Excess dissipation shapes symmetry breaking in non-equilibrium currents

Abstract

Most natural thermodynamic systems operate far from equilibrium, developing persistent currents and organizing into non-equilibrium stationary states (NESSs). Yet, the principles by which such systems self-organize, breaking equilibrium symmetries under external and internal constraints, remain unclear. Here, we establish a general connection between symmetry breaking and dissipation in mesoscopic stochastic systems described by Langevin dynamics. Using a geometric framework based on the inverse diffusion matrix, we decompose the velocity field into excess (gradient) and housekeeping (residual) components. This provides a natural entropy production split: the excess part captures internal reorganization under non-equilibrium conditions, while the housekeeping part quantifies detailed-balance violation due to external forces. We derive an exact equality linking the two, along with an inequality identifying accessible thermodynamics. A weak-noise expansion of the stationary solution reveals the general geometry of the NESS velocity field, enabling a unified classification of steady states. We apply this framework to systems ranging from molecular machines to coupled oscillators, showing how symmetry breaking in trajectory space constrains NESS organization. We further extend our approach to systems with multiplicative noise, deriving how additional symmetry breaking relates to curved (space-dependent) metrics. Finally, we show that both the NESS velocity field and stationary distribution can be derived through variational functionals based on excess dissipation. This work sheds light on the intimate connection between geometric features, dissipative properties, and symmetry breaking, uncovering a classification of NESSs that reflects how emergent organization reflects physical non-equilibrium conditions.