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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Distribution function (physics)</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the distribution function as used in physics. For the related mathematical concepts, see <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> and <a href="Probability_density_function" title="Probability density function">probability density function</a>.</div>
<p>In molecular <a href="Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a> in <a href="Physics" title="Physics">physics</a>, a system's <b>distribution function</b> is a function of seven variables, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t,x,y,z,v_{x},v_{y},v_{z})}">
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<annotation encoding="application/x-tex">{\displaystyle f(t,x,y,z,v_{x},v_{y},v_{z})}</annotation>
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</math></span><img src="./526f3b5b9a848742edf29b461b8b77813191ceb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.311ex; height:3.009ex;" alt="{\displaystyle f(t,x,y,z,v_{x},v_{y},v_{z})}" loading="lazy"></span>, which gives the number of particles per unit volume in single-particle <a href="Phase_space" title="Phase space">phase space</a>.<sup id="cite_ref-m713_1-0" class="reference"><a href="#cite_note-m713-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is the number of particles per unit volume having approximately the <a href="Velocity" title="Velocity">velocity</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} =(v_{x},v_{y},v_{z})}">
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</math></span><img src="./d47345c2e02313cffb6fdaf6c5957f66c6507472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.651ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =(x,y,z)}" loading="lazy"></span> and time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
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</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>. The usual normalization of the distribution function is
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}n(\mathbf {r} ,t)&amp;=\int f(\mathbf {r} ,\mathbf {v} ,t)\,dv_{x}\,dv_{y}\,dv_{z},\\N(t)&amp;=\int n(\mathbf {r} ,t)\,dx\,dy\,dz,\end{aligned}}}">
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where <span class="texhtml"><i>N</i></span> is the total number of particles and <span class="texhtml"><i>n</i></span> is the <a href="Number_density" title="Number density">number density</a> of particles – the number of particles per unit volume, or the <a href="Density" title="Density">density</a> divided by the mass of individual particles.
</p><p>A distribution function may be specialised with respect to a particular set of dimensions. E.g. take the quantum mechanical six-dimensional phase space, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z;p_{x},p_{y},p_{z})}">
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</p><p>Particle distribution functions are often used in <a href="Plasma_physics" class="mw-redirect" title="Plasma physics">plasma physics</a> to describe wave–particle interactions and velocity-space instabilities. Distribution functions are also used in <a href="Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> and <a href="Nuclear_physics" title="Nuclear physics">nuclear physics</a>.
</p><p>The <a href="Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">basic distribution function</a> uses the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
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<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f&amp;=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {mv^{2}}{2kT}}\right)\\[2pt]&amp;=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right).\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f&amp;=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {mv^{2}}{2kT}}\right)\\[2pt]&amp;=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right).\end{aligned}}}</annotation>
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</p><p>Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m((v_{x}-u_{x})^{2}+(v_{y}-u_{y})^{2}+(v_{z}-u_{z})^{2})}">
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</p><p><a href="Plasma_(physics)" title="Plasma (physics)">Plasma</a> theories such as <a href="Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamics</a> may assume the particles to be in <a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>. In this case, the distribution function is <i><a href="Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwellian</a></i>. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used, since plasmas are rarely in thermal equilibrium.
</p><p>The mathematical analogue of a distribution is a <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a>; the time evolution of a measure on a phase space is the topic of study in <a href="Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>.
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</style><cite id="CITEREFHilleryO'ConnellScullyWigner1984" class="citation journal cs1">Hillery, M.; O'Connell, R.F.; Scully, M.O.; Wigner, E.P. (1984). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/0370157384901601">"Distribution functions in physics: Fundamentals"</a></span>. <i>Physics Reports</i>. <b>106</b> (3): <span class="nowrap">121–</span>167. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-1573%2884%2990160-1">10.1016/0370-1573(84)90160-1</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-07-25</span></span>.</cite></span>
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