View source for Boltzmann distribution

The statistical equilibrium distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168101.png" /> of the momenta <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168102.png" /> and coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168103.png" /> for the particles of an ideal gas, the molecules of which obey the laws of classical mechanics, in an external potential field:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168105.png" /> is the Boltzmann constant (a universal constant: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168106.png" /> times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168107.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168108.png" /> is the absolute temperature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b0168109.png" /> is the kinetic energy of the particle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681010.png" /> is the potential energy of the particle in the field, and the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681011.png" /> is defined by normalization over a dimensionless phase volume:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681012.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681013.png" /> is the total number of particles, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681014.png" /> is the Planck constant (a universal constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681015.png" />),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681016.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681017.png" /> can also be defined by the condition of normalization in the space of velocities and coordinates, which is more usual in the kinetic theory of gases:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681018.png" /></td> </tr></table>

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The Boltzmann distribution is a consequence of the [[Boltzmann statistics|Boltzmann statistics]] for an ideal gas, and is a particular case of the [[Gibbs distribution|Gibbs distribution]]

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for an ideal gas, when

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and the canonical Gibbs distribution becomes the product of the Boltzmann distributions for individual particles. The Boltzmann distribution is the limiting case of quantum statistics for an ideal gas at sufficiently high temperatures, when quantum effects can be neglected. The average occupation number of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681022.png" />-th quantum state of a particle is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681024.png" /> is the energy corresponding to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681025.png" />-th quantum state of the particle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681026.png" /> is the chemical potential defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681027.png" />. Formula (2) is valid for temperatures and densities at which the average distance between the particles is larger than the ratio between the Planck constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681028.png" /> and the modulus of the average thermal velocity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681029.png" /></td> </tr></table>

The [[Maxwell distribution|Maxwell distribution]] is a special case of the Boltzmann distribution (1) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681030.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

The distribution function (1) is sometimes referred to as the Maxwell–Boltzmann distribution, the term Boltzmann distribution being reserved for the distribution function (1) integrated over all momenta of particles representing the density of the number of particles at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681032.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681034.png" /> is the density of the number of particles corresponding to the point at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681035.png" />. The relative densities of the number of particles at different points depend on the differences between the potential energies at these points:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681037.png" />. A particular case of (4) yields the barometric formula, which defines the particle densities in the gravity field above the surface of the Earth:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681039.png" /> is the acceleration of gravity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681040.png" /> is the mass of the particle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681041.png" /> is the altitude above the Earth's surface, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681042.png" /> is the density at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681043.png" />.

The Boltzmann distribution of a mixture of several gases with different masses shows that the partial density distributions of the particles for each individual component is independent of that of other components. For a gas in a rotating vessel, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681044.png" /> is the field of the centrifugal forces:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681045.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016810/b01681046.png" /> is the angular velocity of rotation.

For references, see [[Boltzmann statistics|Boltzmann statistics]].