= Kinetic theory
{title2=$f(t,x,v)$}
Kinetic theory describes many-particle systems through a distribution on <phase space>, retaining both position and <velocity>. In the number-density convention, $f(t,\mathbf x,\mathbf v)\,d^3x\,d^3v$ is the expected number of particles in a small position–<velocity> region. For identical particles of <mass> $m$, its <integrals> give the <number density> $n$ and bulk <velocity> $\mathbf u$:
$$
n(t,\mathbf x)=\int_{\mathbb R^3}f\,d^3v,\qquad n\mathbf u=\int_{\mathbb R^3}\mathbf v f\,d^3v.
$$
This $f$ is not normalized to one: its <integral> over the whole <phase space> gives the total expected particle number. Transport moves particles along trajectories, while <collision> or <diffusion> terms change the distribution. The <Boltzmann equation>, <Hamiltonian Liouville equation> and <Fokker-Planck equation> describe different physical assumptions, rather than interchangeable collision laws.
An <ideal gas> is a motivating example: the <pressure> on a wall comes from microscopic <momentum> transfers, even when the bulk <velocity> is zero. The <Maxwell-Boltzmann velocity distribution> separates this random molecular motion from bulk motion; see <ideal gas pressure from wall collisions>. The attractive feature of <kinetic theory> is that a distribution can describe two opposing populations where one average <velocity> hides almost everything. It also tells us when averaging into a <continuum approximation> loses essential information: compare the intermolecular <mean free path> with the apparatus scale using the <Knudsen number>. https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node212.html[Richard Fitzpatrick, Kinetic Theory: fundamental assumptions and wall pressure]{ref}
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