View source for Maxwell–Boltzmann distribution

{{Short description|Specific probability distribution function, important in physics}}
{{Hide in print|1={{Probability distribution
| name       = Maxwell–Boltzmann
| type       = density
| pdf_image  = [[File:Maxwell-Boltzmann distribution pdf.svg|325px]]
| cdf_image  = [[File:Maxwell-Boltzmann distribution cdf.svg|325px]]
| parameters = <math>a>0</math>
| support    = <math>x\in (0;\infty)</math>
| pdf        = <math>\sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/\left(2a^2\right)}}{a^3}</math>
| cdf        = <math>\operatorname{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \frac{x e^{-x^2/\left(2a^2\right)}}{a} </math> where erf is the [[error function]]
| mean       = <math>\mu=2a \sqrt{\frac{2}{\pi}}</math>
| median     =
| mode       = <math>\sqrt{2} a</math>
| variance   = <math>\sigma^2=\frac{a^2(3 \pi - 8)}{\pi}</math>
| skewness   = <math>\gamma_1=\frac{2 \sqrt{2} (16 -5 \pi)}{(3 \pi - 8)^{3/2}}</math>
| kurtosis   = <math>\gamma_2=4\frac{\left(-96+40\pi-3\pi^2\right)}{(3 \pi - 8)^2}</math>
| entropy    = <math>\ln\left(a\sqrt{2\pi}\right)+\gamma-\frac{1}{2}</math>
| mgf        =
| char       =
}}}}
In [[physics]] (in particular in [[statistical mechanics]]), the '''Maxwell–Boltzmann distribution''' is a particular [[probability distribution]] named after [[James Clerk Maxwell]] and [[Ludwig Boltzmann]].

It was first defined and used for describing particle [[speed]]s in [[ideal gas|idealized gases]], where the particles move freely inside a stationary container without interacting with one another, except for very brief [[collision]]s in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only ([[atoms]] or [[molecules]]), and the system of particles is assumed to have reached [[thermodynamic equilibrium]].<ref name="StatisticalPhysics">''Statistical Physics'' (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, {{isbn|9780471915331}}</ref> The energies of such particles follow what is known as [[Maxwell-Boltzmann statistics]], and the statistical distribution of speeds is derived by equating particle energies with [[kinetic energy]].

Mathematically, the Maxwell–Boltzmann distribution is the [[chi distribution]] with three [[degrees of freedom]] (the components of the [[velocity]] vector in [[Euclidean space]]), with a [[scale parameter]] measuring speeds in units proportional to the square root of <math>T/m</math> (the ratio of temperature and particle mass).<ref>University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, {{isbn|978-0-321-50130-1}}</ref>

The Maxwell–Boltzmann distribution is a result of the [[kinetic theory of gases]], which provides a simplified explanation of many fundamental gaseous properties, including [[pressure]] and [[diffusion]].<ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, {{isbn|3-527-26954-1}} (Verlagsgesellschaft), {{isbn|0-89573-752-3}} (VHC Inc.)</ref> The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (the [[Magnitude (mathematics)|magnitude]] of the velocity) of the particles. A particle speed probability distribution indicates which speeds are more likely: a particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases applies to the classical [[ideal gas]], which is an idealization of real gases. In real gases, there are various effects (e.g., [[van der Waals interaction]]s, [[vortex|vortical]] flow, [[special relativity|relativistic]] speed limits, and quantum [[exchange interaction]]s) that can make their speed distribution different from the Maxwell–Boltzmann form. However, [[Rarefaction|rarefied]] gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. Ideal [[plasma (physics)|plasmas]], which are ionized gases of sufficiently low density, frequently also have particle distributions that are partially or entirely Maxwellian.<ref>N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press, Inc., 1986, among many other texts on basic plasma physics</ref>

The distribution was first derived by Maxwell in 1860 on heuristic grounds.<ref name=Maxwell>See:
*  Maxwell, J.C. (1860 A): ''Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Series, vol.19, pp.19-32. [https://www.biodiversitylibrary.org/item/53795#page/33/mode/1up] 
*  Maxwell, J.C. (1860 B): ''Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'', 4th Ser., vol.20, pp.21-37. [https://www.biodiversitylibrary.org/item/20012#page/37/mode/1up]</ref> Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution.

The distribution can be derived on the ground that it maximizes the entropy of the system. A list of derivations are:

# [[Maximum entropy probability distribution#Distributions with measured constants|Maximum entropy probability distribution]] in the phase space, with the constraint of [[Conservation of energy|conservation of average energy]] <math>\langle H \rangle = E</math>;
# [[Canonical ensemble]].

== Distribution function ==
Assuming the system of interest contains a large number of particles, the fraction of the particles within an infinitesimal element of three-dimensional velocity space, <math>d^3v</math>, centered on a velocity vector of magnitude <math>v</math>, is <math>f(v) d^3v</math>, in which

:<math> f(v) ~\mathrm{d}^3v = \left(\frac{m}{2 \pi kT}\right)^{3/2}\,  e^{- \frac{mv^2}{2kT}} ~\mathrm{d}^3v, </math>

where <math>m</math> is the particle mass and <math>kT</math> is the product of [[Boltzmann's constant]] and [[thermodynamic temperature]].

[[Image:MaxwellBoltzmann-en.svg|right|thumb|340px|The speed probability density functions of the speeds of a few [[noble gas]]es at a temperature of 298.15&nbsp;K (25&nbsp;°C). The ''y''-axis is in s/m so that the area under any section of the curve (which represents the probability of the speed being in that range) is dimensionless.]]
One can write the element of velocity space as d<math>^3v</math> = d<math>v_x</math>d<math>v_y</math>d<math>v_z</math>, for velocities in a standard Cartesian coordinate system, or as d<math>^3v</math> = <math>v^2</math>d<math>v</math>d<math>\Omega</math> in a standard spherical coordinate system, where d<math>\Omega</math> is an element of solid angle. Here <math>f(v)</math> is given as a probability distribution function, properly normalized so that <math>\int f(v)</math>d<math>^3v</math> over all velocities equals one.  In plasma physics, the probability distribution is often multiplied by the particle density, so that the integral of the resulting distribution function equals the density.

Undergraduate students are likely to work with the Maxwellian distribution function for particles moving in only one direction. If this direction is <math>x</math>, then one has 
:<math> f(v_x) ~\mathrm{d}v_x = \left(\frac{m}{2 \pi kT}\right)^{1/2}\,  e^{- \frac{mv_x^2}{2kT}} ~\mathrm{d}v_x, </math> 
which can be obtained by integrating the three-dimensional form given above over <math>v_y</math> and <math>v_z</math>.

Recognizing the symmetry of <math>f(v)</math>, one can integrate over solid angle and write a probability distribution of speeds as the function<ref>[[Harald J. W. Mueller-Kirsten|H.J.W. Müller-Kirsten]] (2013), ''Basics of Statistical Physics'', 2nd ed., [[World Scientific]], {{isbn|978-981-4449-53-3}}, Chapter 2.</ref>

:<math> f(v) ~\mathrm{d}v= \left(\frac{m}{2 \pi kT}\right)^{3/2}\, 4\pi v^2 e^{- \frac{mv^2}{2kT}} ~\mathrm{d}v, </math>

This [[probability density function]] gives the probability, per unit speed, of finding the particle with a speed near <math>v</math>. This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter <math>a=\sqrt{kT/m}</math>. 
The Maxwell–Boltzmann distribution is equivalent to the [[chi distribution]] with three degrees of freedom and [[scale parameter]] <math>a=\sqrt{kT/m}</math>.

The simplest [[ordinary differential equation]] satisfied by the distribution is:

:<math>k T v f'(v)+f(v) (m v^2-2 kT)=0,</math>
:<math>f(1)=\sqrt{\frac{2}{\pi }} e^{-\frac{m}{2 k T}} \left(\frac{m}{k T}\right)^{3/2}</math>

or in unitless presentation:

:<math>a^2 x f'(x)+\left(x^2-2 a^2\right) f(x)=0, </math>
:<math>f(1)=\frac{\sqrt{\frac{2}{\pi }} e^{-\frac{1}{2 a^2}}}{a^3}.</math>
<!--Note that a distribution (function) is not the same as the probability. The distribution (function) stands for an average number, as in all three kinds of statistics (Maxwell–Boltzmann, [[Bose–Einstein statistics|Bose–Einstein]], [[Fermi–Dirac statistics|Fermi–Dirac]]).--> 
With the [[Darwin–Fowler method]] of mean values the Maxwell–Boltzmann distribution is obtained as an exact result.

== Relaxation to the 2D Maxwell-Boltzmann distribution ==
[[File:Simulation of gas for relaxation demonstration.gif|thumb|471x471px|Simulation of a 2D gas relaxing towards a Maxwell-Boltzmann speed distribution]]
For particles confined to move in a plane, the speed distribution is given by

<math>P(s < |\vec{v}| < s + ds) = \frac{ms}{kT}\exp\bigg(-\frac{ms^2}{2kT}\bigg) ds  </math>

This distribution is used for describing systems in equilibrium. However, most systems do not start out in their equilibrium state. The evolution of a system towards its equilibrium state is governed by the [[Boltzmann equation]]. The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a [[molecular dynamics]] (MD) simulation in which 900 [[Hard spheres|hard sphere]] particles are constrained to move in a rectangle. They interact via [[Elastic collision|perfectly elastic collisions]]. The system is initialized out of equilibrium, but we see that the velocity distribution (in blue) quickly converges to the 2D Maxwell–Boltzmann distribution (in orange).

==Typical speeds==
The [[expectation value|mean]] speed <math> \langle v \rangle</math>, 
most probable speed ([[Mode (statistics)|mode]]) {{mvar|v<sub>p</sub>}}, 
and root-mean-square speed  <math>\sqrt{\langle v^2 \rangle}</math> 
can be obtained from properties of the Maxwell distribution.

This works well for nearly [[ideal gas|ideal]], [[noble gas|monatomic]] gases like [[helium]], but also for [[Molecule|molecular gas]]es like diatomic [[oxygen]]. 
This is because despite the larger [[heat capacity]] (larger internal energy at the same temperature) due to their larger number of [[Equipartition theorem|degrees of freedom]],  their [[Translation (physics)|translational]] [[kinetic energy]] (and thus their speed) is unchanged.<ref>
{{cite book
 | title = College Physics, Volume 1
 | edition = 9th
 | author1=Raymond A. Serway |author2=Jerry S. Faughn |author3=Chris Vuille  |last-author-amp=yes | publisher = 
 | year = 2011
 | isbn = 9780840068484
 | page = 352
 | url = https://books.google.com/books?id=HLxV-IKYO5IC&pg=PA352
 }}</ref>

{{unordered list
|1= The most probable speed, {{mvar|v<sub>p</sub>}}, is the speed most likely to be possessed by any molecule (of the same mass {{mvar|m}}) in the system and corresponds to the maximum value or [[mode (statistics)|mode]] of {{mvar|f(v)}}.  To find it, we calculate the [[derivative]] {{mvar|df/dv}}, set it to zero and solve for {{mvar|v}}:

:<math>\frac{df(v)}{dv} = -8\pi \left(\frac{m}{2 \pi kT}\right)^{3/2}\ v\ e^{-\frac{mv^2}{2kT}} \left(\frac{mv^2}{2kT}-1\right) = 0</math>

with the solution:
:<math>\frac{mv_p^2}{2kT} = 1 </math>
:<math>v_p = \sqrt { \frac{2kT}{m} } = \sqrt { \frac{2RT}{M} }</math>
(the second solution <math>v=0</math> representing the "least probable speed").

{{mvar|R}} is the [[gas constant]] and {{mvar|M}} is molar mass of the substance, and thus may be calculated as a product of particle mass, {{math|''m''}}, and [[Avogadro constant]], {{math|''N''<sub>''a''</sub>}}:
:<math>M = m N_a</math>

For diatomic nitrogen (N<sub>2</sub>, the primary component of [[air]])<ref>The calculation is unaffected by the Nitrogen being diatomic. Despite the larger [[heat capacity]] (larger internal energy at the same temperature) of diatomic gases relative to monatomic gases, due to their larger number of [[Equipartition theorem|degrees of freedom]],  <math>{{3RT}\over{M_m}}</math> is still the mean [[Translation (physics)|translational]] [[kinetic energy|kinetic]] energy. Nitrogen being diatomic only affects the value of the molar mass {{mvar|M}} {{=}} {{val|28|u=g/mol}}.
See e.g. K. Prakashan, ''Engineering Physics'' (2001), [https://books.google.com/books?id=6C0R1qpAk7EC&pg=SA2-PA278 2.278].</ref> 
at [[room temperature]] ({{val|300|u=K}}), this gives 
:<math>v_p \approx  \sqrt{\frac{2\cdot{}8.31\ \text{J}\cdot{}\text{mol}^{-1}\text{K}^{-1}\ 300\ \text{K}}{0.028\ \text{kg}\cdot{}\text{mol}^{-1}}} \approx  422\ \text{m/s}.</math>
|2= The mean speed is the [[expected value]] of the speed distribution, setting <math>b=\frac{1}{2a^2}=\frac{m}{2kT}</math>:

:<math> \langle v \rangle = \int_0^{\infty} v \, f(v) \, dv</math>
::<math> =  4 \pi \left (\frac{b}{ \pi  } \right )^\frac{3}{2} \int_{0}^{\infty} v^3\ e^{-bv^{2}}dv\,\!</math>
::<math> = 4 \pi \left (\frac{b}{  \pi } \right )^\frac{3}{2}  \frac{1}{2b^2} = \sqrt{\frac{4}{\pi b}} \,\!</math>
::<math>= \sqrt { \frac{8kT}{\pi m}}= \sqrt { \frac{8RT}{\pi M}} = \frac{2}{\sqrt{\pi}} v_p  </math>

|3= The mean square speed <math>\langle v^2 \rangle</math> is the second-order [[Moment (mathematics)|raw moment]] of the speed distribution. The "root mean square speed" <math> v_\mathrm{rms}</math>  is the square root of the mean square speed, corresponding to the speed of a particle with median [[kinetic energy]], setting <math>b=\frac{1}{2a^2}=\frac{m}{2kT}</math>:
:<math> v_\mathrm{rms} = \sqrt{\langle v^2 \rangle} = \left(\int_0^{\infty} v^2 \, f(v) \, dv  \right)^{1/2}</math>
<!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
::<math> = \left( 4 \pi \left (\frac{b}{  \pi } \right )^\frac{3}{2} \int_{0}^{\infty}  v^4\ e^{-bv^2}dv\right)^{1/2}\,\!</math>
::<math> = \left(4 \pi \left (\frac{b}{\pi}\right )^\frac{3}{2} \frac{3}{8}  \sqrt{\frac{\pi}{b^5}} \right)^{1/2} = \left( \frac{3}{2b} \right)^{1/2}\,\!</math>
:<math>= \sqrt { \frac{3kT}{m}}= \sqrt { \frac{3RT}{M} } = \sqrt{ \frac{3}{2} } v_p  </math>
}}

In summary, the typical speeds are related as follows:
:<math>v_p \approx 88.6\%\ \langle v \rangle  < \langle v \rangle < 108.5\%\  \langle v \rangle \approx v_\mathrm{rms}. </math>

The root mean square speed is directly related to the [[speed of sound]] {{mvar|c}} in the gas, by
:<math>c = \sqrt{\frac{\gamma}{3}}\ v_\mathrm{rms}  = \sqrt{\frac{f+2}{3f}}\ v_\mathrm{rms} = \sqrt{\frac{f+2}{2f}}\ v_p ,</math>
where  <math>\gamma = 1 + \frac{2}{f}</math> is the [[adiabatic index]], 
{{mvar|f}} is the number of [[degrees of freedom]] of the individual gas molecule. 
For the example above, diatomic nitrogen (approximating [[air]]) at {{val|300|u=K}}, <math>f = 5</math><ref>
Nitrogen at room temperature is considered a "rigid" diatomic gas, with two rotational degrees of freedom additional to the three translational ones, and the vibrational degree of freedom not accessible.</ref> and
:<math>c = \sqrt{\frac{7}{15}}v_\mathrm{rms} \approx 68\%\ v_\mathrm{rms} \approx 84\%\ v_p  \approx 353\ \mathrm{m/s}, </math>
the true value for air can be approximated by using the average molar weight of [[Atmospheric chemistry|air]] ({{val|29|u=g/mol}}), yielding 
{{val|347|u=m/s}} at {{val|300|u=K}} (corrections for variable [[humidity]] are of the order of 0.1% to 0.6%).

The average relative velocity
:<math> v_{\rm rel} \equiv \langle |\vec{v}_1-\vec{v}_2| \rangle = \int\!d^3v_1d^3v_2|\vec{v}_1-\vec{v}_2|f(\vec{v}_1)f(\vec{v}_2) = \frac{4}{\sqrt{\pi}}\frac{kT}{m} = 2\langle v \rangle </math>

where the three-dimensional velocity distribution is

:<math> f(\vec{v}) \equiv \frac{1}{(2\pi kT/m)^{3/2}}e^{-\frac12 m\vec{v}^2/kT}. </math>

The integral can easily be done by changing to coordinates <math> \vec{u} = \vec{v}_1-\vec{v}_2 </math> and <math> \vec{U} = \frac{\vec{v}_1+\vec{v}_2}{2}.</math>

==Derivation and related distributions==
===Maxwell–Boltzmann statistics===
{{main|Maxwell–Boltzmann statistics#Derivations|Boltzmann distribution}}
The original derivation in 1860 by [[James Clerk Maxwell]] was an argument based on molecular collisions of the [[Kinetic theory of gases]] as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium.<ref name=Maxwell/><ref>{{Cite journal | last1 = Gyenis | first1 = Balazs | doi = 10.1016/j.shpsb.2017.01.001 | title = Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium | journal = Studies in History and Philosophy of Modern Physics | volume = 57 | pages = 53–65 | year = 2017| arxiv = 1702.01411 | bibcode = 2017SHPMP..57...53G }}</ref> After Maxwell, [[Ludwig Boltzmann]] in 1872<ref>Boltzmann, L., "Weitere studien über das Wärmegleichgewicht unter Gasmolekülen." ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Classe'', '''66''', 1872, pp. 275–370.</ref> also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (see [[H-theorem]]). He later (1877)<ref>Boltzmann, L., "Über die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht." ''Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe''. Abt. II, '''76''', 1877, pp. 373–435. Reprinted in ''Wissenschaftliche Abhandlungen'', Vol. II, pp. 164–223, Leipzig: Barth, 1909.</ref> derived the distribution again under the framework of [[statistical thermodynamics]]. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as [[Maxwell–Boltzmann statistics]] (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle [[Microstate (statistical mechanics)|microstate]]. Under certain assumptions, the logarithm of the fraction of particles in a given microstate is proportional   to the ratio of the energy of that state to the temperature of the system:
:<math>-\log \left(\frac{N_i}{N}\right) \propto \frac{E_i}{T}.</math>
The assumptions of this equation are that the particles do not interact, and that they are classical; this means that each particle's state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium.<ref name="StatisticalPhysics" /><ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, {{isbn|0-07-051400-3}}</ref>

This relation can be written as an equation by introducing a normalizing factor:
{{NumBlk|:|<math>
\frac{N_i} N = \frac{\exp(-E_i/kT) } { \sum_j \exp(-E_j/kT) }</math>
|{{EquationRef|1}}}}

where:
* {{mvar|N<sub>i</sub>}} is the expected number of particles in the single-particle microstate {{mvar|i}},
* {{mvar|N}} is the total number of particles in the system,
* {{mvar|E<sub>i</sub>}}  is the energy of microstate {{mvar|i}},
* the sum over index {{mvar|j}} takes into account all microstates, 
*{{mvar|T}} is the equilibrium temperature of the system,
*{{mvar|k}} is the [[Boltzmann constant]].
The denominator in Equation ({{EquationNote|1}}) is simply a normalizing factor so that the ratios <math>N_i:N</math>  add up to unity — in other words it is a kind of [[partition function (statistical mechanics)|partition function]] (for the single-particle system, not the usual partition function of the entire system).

Because velocity and speed are related to energy, Equation ({{EquationNote|1}}) can be used to derive relationships between temperature and the speeds of gas particles. All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions.

===Distribution for the momentum vector===

The potential energy is taken to be zero, so that all energy is in the form of kinetic energy.
The relationship between [[Kinetic energy#Kinetic energy of rigid bodies|kinetic energy and momentum]] for massive non-[[special relativity|relativistic]] particles is

{{NumBlk|:|<math>E=\frac{p^2}{2m}</math>|{{EquationRef|2}}}}

where ''p''<sup>2</sup> is the square of the momentum vector 
'''p'''&nbsp;=&nbsp;[''p''<sub>''x''</sub>,&nbsp;''p''<sub>''y''</sub>,&nbsp;''p''<sub>''z''</sub>]. We may therefore rewrite Equation ({{EquationNote|1}}) as:

{{NumBlk|:|<math>
\frac{N_i}{N} = 
\frac{1}{Z} 
\exp \left[
-\frac{p_{i, x}^2 + p_{i, y}^2 + p_{i, z}^2}{2mkT}
\right]</math>
|{{EquationRef|3}}}}

where ''Z'' is the [[partition function (statistical mechanics)|partition function]], corresponding to the denominator in Equation ({{EquationNote|1}}). Here ''m'' is the molecular mass of the gas, ''T'' is the thermodynamic temperature and ''k'' is the [[Boltzmann constant]].  This distribution of <math>N_i:N</math> is [[Proportionality (mathematics)|proportional]] to the [[probability density function]] ''f''<sub>'''p'''</sub> for finding a molecule with these values of momentum components, so:

{{NumBlk|:|<math>
f_\mathbf{p} (p_x, p_y, p_z) 
\propto
\exp \left[
-\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}
\right]</math>|{{EquationRef|4}}}}

The [[normalizing constant]]  can be determined by recognizing that the probability of a molecule having ''some'' momentum must be 1.  
Integrating  the exponential in ({{EquationNote|4}}) over all ''p''<sub>''x''</sub>, ''p''<sub>''y''</sub>, and ''p''<sub>''z''</sub> yields a factor of 
:<math>\iiint_{-\infty}^{+\infty} \exp \left[ -\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}\right] dp_x\ dp_y\ dp_z  = {(\sqrt{\pi} \sqrt{2mkT})^3}</math>

So that the normalized distribution function is:
{{Equation box 1 |indent=: |equation=
<math>
f_\mathbf{p} (p_x, p_y, p_z) =
\left( 2 \pi mkT \right)^{-3/2}
\exp \left[
-\frac{p_x^2 + p_y^2 + p_z^2}{2mkT}
\right]</math>
|cellpadding |border |border colour = #50C878 |background colour = #ECFCF4|ref=6}}

The distribution is seen to be the product of three independent [[normal distribution|normally distributed]] variables <math>p_x</math>, <math>p_y</math>, and <math>p_z</math>, with variance <math>mkT</math>. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with <math>a=\sqrt{mkT}</math>.
The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using the [[H-theorem]] at equilibrium within the [[Kinetic theory of gases]] framework.

===Distribution for the energy===

The energy distribution is found imposing
{{NumBlk|:|<math>
  f_E(E) dE = f_p(\textbf p) d^3 \textbf p,
</math>|{{EquationRef|7}}}}
where <math>d^3 \textbf p</math> is the infinitesimal phase-space volume of momenta corresponding to the energy interval <math>dE</math>.
Making use of the spherical symmetry of the energy-momentum dispersion relation <math>E=| \textbf p|^2/2m</math>,
this can be expressed in terms of <math>dE</math> as
{{NumBlk|:|<math>
  d^3 \textbf p = 4 \pi | \textbf p |^2 d |\textbf p| = 4 \pi m \sqrt{2mE} dE.
</math>|{{EquationRef|8}}}}
Using then ({{EquationNote|8}}) in ({{EquationNote|7}}), and expressing everything in terms of the energy <math>E</math>, we get
:<math>
  f_E(E) dE = \frac{1}{(2\pi m k T)^{3/2}} e^{-E/kT} 4 \pi m \sqrt{2mE} dE = 2 \sqrt{\frac{E}{\pi}} \left( \frac{1}{kT} \right)^{3/2} \exp\left(\frac{-E}{kT} \right) dE
</math>
and finally

{{Equation box 1 |indent=: |equation=
<math>f_E(E) = 2 \sqrt{\frac{E}{\pi}} \left( \frac{1}{kT} \right)^{3/2} \exp\left(\frac{-E}{kT} \right)</math>
|cellpadding |border |border colour = #50C878 |background colour = #ECFCF4|ref=9}}

Since the energy is proportional to the sum of the squares of the three normally distributed momentum components, this distribution is a [[gamma distribution]]; in particular, it is a [[chi-squared distribution]] with three degrees of freedom.

By the [[equipartition theorem]], this energy is evenly distributed among all three degrees of freedom, so that the energy per degree of freedom is distributed as a chi-squared distribution with one degree of freedom:<ref>{{cite book
|title=Statistical thermodynamics: fundamentals and applications
|first1=Normand M.
|last1=Laurendeau
|publisher=Cambridge University Press
|year=2005
|isbn=0-521-84635-8
|page=434

|url=https://books.google.com/books?id=QF6iMewh4KMC}}, [https://books.google.com/books?id=QF6iMewh4KMC&pg=PA434 Appendix N, page 434]
</ref>
:<math>
f_\epsilon(\epsilon)\,d\epsilon= \sqrt{\frac{1 }{\pi \epsilon kT}}~\exp\left[\frac{-\epsilon}{kT}\right]\,d\epsilon
</math>

where <math>\epsilon</math> is the energy per degree of freedom. At equilibrium, this distribution will hold true for any number of degrees of freedom. For example, if the particles are rigid mass dipoles of fixed dipole moment, they will have three translational degrees of freedom and two additional rotational degrees of freedom. The energy in each degree of freedom will be described according to the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five degrees of freedom. This has implications in the theory of the [[specific heat]] of a gas.

The Maxwell–Boltzmann distribution can also be obtained by considering the gas to be a type of [[gas in a box|quantum gas]] for which the approximation ''&epsilon; >> k T'' may be made.

===Distribution for the velocity vector===

Recognizing that the velocity probability density ''f''<sub>'''v'''</sub> is proportional to the momentum probability density function by

:<math>
f_\mathbf{v} d^3v = f_\mathbf{p} \left(\frac{dp}{dv}\right)^3 d^3v
</math>

and using '''p''' = m'''v''' we get

{{Equation box 1 |indent=: |equation=
<math>
f_\mathbf{v} (v_x, v_y, v_z) =
\left(\frac{m}{2 \pi kT} \right)^{3/2}
\exp \left[-
\frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}
\right]
</math>
|cellpadding |border |border colour = #50C878 |background colour = #ECFCF4}}

which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element [''dv''<sub>''x''</sub>,&nbsp;''dv''<sub>''y''</sub>,&nbsp;''dv''<sub>''z''</sub>] about velocity '''v'''&nbsp;=&nbsp;[''v''<sub>''x''</sub>,&nbsp;''v''<sub>''y''</sub>,&nbsp;''v''<sub>''z''</sub>] is

:<math>
f_\mathbf{v} \left(v_x, v_y, v_z\right)\, dv_x\, dv_y\, dv_z.
</math>

Like the momentum, this distribution is seen to be the product of three independent [[normal distribution|normally distributed]] variables <math>v_x</math>, <math>v_y</math>, and <math>v_z</math>, but with variance <math>\frac{kT}{m}</math>.  
It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity
[''v''<sub>''x''</sub>,&nbsp;''v''<sub>''y''</sub>,&nbsp;''v''<sub>''z''</sub>] is the product of the distributions for each of the three directions:

:<math>
f_\mathbf{v} \left(v_x, v_y, v_z\right) = f_v (v_x)f_v (v_y)f_v (v_z)
</math>

where the distribution for a single direction is

:<math>
f_v (v_i) =
\sqrt{\frac{m}{2 \pi kT}}
\exp \left[
\frac{-mv_i^2}{2kT}
\right].
</math>

Each component of the velocity vector has a [[normal distribution]] with mean <math>\mu_{v_x} = \mu_{v_y} = \mu_{v_z} = 0</math> and standard deviation <math>\sigma_{v_x} = \sigma_{v_y} = \sigma_{v_z} = \sqrt{\frac{kT}{m}}</math>, so the vector has a 3-dimensional normal distribution, a particular kind of [[multivariate normal distribution]], with mean <math> \mu_{\mathbf{v}} = {\mathbf{0}} </math> and standard deviation <math>\sigma_{\mathbf{v}} = \sqrt{\frac{3kT}{m}}</math>.

===Distribution for the speed===
The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above.  Note that the speed is

:<math>v = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>

and the [[volume element]] in [[spherical coordinates]]

:<math> dv_x\, dv_y\, dv_z = v^2 \sin \theta\, dv\, d\theta\, d\phi = v^2   dv\,  d\Omega</math> 
where <math>\phi</math> and <math>\theta</math> are the [[Spherical coordinate system|spherical coordinate]] angles of the velocity vector. [[Spherical coordinate system#Integration and differentiation in spherical coordinates|Integration]] of the probability density function of the velocity  over the solid angles <math>d\Omega</math> yields an  additional factor of <math>4\pi</math>.
The speed distribution with substitution of the speed for the sum of the squares of the vector components:
{{Equation box 1 |indent=: |equation=
<math>
f (v) =
\left(\frac{2}{\pi} \right)^{1/2}
\left(\frac{m}{ kT} \right)^{3/2}
v^2
\exp \left[-
\frac{mv^2}{2kT}
\right].
</math>
}}

== In n-dimensional space ==
In n-dimensional space, Maxwell-Boltzman distribution becomes:

<math> f(v) ~\mathrm{d}^nv = \left(\frac{m}{2 \pi kT}\right)^{n/2}\,  e^{- \frac{m|v|^2}{2kT}} ~\mathrm{d}^nv </math>

Speed distribution becomes:

<math> f(v) ~\mathrm{d}v = \text{const.} \times  e^{- \frac{mv^2}{2kT}} \times v^{n-1} ~\mathrm{d}v </math>

The following integral result is useful:

<math>\begin{align}
\int_{0}^{+\infty} v^a e^{-\frac{mv^2}{2kT}} dv  
&= \left[\frac{2kT}{m}\right]^{(a+1)/2} \int_{0}^{+\infty} e^{-x}x^{\frac{a}{2}}dx^{\frac{1}{2}}\\
&= \left[\frac{2kT}{m}\right]^{(a+1)/2} \int_{0}^{+\infty} e^{-x}x^{\frac{a}{2}}\frac{x^{-\frac{1}{2}}}{2}dx\\
&= \left[\frac{2kT}{m}\right]^{(a+1)/2} \frac{\Gamma (\frac{a+1}{2})}{2}
\end{align}</math>

where <math> \Gamma(z)</math>is the [[Gamma function]]. This result can be used to calculate the [[Moment (mathematics)|moments]] of speed distribution function:

<math> \begin{align}
\langle v \rangle 
&= \frac{\int_{0}^{+\infty} v \cdot v^{n-1} e^{-\frac{mv^2}{2kT}} dv}
{\int_{0}^{+\infty} v^{n-1} e^{-\frac{mv^2}{2kT}} dv} \\
&= \left[\frac{2kT}{m}\right]^{1/2} \frac{\Gamma (\frac{n+1}{2})}{\Gamma (\frac{n}{2})}
\end{align}</math>

which is the [[expectation value|mean]] speed itself <math>v_{\text{avg}} = \langle v \rangle = \left[\frac{2kT}{m}\right]^{1/2} \frac{\Gamma (\frac{n+1}{2})}{\Gamma (\frac{n}{2})}</math>.

<math> \begin{align}
\langle v^2 \rangle 
&= \frac{\int_{0}^{+\infty} v^2 \cdot v^{n-1} e^{-\frac{mv^2}{2kT}} dv}
{\int_{0}^{+\infty} v^{n-1} e^{-\frac{mv^2}{2kT}} dv} \\
&= \left[\frac{2kT}{m}\right] \frac{\Gamma (\frac{n+2}{2})}{\Gamma (\frac{n}{2})} \\
&= \left[\frac{2kT}{m}\right] \frac{n}{2} = \frac{nkT}{m}
\end{align}</math>

which gives root-mean-square speed  <math>v_{\text{rms}} = \sqrt{\langle v^2 \rangle} = \left[\frac{nkT}{m}\right]^{1/2} </math>.

The derivative of speed distribution function:

<math>\frac{df(v)}{dv} = \text{const.} \times \ e^{-\frac{mv^2}{2kT}} \left(-\frac{mv}{kT} v^{n-1}+(n-1)v^{n-2}\right) = 0 </math>

This yields the most probable speed ([[Mode (statistics)|mode]]) <math>v_{\text{p}} = \left[\frac{(n-1)kT}{m}\right]^{1/2}</math>.

==See also==
* [[Quantum Boltzmann equation]]
* [[Maxwell–Boltzmann statistics]]
* [[Maxwell–Jüttner distribution]]
* [[Boltzmann distribution]] 
* [[Boltzmann factor]]
* [[Rayleigh distribution]]
* [[Kinetic theory of gases]]

==References==

{{reflist}}

==Further reading==

* Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, {{isbn|0-7167-8964-7}}
* Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, {{isbn|978-1-4200-7368-3}}
* Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, {{isbn|0-356-03736-3}}
* Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, {{isbn|0-201-05229-6}}
* Ward, CA & Fang, G 1999, 'Expression for predicting liquid evaporation flux: Statistical rate theory approach', Physical Review E, vol. 59, no. 1, pp.&nbsp;429–40.
* Rahimi, P & Ward, CA 2005, 'Kinetics of Evaporation: Statistical Rate Theory Approach', International Journal of Thermodynamics, vol. 8, no. 9, pp.&nbsp;1–14.

==External links==
* [http://demonstrations.wolfram.com/TheMaxwellSpeedDistribution/ "The Maxwell Speed Distribution"] from The Wolfram Demonstrations Project at [[Mathworld]]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Maxwell-Boltzmann Distribution}}
[[Category:Continuous distributions]]
[[Category:Gases]]
[[Category:Ludwig Boltzmann]]
[[Category:James Clerk Maxwell]]
[[Category:Normal distribution]]
[[Category:Particle distributions]]