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 an 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.

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• Fluctuation theorems
• Nonequilibrium & irreversible thermodynamics
• Stochastic thermodynamics