= Distribution function
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{title2=$f(\mathbf x,\mathbf p)$}
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= Phase-space distribution function
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= Phase-space distribution
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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, $f(\mathbf x,\mathbf p,t)\,d^3x\,d^3p$ 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 $(2\pi\hbar)^{-3}$ in the integration measure. Stating the convention is essential before comparing a distribution formula or its moments with another source.
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