Phase space (classical)

For a classical mechanical system with $n$ degrees of freedom, the phase space is the $2n$-dimensional space

(or, more generally, a $2n$-dimensional symplectic manifold ) whose points are pairs $(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n)$ of generalized positions $q_i$ and conjugate momenta $p_i$.

A single point in phase space specifies the complete instantaneous microstate of the system.

Examples

• Single particle in 3D: Phase space is $\mathbb{R}^6$ with coordinates $(x, y, z, p_x, p_y, p_z)$.
• $N$ particles in 3D: Phase space is $\mathbb{R}^{6N}$ with coordinates $(q_1, \ldots, q_{3N}, p_1, \ldots, p_{3N})$.

Role in statistical mechanics

In the microcanonical , canonical , and grand canonical ensembles, equilibrium distributions are probability measures on phase space. The phase space volume element $d^nq\,d^np$ (or $d\Gamma$) is the natural measure for integration.