Machian fractons, Hamiltonian attractors, and nonequilibrium steady states

Abstract

We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N, despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of "order-by-disorder"borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics.