particledistribution

Particledistribution is a term used in physics to describe how particles are distributed in phase space, typically through a distribution function f(x, p, t) or f(p, t). It encodes the number density of particles as a function of position x, momentum p, and time t.

Formally, f(x, p, t) is a density in phase space, and f(x, p, t) d^3x d^3p gives the expected number of particles in the phase-space element d^3x d^3p. Marginal densities obtained by integrating over momentum or position yield observable quantities such as the number density n(x, t) = ∫ f(x, p, t) d^3p and the current density j(x, t) = ∫ (p/m) f(x, p, t) d^3p.

Equilibrium distributions vary by statistics. In classical limits, the Maxwell-Boltzmann distribution describes f with energy ε(p) in thermal equilibrium. For quantum particles, f is governed by Fermi-Dirac statistics (fermions with Pauli exclusion) or Bose-Einstein statistics (bosons capable of condensation). These distributions depend on temperature and chemical potential and reduce to Maxwell-Boltzmann in appropriate limits.

Evolution and dynamics are described by the Boltzmann equation, which accounts for free streaming and external forces as well as collisions: ∂f/∂t + v·∇_x f + F·∇_p f = (∂f/∂t)_coll. In the collisionless limit, this reduces to the Liouville equation. Relativistic generalizations apply in high-energy or cosmological contexts.

Applications span plasma physics, astrophysics, cosmology, and condensed matter. They underlie models of particle beams, cosmic-ray spectra, electron distributions in solids, and numerical simulations such as particle-in-cell or Monte Carlo methods. Notes: f is a phase-space density, not a probability density unless the total particle number is normalized to one. Moments of f yield density, current, and energy density; the quantum case often employs the Wigner distribution as a related concept.