On the Gallavotti-Cohen symmetry and the fluctuation theorem for stochastic processes

We directly extend Kurchans’ work on the fluctuation theorem to general finite state space Markov jump processes and general Langevin dynamics. All systems are treated in a unified manner and we show that this directly reproduces (and corrects) the results of Lebowitz and Spohn in the Langevin case. For the jump process the quantity, for which the fluctuation theorem holds, is different from the ones previously studied by Lebowitz and Spohn. It is also pointed out that the fluctuation theorem, for the quantities considered here, can not be viewed as the result of a simple relation ship between the measure of a path and the measure of the corresponding time inverted path. Such relationship only holds on average and we must interpret the fluctuation theorem as the result of time reversal of the whole ensemble of paths. For the case of Langevin dynamics we give the Stratonovich differentials of the considered quantity, enabling us to interpret it as work over temperature.