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A phase-space distribution function describes the population of particle states as a function of position and momentum. In a classical one-particle normalization, $f(\mathbf{x},\mathbf{p},t)d^{3}x{d}^{3}p$ is the expected number of particles in the small phase-space cell. Integrating over momentum gives the local number density; integrating over position and momentum gives the total particle number.

A spatially uniform gas can have a nontrivial momentum distribution even though its spatial number density is constant. Moments of $f$ give mean velocity, momentum flux and kinetic energy density, connecting microscopic transport to fluid mechanics. Different conventions can instead normalize $f$ to probability or include factors such as $(2\pi \hslash {)}^{-3}$ in the integration measure. Stating the convention is essential before comparing a distribution formula or its moments with another source.

A momentum distribution describes the statistical weight of particle momenta, with its normalization stated. A one-particle distribution $n(p)$ can be normalized to one as a probability density or to the particle number; these conventions differ by a factor.

For a pure state with momentum-space wavefunction $\tilde{\psi }(p)$, the probability density is $|\tilde{\psi }(p){|}^2$ under the chosen Fourier normalization. In many-body systems, the one-body density matrix determines the single-particle momentum occupation. A narrow momentum peak can indicate coherence, but finite size, interactions and the measurement protocol also affect the observed profile.