 Distribution function (physics)

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.

In physics, a distribution function describes how a quantity is distributed over a range of values or states. It is often used in various fields, including statistical mechanics, thermodynamics, and quantum mechanics, to describe the statistical properties of systems consisting of many particles. ### Key Contexts: 1. **Statistical Mechanics**: In statistical mechanics, the distribution function characterizes the probability of finding particles within certain states defined by parameters such as energy, momentum, or position.