therefore equal to the spontaneous entropy production in the system. The spontaneous entropy production plays a key role in linear irreversible thermodynamics – see de Groot and Mazur "Non-equilibrium thermodynamics" Dover. The fluctuation theorem (FT) is valid for arbitrary averaging times, t. Let's apply the FT in the long time limit while simultaneously reducing the field so that the product is held constant, Because of the particular way we take the double limit, the negative of the mean value of the flux remains a fixed number of standard deviations away from the mean as the averaging time increases (narrowing the distribution) and the field decreases. This means that as the averaging time gets longer the distribution near the mean flux and its negative, is accurately described by the central limit theorem. This means that the distribution is Gaussian near the mean and its negative so that Combining these two relations yields (after some tedious algebra!) the exact Green–Kubo relation for the linear zero field transport coefficient, namely, Here are the details of the proof of Green–Kubo relations from the FT. A proof using only elementary quantum mechanics was given by Zwanzig. Summary This shows the fundamental importance of the fluctuation theorem (FT) in nonequilibrium statistical mechanics. The FT gives a generalisation of the second law of thermodynamics. It is then easy to prove the second law inequality and the Kawasaki identity. When combined with the central limit theorem, the FT also implies the Green–Kubo relations for linear transport coefficients close to equilibrium. The FT is, however, more general than the Green–Kubo Relations because, unlike them, the FT applies to fluctuations far from equilibrium. In spite of this fact, no one has yet been able to derive the equations for nonlinear response theory from the FT. The FT does not imply or require that the distribution of time-averaged dissipation is Gaussian. There are many examples known when the distribution is non-Gaussian and yet the FT still correctly describes the probability ratios. See also Density matrix Fluctuation theorem Fluctuation–dissipation theorem Green's function (many-body theory) Lindblad equation Linear response function References Theoretical physics Thermodynamic equations Statistical mechanics Non-equilibrium thermodynamics