# 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 \([momentum][position]=[\hbar]\)).

## Keywords

Hamiltonian System Lyapunov Exponent Entropy Production Microcanonical Ensemble Lagrange Parameter## Preview

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