SQUARE45

Let

\mathcal{H}: (\boldsymbol{\rho}, t) \rightarrow \mathbb{R}

be the Hamiltonian of the system, defined on the phase space

\Gamma \subset \mathbb{R}^{6N}

. The system is constrained to the energy shell

\Omega_{E, \epsilon} = \left\{ \boldsymbol{\rho} \in \Gamma \mid E \le \mathcal{H}(\boldsymbol{\rho}) \le E + \epsilon \right\}

. The number of accessible microstates,

W(E, V, N)

, is given by the phase space volume integral:

W(E, V, N) = \int_{\Omega_{E, \epsilon}} \frac{d^{3N} \boldsymbol{q}}{h^{3N} N!} \delta\left(\mathcal{H}(\boldsymbol{q}, \boldsymbol{p}) - E\right) \cdot \frac{1}{\epsilon} dE' d^{3N} \boldsymbol{p}

where

\boldsymbol{q} = (q_1, \dots, q_N)

and

\boldsymbol{p} = (p_1, \dots, p_N)

are the generalized coordinates and momenta, respectively. The entropy

S

is then defined via the Boltzmann relation:

S(E, V, N) = k_B \ln W(E, V, N)

where

k_B

is the Boltzmann constant.

Microcanonical Ensemble