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 $\rho(\mathbf{p}, \mathbf{q}, t) = \delta(\mathbf{p} - \mathbf{p}_0(t))\delta(\mathbf{q} - \mathbf{q}_0(t))$. Now, if we build a general density $\rho$ as superpositions of "δs" as $\rho = \Sigma_i p_i\delta_{\mathbf{p}_i(t),\mathbf{q_i}(t)}$, we find that $\rho$ verifies Liouville equation:
$$ \frac{\partial \rho}{\partial t} = - \{H, \rho\} \tag{6} $$So we say classically that (6) describes the evolution of any probability density $\rho$.
Now, starting from a density $\rho$ that verifies (6), we can look at the evolution of the expectation value $< f >_t$ of an observable $f(\mathbf{p}, \mathbf{q}, t)$ defined as:
$$ < f(\mathbf{p}, \mathbf{q}, t) >_t = \int d^3\mathbf{p} d^3\mathbf{q} \rho(\mathbf{q}, \mathbf{p}, t) f(\mathbf{p}, \mathbf{q}, t) $$After a few algebra, we find:
$$ \frac{d}{dt} < f >_t = < \frac{\partial f}{\partial t} >_t + < \{H, f\} >_t $$Another important fact in Classical statistical mechanics is Boltzmann distribution.
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Author of the notes: Antonio J. Pan-Collantes
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