in the conjugate flux, where (with k the Boltzmann constant), and V is the system volume. The integral is over the equilibrium flux autocovariance function. At zero time the autocovariance is positive since it is the mean square value of the flux at equilibrium. Note that at equilibrium the mean value of the flux is zero by definition. At long times the flux at time t, J(t), is uncorrelated with its value a long time earlier J(0) and the autocorrelation function decays to zero. This remarkable relation is frequently used in molecular dynamics computer simulation to compute linear transport coefficients; see Evans and Morriss, "Statistical Mechanics of Nonequilibrium Liquids", Academic Press 1990. Nonlinear response and transient time correlation functions In 1985 Denis Evans and Morriss derived two exact fluctuation expressions for nonlinear transport coefficients—see Evans and Morriss in Mol. Phys, 54, 629(1985). Evans later argued that these are consequences of the extremization of free energy in Response theory as a free energy minimum. Evans and Morriss proved that in a thermostatted system that is at equilibrium at t = 0, the nonlinear transport coefficient can be calculated from the so-called transient time correlation function expression: where the equilibrium () flux autocorrelation function is replaced by a thermostatted field dependent transient autocorrelation function. At time zero but at later times since the field is applied . Another exact fluctuation expression derived by Evans and Morriss is the so-called Kawasaki expression for the nonlinear response: The ensemble average of the right hand side of the Kawasaki expression is to be evaluated under the application of both the thermostat and the external field. At first sight the transient time correlation function (TTCF) and Kawasaki expression might appear to be of limited use—because of their innate complexity. However, the TTCF is quite useful in computer simulations for calculating transport coefficients. Both expressions can be used to derive new and useful fluctuation expressions quantities like specific heats, in nonequilibrium steady states. Thus they can be used as a kind of partition function for nonequilibrium steady states. Derivation from the fluctuation theorem and the central limit theorem For a thermostatted steady state, time integrals of the dissipation function are related to the dissipative flux, J, by the equation We note in passing that the long time average of the dissipation function is a product of the thermodynamic force and the average conjugate thermodynamic flux. It is