TABLE OF CONTENTS The Green Kubo (G K) development of a time-correlation function expression for the thermal conductivity (the G K approach) is based in classical statistical thermodynamics. Multiple methods can be used to arrive at the final result [361]. Similar approaches can be used to develop expressions for the self-diffusion coefficient, the shear viscosity, and the bulk viscosity. These are all transport coefficients that cannot be obtained by applying a perturbation to the system Hamiltonian, as can be done for some other properties (e.g., the electrical conductivity) for which there is a real force that drives the transport. Here, the method of Helfand [142] as outlined in [239], is presented step-by-step [232].
The G K approach is valid in the case of small disturbances from equilibrium and for long times (i.e., the hydrodynamic limit). The key aspect of the derivation is the introduction of a microscopic description of a system to the solution of the macroscopic governing equation.
For the thermal conductivity, a canonical ensemble of particles (i.e., the NVT ensemble) is considered, and the energy equation (similar to the macroscopic energy equation of Table 1.1 for no net flow and no radiation heat transfer, with conduction heat flux given by the Fourier law, using the thermal conductivity tensor K) is written as
where n is the number of particles per unit volume and c ? is the specific heat per particle.
Here the independent variable E ? ( x, t) is...
