The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for a transport coefficient
(sometimes termed a "gross variable", as in [1]): One intuitive way to understand this relation is that relaxations resulting from random fluctuations in equilibrium are indistinguishable from those due to an external perturbation in linear response.
In addition, they allow one to measure the transport coefficient without perturbing the system out of equilibrium, which has found much use in molecular dynamics simulations.
[3] Thermodynamic systems may be prevented from relaxing to equilibrium because of the application of a field (e.g. electric or magnetic field), or because the boundaries of the system are in relative motion (shear) or maintained at different temperatures, etc.
The standard example of a mechanical transport process is Newton's law of viscosity, which states that the shear stress
is the rate of change streaming velocity in the x-direction, with respect to the y-coordinate,
Newton's law of viscosity states As the strain rate increases we expect to see deviations from linear behavior Another well known thermal transport process is Fourier's law of heat conduction, stating that the heat flux between two bodies maintained at different temperatures is proportional to the temperature gradient (the temperature difference divided by the spatial separation).
Except in special cases, this matrix is symmetric as expressed in the Onsager reciprocal relations.
In the 1950s Green and Kubo proved an exact expression for linear transport coefficients which is valid for systems of arbitrary temperature T, and density.
They proved that linear transport coefficients are exactly related to the time dependence of equilibrium fluctuations in the conjugate flux, where
At zero time the autocovariance is positive since it is the mean square value of the flux at equilibrium.
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.
At first sight the transient time correlation function (TTCF) and Kawasaki expression might appear to be of limited use—because of their innate complexity.
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.
[6] This shows the fundamental importance of the fluctuation theorem (FT) in nonequilibrium statistical mechanics.
When combined with the central limit theorem, the FT also implies the Green–Kubo relations for linear transport coefficients close to 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.