Classical Statistical Mechanics

When in the XIX century a major problem of theoretical physics was the description of complex systems, with $10^{23}$ degrees of freedom, as required for the foundations of thermodynamics and statistical mechanics, it became clear that new mathematical ideas were needed; one could not reasonably think to consider a Cauchy problem with $10^{23}$ initial data. This led to abandon the idea that a physical state is described by a point in phase space and to rather describe a state as a probability measure on the phase space. In this way probability theory and random variables entered in a crucial and philosophically important way into the framework of Physics, at the basis of Classical Statistical Mechanics.

What follows is from "From Classical to Quantum Mechanics:”How to translate physical ideas into
mathematical language” H. Bergeron", in Calibre.

The Hamiltonian equations for a particle

\frac{\partial p}{\partial t} = - \frac{\partial H}{\partial q},

\frac{\partial q}{\partial t} = \frac{\partial H}{\partial p} .

correspond to the ideal case of a particle perfectly localized in Phase Space and we can represent this situation by the probability density $ρ (p, q, t) = δ (p - p_{0} (t)) δ (q - q_{0} (t))$ . Now, if we build a general density $ρ$ as superpositions of "δs" as $ρ = Σ_{i} p_{i} δ_{p_{i} (t), q_{i} (t)}$ , we find that $ρ$ verifies Liouville equation:

\begin{matrix} (6) & \frac{\partial ρ}{\partial t} = - {H, ρ} \end{matrix}

So we say classically that (6) describes the evolution of any probability density $ρ$ .

Now, starting from a density $ρ$ that verifies (6), we can look at the evolution of the expectation value $< f >_{t}$ of an observable $f (p, q, t)$ defined as:

< f (p, q, t) >_{t} = \int d^{3} p d^{3} q ρ (q, p, t) f (p, q, t)

After a few algebra, we find:

\frac{d}{d t} < f >_{t} =< \frac{\partial f}{\partial t} >_{t} + < {H, f} >_{t}

Another important fact in Classical statistical mechanics is Boltzmann distribution.