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We study two non-equilibrium work fluctuation theorems, the Crooks' theoremand the Jarzynski equality, for a test system coupled to a spatially extendedheat reservoir whose degrees of freedom are explicitly modeled. The sufficientconditions for the validity of the theorems are discussed in detail andcompared to the case of classical Hamiltonian dynamics. When the conditions aremet the fluctuation theorems are shown to hold despite the fact that theimmediate vicinity of the test system goes out of equilibrium during anirreversible process. We also study the effect of the coupling to the heatreservoir on the convergence of $<\exp(-\beta W)>$ to its theoretical meanvalue, where $W$ is the work done on the test system and $\beta$ is the inversetemperature. It is shown that the larger the local heating, the slower theconvergence.

Non-equilibrium fluctuation theorems in the presence of local heating