Generalization of the Boltzmann distribution to systems out of equilibrium

The Boltzmann distribution predicts the collective behavior of systems at thermodynamic equilibrium as a function of their constituent parts. Yet most systems in nature–especially living systems– are not at equilibrium, and a unified theory of their behavior does not currently exist. I will show that the Boltzmann distribution is a special case of a general probability flow equation (PFE) that governs stochastic systems, even if far from equilibrium. The PFE is an analog of the voltage equation governing electronic circuits, where resistors, batteries, node voltages, and path currents correspond to equilibrium rate constants, driven rate constants, probabilities, and probability flows, respectively. I will discuss how this new approach can be used to recapitulate known properties of weakly driven systems as well as derive new relations applicable to strongly-driven systems. These relations include general limits on performance and efficiency which are independent of system details; experimental data are used to show that living systems often operate at those limits.