reservoir at temperature T, then the dissipation function is the total electric current density J multiplied by the voltage drop across the circuit, , and the system volume V, divided by the absolute temperature T, of the heat reservoir times Boltzmann's constant. Thus the dissipation function is easily recognised as the Ohmic work done on the system divided by the temperature of the reservoir. Close to equilibrium the long time average of this quantity is (to leading order in the voltage drop), equal to the average spontaneous entropy production per unit time. However, the fluctuation theorem applies to systems arbitrarily far from equilibrium where the definition of the spontaneous entropy production is problematic. Relation to Loschmidt's paradox The second law of thermodynamics, which predicts that the entropy of an isolated system out of equilibrium should tend to increase rather than decrease or stay constant, stands in apparent contradiction with the time-reversible equations of motion for classical and quantum systems. The time reversal symmetry of the equations of motion show that if one films a given time dependent physical process, then playing the movie of that process backwards does not violate the laws of mechanics. It is often argued that for every forward trajectory in which entropy increases, there exists a time reversed anti trajectory where entropy decreases, thus if one picks an initial state randomly from the system's phase space and evolves it forward according to the laws governing the system, decreasing entropy should be just as likely as increasing entropy. It might seem that this is incompatible with the second law of thermodynamics which predicts that entropy tends to increase. The problem of deriving irreversible thermodynamics from time-symmetric fundamental laws is referred to as Loschmidt's paradox. The mathematical derivation of the fluctuation theorem and in particular the second law inequality shows that, for a nonequilibrium process, the ensemble averaged value for the dissipation function will be greater than zero. This result requires causality, i.e. that cause (the initial conditions) precede effect (the value taken on by the dissipation function). This is clearly demonstrated in section 6 of that paper, where it is shown how one could use the same laws of mechanics to extrapolate backwards from a later state to an earlier state, and in this case the fluctuation theorem would lead us to predict the ensemble average dissipation function to be negative, an anti-second law. This second prediction,