A nonequilibrium distribution for stochastic thermodynamics

We extend the canonical Gibbs distribution, originally formulated for systems at equilibrium, to systems driven out of equilibrium. The stochastic dynamics of a small system are described by a probability distribution over discrete energy levels. Within this framework, we derive a microscopic expression for work and introduce a microscopic definition of entropy production (defined here in terms of the uncompensated heat of Clausius) during a nonequilibrium stochastic process. Work and entropy production share a common origin arising from variations of the system energy. The proposed framework allows us to recover the nonequilibrium work relation and to establish a new equivalent identity for the heat exchanged during a work protocol. Finally, we show that the fluctuations of work and heat governed by the extended canonical distribution follow directly from the fluctuation theorem for entropy production.