Learning with Boltzmann–Gibbs Statistical Mechanics

where $x/\sigma \in{\mathbb R}^D$, $D \ge 1$ being the dimension of the full space of microscopic states (called Gibbs $\Gamma$ phase-space} for classical Hamiltonian systems). Typically x carries physical units. The constant $\sigma$ carries the same physical units as x, so that $x/\sigma$ is a dimensionless quantity (we adopt from now on the notation $[x]=[\sigma]$, hence $[x/\sigma]=1$). For example, if we are dealing with an isolated classical N-body Hamiltonian system of point masses interacting among them in d dimensions, we may use $\sigma=\hbar^{Nd}$. This standard choice comes of course from the fact that, at a sufficiently small scale, Newtonian mechanics becomes incorrect and we must rely on quantum mechanics. In this case, $D=2dN$, where each of the d pairs of components of momentum and position of each of the N particles has been taken into account (we recall that $[momentum][position]=[\hbar]$).