The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions: Thermal and mechanical transport processes 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. This generates two classes of nonequilibrium system: mechanical nonequilibrium systems and thermal nonequilibrium systems. The standard example of an electrical transport process is Ohm's law, which states that, at least for sufficiently small applied voltages, the current I is linearly proportional to the applied voltage V, As the applied voltage increases one expects to see deviations from linear behavior. The coefficient of proportionality is the electrical conductance which is the reciprocal of the electrical resistance. The standard example of a mechanical transport process is Newton's law of viscosity, which states that the shear stress is linearly proportional to the strain rate. The strain rate 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). Linear constitutive relation Regardless of whether transport processes are stimulated thermally or mechanically, in the small field limit it is expected that a flux will be linearly proportional to an applied field. In the linear case the flux and the force are said to be conjugate to each other. The relation between a thermodynamic force F and its conjugate thermodynamic flux J is called a linear constitutive relation, L(0) is called a linear transport coefficient. In the case of multiple forces and fluxes acting simultaneously, the fluxes and forces will be related by a linear transport coefficient matrix. 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