Statistical Mechanics

Classical Mechanics

The movement of a particle is given by an ODE:

\[ \frac{d}{dt}\begin{bmatrix} x \\ p \end{bmatrix} = \begin{bmatrix} \frac{\partial }{\partial p}H(x,p) \\ -\frac{\partial }{\partial x}H(x,p) \end{bmatrix} \]

Harmonic oscillator

\[ H(x,p) = \frac{p^2}{2} + \frac{x^2}{2} \]

Statistical Mechanics

We are interested in stationary distributions on phase space (i.e. fixed points on the information manifold under the flow of the dynamics).

Canonical distribution

\[ p(x, v) \propto e^{-H(x, v)} \]

Example of harmonic oscillator

\[ p(x,v) \propto e^{-\frac{1}{2}v^2 - \frac{1}{2}x^2} \]

i.e a Gaussian distribution!

[illustration]

Microcanonical distribution

\[ p(x, v) \propto \delta(H(x, v) - E) \]

for some fixed \(E\).

Example of harmonic oscillator

[illustration]