Entropy production and collective excitations of crystals out of equilibrium. The concept of entropons

We study the collective vibrational excitations of crystals under out-of-equilibrium steady conditions that give rise to entropy production. Their excitation spectrum comprises equilibriumlike phonons of thermal origin and additional collective excitations called entropons because each of them represents a mode of spectral entropy production. Entropons coexist with phonons and dominate them when the system is far from equilibrium while they are negligible in near-equilibrium regimes. The concept of entropons has been recently introduced and verified in a special case of crystals formed by self-propelled particles. Here we show that entropons exist in a broader class of active crystals that are intrinsically out of equilibrium and characterized by the lack of detailed balance. After a general derivation, several explicit examples are discussed, including crystals consisting of particles with alignment interactions and frictional contact forces.