl semimetals breaking Lorentz reciprocity