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We calculate the probability distribution of work for an exactly solvablemodel of a system interacting with its environment. The system of interest is aharmonic oscillator with a time dependent control parameter, the environment ismodeled by $N$ independent harmonic oscillators with arbitrary frequencies, andthe system-environment coupling is bilinear and not necessarily weak. Theinitial conditions of the combined system and environment are sampled from amicrocanonical distribution and the system is driven out of equilibrium bychanging the control parameter according to a prescribed protocol. In the limitof infinitely large environment, i.e. $N \rightarrow \infty$, we recover thenonequilibrium work relation and Crooks's fluctuation relation. Moreover, themicrocanonical Crooks relation is verified for finite environments. Finally, weshow the equivalence of multi-time correlation functions of the system in theinfinite environment limit for canonical and microcanonical ensembles.

Microcanonical work and fluctuation relations for an open system: An exactly solvable model