SQUARE45

Let the system's configuration space be

\mathcal{Q} \subset \mathbb{R}^{3N}

and the corresponding phase space be

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

. The generalized coordinates are defined by the vector

\mathbf{q} = (q_1, q_2, \dots, q_{3N})

and the conjugate momenta

\mathbf{p} = (p_1, p_2, \dots, p_{3N})

. The number of degrees of freedom,

f

, is the dimension of the configuration space,

f = \dim(\mathcal{Q})

. For a system of

N

particles interacting via a potential

V(\mathbf{q})

, the Hamiltonian is given by

\mathcal{H}(\mathbf{q}, \mathbf{p}) = \sum_{i=1}^{N} \frac{\mathbf{p}_i^2}{2m_i} + V(\mathbf{q})

. The canonical volume element in phase space is

d\Gamma = d^{3N}\mathbf{q} d^{3N}\mathbf{p}

. The partition function

Z

is defined by the integral over the phase space:

Z = \frac{1}{h^{3N} N!} \int_{\Gamma} e^{-\beta \mathcal{H}(\mathbf{q}, \mathbf{p})} d\Gamma

where

\beta = 1/k_B T

. The number of degrees of freedom

f

dictates the exponent of the momentum integral, such that the classical limit of the partition function yields

Z \propto (\frac{1}{\hbar \beta})^{f/2} \cdot \text{constant}

Degrees of Freedom