Boltzmann-Gibbs ensemble theory

Most investigations on nonequilibrium systems were initially carried out in the NESS. It is widely believed that NESSs are among the best candidate nonequilibrium systems to possibly extend the Boltzmann-Gibbs ensemble theory beyond equilibrium [50, 51]. 

There have apparently been two parallel developments of the statistical mechanics theory, which are typified by the work of Boltzmann [6] and Gibbs [33]. The main difference between the two approaches lies in the choice of statistical unit [15]. In the method of Boltzmann the statistical unit is the molecule and the statistical ensemble is a large number of molecules constituting a system, thus the system properties are said to be calculated as... 

F is thus expressed as the statistical average of the random variable exp[-l-E(x)/fe3T] with the Boltzmann PD. Notice that F (like S) is defined up to an additive constant. < E>, S, and F are extensive variables that is, they are proportional to the number of particles N. One can also define the Gibbs free energy G = F + PV of the (NPT) ensemble. Even though G is calculated in simulations more often than F, for the sake of simplicity (and without a loss of generality) we shall describe the general theory and the various methods with respect to F. 

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