2.3 Equilibrium and transport properties
According to equilibrium statistical mechanics, in a uniform temperature
fluid, the molecular velocities will be Maxwellian, and the energy
components related to the various degrees of freedom will satisfy the
equipartition principle. Thus, the equilibrium probability density
function (PDF) of each of the cartesian components of the particle in
the above example Ui, will follow the Maxwell-Boltzmann
(MB) distribution. Another important application of the Onsager
regression relationship (Eq. 6) is the emergence of a class of
relationships that relate transport properties to correlation functions
known as the Green-Kubo relationships [13, 17]. These relationships
are also a consequence of the fluctuation-dissipation theorem. Thus,
γ =(1/d) ∫ dt 〈A(0) • A(t)〉. (Eq. 10)
Here, γ is the transport coefficient of interest, t is time, d is the
dimensionality, A is the current that drives it. The integrand of (Eq.
10) is the autocorrelation function (ACF) of quantity A. One can
calculate the transport coefficients such as diffusion D, shear
viscosity ηs, and thermal conductivity k using the
Green-Kubo formula.