A phase-space distribution function describes the population of particle states as a function of position and momentum. In a classical one-particle normalization, 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 give mean velocity, momentum flux and kinetic energy density, connecting microscopic transport to fluid mechanics. Different conventions can instead normalize to probability or include factors such as 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 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 , the probability density is 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.
A phase-space distribution function constant inside a prescribed region and zero outside. The one-dimensional example in the title has a velocity plateau and distributional derivatives at its edges. Its simple kinetic response can support undamped collective modes, illustrating why absence of growing modes does not imply Landau damping. For electrons with fixed ions, , so collective roots lie outside the populated velocity interval.
An interval of particle velocities on which an equilibrium phase-space distribution function is constant. Its zero derivative removes the resonant imaginary part of the linear dielectric function for waves with phase velocity inside the interval. This permits a completely undamped electron acoustic wave in suitable single-population, nonthermal electron distributions. Zero slope alone is necessary but does not guarantee that the real dispersion relation has a root.
For identical particles with symmetric, unit-normalized phase-space distribution function , the number-normalized two-particle marginal is . Its integral over is and its full integral is . The factorial convention counts ordered distinct pairs and absorbs the combinatorial coefficient in the first BBGKY hierarchy equation. It does not imply statistical independence of the pair.
The spatial marginal of a number-normalized two-particle distribution function. It counts ordered distinct pairs, so its integral over both positions is . Exchange symmetry gives . In a homogeneous isotropic fluid it is , where is the radial distribution function.
The number density is the number of particles per unit volume. For a homogeneous species with internal states and phase-space distribution in natural units,
Bacterial density is the number of bacteria per unit length, area or volume in a spatial model. It is a field variable, distinct from the organisms themselves.
Column density is the number density integrated along a line of sight. Multiplication by an absorption cross-section gives an optical depth. A small column can result from a small local density, a small path length, or a small fraction of atoms in the absorbing state, so density inference requires these quantities to be constrained.
Electron number density is the number of free electrons per unit volume. In fully ionized stellar matter, , with mass density and mean molecular weight per electron .
The energy density is energy per unit volume. A homogeneous particle species with one-particle energy has
Energy density per unit spectral coordinate, such as per frequency or per wavelength. Conservation of energy in corresponding bins requires ; therefore their peak positions need not map by .
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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.