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

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