Microcanonical ensemble out of equilibrium

Introduced by Boltzmann under the name “monode,” the microcanonical ensemble serves as the fundamental representation of equilibrium thermodynamics in statistical mechanics by counting all possible realizations of a system's states. Ensemble theory connects this idea with probability and information theory, leading to the notion of Shannon-Gibbs entropy and, ultimately, to the principle of maximum caliber describing of systems—in and out of equilibrium. While the latter phenomenological generalization reproduces many results of nonequilibrium thermodynamics, given a proper choice of observables, its physical justification remains an open area of research. What is the microscopic origin and physical interpretation of this variational approach? What guides the choice of relevant observables? We address these questions by extending Boltzmann's method to a microcanonical caliber principle and counting realizations of a system's trajectories—all assumed equally probable. Maximizing the microcanonical caliber under the imposed constraints, we systematically develop generalized local detailed-balance relations, clarify the statistical origins of inhomogeneous transport, and provide an independent derivation of key equations from stochastic thermodynamics. This approach introduces a ensemble theory for nonequilibrium steady states in spatially extended and active systems. While verifying the equivalence of ensembles, e.g., those of Norton and Thévenin, our framework contests other common assumptions about nonequilibrium regimes, with supporting evidence provided by stochastic simulations. Our theory suggests further connections to the first principles of microscopic dynamics in classical statistical mechanics, which are essential for investigating systems where the necessary conditions for thermodynamic behavior are not satisfied.