View source for Maxwell–Boltzmann distribution

{{distinguish|Maxwell–Boltzmann statistics}}
{{Short description|Specific probability distribution function, important in physics}}
{{about|particle energy levels and velocities|system energy states|Boltzmann distribution}}
{{Probability distribution
| name       = Maxwell–Boltzmann distribution
| 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}{a^3} \, \exp\left(\frac{-x^2}{2a^2} \right)</math> 
(where {{math|exp}} is the [[exponential function]])
| cdf        = <math>\operatorname{erf}\left(\frac{x}{\sqrt{2} a}\right) -\sqrt{\frac{2}{\pi}} \, \frac{x}{a} \, \exp\left(\frac{-x^2}{2a^2} \right)</math>
(where {{math|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}} \approx 0.48569</math>
| kurtosis   = <math>\gamma_2=\frac{4(-96+40\pi-3\pi^2)}{(3 \pi - 8)^2} \approx 0.10816</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''', or '''Maxwell(ian) 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">{{ cite book | last = Mandl | first = Franz | title = Statistical Physics | date = 2008 | publisher = John Wiley & Sons | isbn = 978-0471915331 | edition = 2nd |series = Manchester Physics | location = Chichester }}</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>{{ cite book | last1 = Young |first1 = Hugh D. | last2 = Friedman | first2 = Roger A. | last3 = Ford | first3 = Albert Lewis | last4 = Sears | first4 = Francis Weston | last5 = Zemansky | first5 = Mark Waldo | title = Sears and Zemansky's University Physics: With Modern Physics |date = 2008 | publisher = Pearson, Addison-Wesley | isbn = 978-0-321-50130-1 | edition = 12th | location = San Francisco }}</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), [[Rita G. Lerner|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 randomly chosen 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. This is also true for ideal [[plasma (physics)|plasmas]], which are ionized gases of sufficiently low density.<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="MaxwellA">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]</ref><ref name="MaxwellB">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 ==
For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space {{math|''d''{{sup|&thinsp;3}}'''v'''}}, centered on a velocity vector <math> \mathbf{v} </math> of magnitude <math>v</math>, is given by
<math display="block"> f(\mathbf{v}) ~ d^3\mathbf{v} = \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{{3}/{2}} \, \exp\left(-\frac{mv^2}{2k_\text{B}T}\right) ~ d^3\mathbf{v}, </math>
where:
*{{mvar|m}} is the particle mass;
*{{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]];
*{{mvar|T}} is [[thermodynamic temperature]];
*<math> f(\mathbf{v}) </math> is a probability distribution function, properly normalized so that <math display="inline">\int f(\mathbf{v}) \, d^3\mathbf{v}</math> over all velocities is unity.

[[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 <math>d^3\mathbf{v} = dv_x \, dv_y \, dv_z</math>, for velocities in a standard Cartesian coordinate system, or as <math>d^3\mathbf{v} = v^2 \, dv \, d\Omega</math> in a standard spherical coordinate system, where <math>d\Omega = \sin{v_\theta} ~ dv_\phi ~ dv_\theta</math> is an element of solid angle and <math display="inline">v^2 = |\mathbf{v}|^2 = v_x^2 + v_y^2 + v_z^2</math>. 

Alternately, the distribution function can also be written in momentum space as<math display="block">f(\mathbf{p}) \, d^3\mathbf{p} = \left[\frac{1}{2\pi m k_\text{B}T}\right]^{3/2} \exp\left(-\frac{p^2}{2 m k_\text{B}T}\right) \, d^3\mathbf{p}</math>where <math>\mathbf{p} = m \mathbf{v}</math> is the momentum vector. 

The Maxwellian distribution function for particles moving in only one direction, if this direction is {{mvar|x}}, is a [[normal distribution]] with a standard deviation of <math display="inline">\sqrt{k_\text{B}T / m}</math>:
<math display="block"> f(v_x) ~dv_x = \sqrt{\frac{m}{2 \pi k_\text{B}T}} \, \exp\left(-\frac{mv_x^2}{2k_\text{B}T}\right) ~ dv_x, </math>
which can be obtained by integrating the three-dimensional form given above over {{mvar|v{{sub|y}}}} and {{mvar|v{{sub|z}}}}.

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>{{cite book | last = Müller-Kirsten | first = H. J. W. | author-link = Harald J. W. Mueller-Kirsten | url = https://www.worldcat.org/title/822895930 | title = Basics of Statistical Physics | date = 2013 | publisher = [[World Scientific]] | isbn = 978-981-4449-53-3 | edition = 2nd | oclc = 822895930 | chapter = 2 }}</ref>

<math display="block"> f(v) = \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{{3}/{2}} \, 4\pi v^2 \exp\left(-\frac{mv^2}{2k_\text{B}T}\right). </math>

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

The simplest [[ordinary differential equation]] satisfied by the distribution is:
<math display="block">\begin{align}
0 &= k_\text{B}T v f'(v) + f(v) \left(mv^2 - 2k_\text{B}T\right), \\[4pt]
f(1) &= \sqrt{\frac{2}{\pi}} \, \biggl[\frac{m}{k_\text{B} T}\biggr]^{3/2} \exp\left(-\frac{m}{2k_\text{B}T}\right);
\end{align}</math>

or in [[unitless]] presentation:
<math display="block">\begin{align}
0 &= a^2 x f'(x) + \left(x^2-2 a^2\right) f(x), \\[4pt]
f(1) &= \frac{1}{a^3} \sqrt{\frac{2}{\pi }} \exp\left(-\frac{1}{2 a^2} \right).
\end{align}</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.

[[File:Simulation of gas for relaxation demonstration.gif|thumb|471x471px|Simulation of a 2D gas relaxing towards a Maxwell–Boltzmann speed distribution]]

== Relaxation to the 2D Maxwell–Boltzmann distribution ==
For particles confined to move in a plane, the speed distribution is given by

<math display="block">P(s < |\mathbf{v}| < s {+} ds) = \frac{ms}{k_\text{B}T}\exp\left(-\frac{ms^2}{2k_\text{B}T}\right) 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&nbsp;[[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 the velocity distribution (in blue) quickly converges to the 2D Maxwell–Boltzmann distribution (in orange).

== Typical speeds ==
[[File:Stellar MB.png|alt=Solar Atmosphere Maxwell–Boltzmann Distribution.|thumb|400px|The Maxwell–Boltzmann distribution corresponding to the solar atmosphere. Particle masses are one [[proton mass]], {{math|1=''m''<sub>p</sub> = {{val|1.67|e=-27|u=kg}} ≈ {{val|1|ul=Da}}}}, and the temperature is the effective temperature of the [[Photosphere|Sun's photosphere]], {{math|1=''T'' = 5800 K}}. <math>\tilde{V}</math>, <math>\bar{V}</math>, and {{math|''V''<sub>rms</sub>}} mark the most probable, mean, and root mean square velocities, respectively. Their values are <math>\tilde{V}</math> ≈ {{val|9.79|u=km/s}}, <math>\bar{V}</math> ≈ {{val|11.05|u=km/s}}, and {{math|''V''<sub>rms</sub> ≈ {{val|12.00|u=km/s}}}}.]]
The [[expectation value|mean]] speed <math> \langle v \rangle</math>, most probable speed ([[Mode (statistics)|mode]]) {{math|''v''<sub>p</sub>}}, and root-mean-square speed <math display="inline">\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 | first1 = Raymond A. | last1 = Serway | first2 = Jerry S. | last2 = Faughn | first3 = Chris | last3 = Vuille | name-list-style=amp | year = 2011 | isbn = 9780840068484 | page = 352 | publisher = Cengage Learning | url = https://books.google.com/books?id=HLxV-IKYO5IC&pg=PA352 }}</ref>

{{bulleted list
| The most probable speed, {{math|''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 the [[mode (statistics)|mode]] of {{math|''f''(''v'')}}. To find it, we calculate the [[derivative]] {{tmath|\tfrac{df}{dv},}} set it to zero and solve for {{mvar|v}}: <math display="block">
\frac{df(v)}{dv} = -8\pi \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{3/2} \, v \, \left[\frac{mv^2}{2k_\text{B}T}-1\right] \exp\left(-\frac{mv^2}{2k_\text{B}T}\right) = 0
</math> with the solution: <math display="block">
\frac{mv_\text{p}^2}{2k_\text{B}T} = 1; \quad v_\text{p} = \sqrt{ \frac{2k_\text{B}T}{m} } = \sqrt{ \frac{2RT}{M} }
</math>where:
*{{mvar|R}} is the [[gas constant]];
*{{mvar|M}} is molar mass of the substance, and thus may be calculated as a product of particle mass, {{mvar|m}}, and [[Avogadro constant]], {{math|''N''<sub>A</sub>}}: <math>M = m N_\mathrm{A}.</math>
For diatomic nitrogen ({{chem2|N2}}, the primary component of [[air]])<ref group="note">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>\frac{3RT}{M_\text{m}}</math> is still the mean [[Translation (physics)|translational]] [[kinetic energy]]. Nitrogen being diatomic only affects the value of the molar mass {{math|1=''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 display="block">v_\text{p} \approx \sqrt{\frac{2 \cdot 8.31\, \mathrm{J {\cdot} {mol}^{-1} K^{-1}} \ 300\, \mathrm{K}}{0.028\, \mathrm{ {kg} {\cdot} {mol}^{-1} }}} \approx 422\, \mathrm{m/s}.</math>
| The mean speed is the [[expected value]] of the speed distribution, setting <math display="inline">b= \frac{1}{2a^2} = \frac{m}{2k_\text{B}T}</math>: <math display="block">\begin{align}
 \langle v \rangle
&= \int_0^{\infty} v \, f(v) \, dv \\[1ex]
&= 4\pi \left[ \frac{b}{\pi} \right]^{3/2} \int_0^\infty v^3 e^{-b v^2} dv
= 4\pi \left[ \frac{b}{\pi} \right]^{3/2} \frac{1}{2b^2} \\[1.4ex]
&= \sqrt{\frac{4}{\pi b}}
= \sqrt{ \frac{8k_\text{B}T}{\pi m}}
= \sqrt{ \frac{8RT}{\pi M}}
= \frac{2}{\sqrt{\pi}} v_\text{p}
\end{align}</math>
| 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_\text{rms}</math> is the square root of the mean square speed, corresponding to the speed of a particle with average [[kinetic energy]], setting <math display="inline">b = \frac{1}{2a^2} = \frac{m}{2k_\text{B}T}</math>: <math display="block">\begin{align}
v_\text{rms}
& = \sqrt{\langle v^2 \rangle}
= \left[\int_0^{\infty} v^2 \, f(v) \, dv \right]^{1/2} \\[1ex]
& = \left[ 4 \pi \left (\frac{b}{\pi } \right)^{3/2} \int_{0}^{\infty} v^4 e^{-bv^2} dv\right]^{1/2} \\[1ex]
& = \left[ 4 \pi \left (\frac{b}{\pi}\right )^{3/2} \frac{3}{8} \left(\frac{\pi}{b^5}\right)^{1/2} \right]^{1/2}
= \sqrt{ \frac{3}{2b} } \\[1ex]
&= \sqrt { \frac{3k_\text{B}T}{m}}
= \sqrt { \frac{3RT}{M} }
= \sqrt{ \frac{3}{2} } v_\text{p}
\end{align}</math>
}}

In summary, the typical speeds are related as follows:
<math display="block">v_\text{p} \approx 88.6\%\ \langle v \rangle < \langle v \rangle < 108.5\%\ \langle v \rangle \approx v_\text{rms}. </math>

The root mean square speed is directly related to the [[speed of sound]] {{mvar|c}} in the gas, by
<math display="block">c = \sqrt{\frac{\gamma}{3}} \ v_\mathrm{rms} = \sqrt{\frac{f+2}{3f}}\ v_\mathrm{rms} = \sqrt{\frac{f+2}{2f}}\ v_\text{p} ,</math>
where <math display="inline">\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 group="note">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 display="block">c = \sqrt{\frac{7}{15}}v_\mathrm{rms} \approx 68\%\ v_\mathrm{rms} \approx 84\%\ v_\text{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 display="block">\begin{align}
v_\text{rel} \equiv \langle |\mathbf{v}_1 - \mathbf{v}_2| \rangle
&= \int \! d^3\mathbf{v}_1 \, d^3\mathbf{v}_2 \left|\mathbf{v}_1 - \mathbf{v}_2\right| f(\mathbf{v}_1) f(\mathbf{v}_2) \\[2pt]
&= \frac{4}{\sqrt{\pi}}\sqrt{\frac{k_\text{B}T}{m}}
= \sqrt{2}\langle v \rangle 
\end{align}</math>
where the three-dimensional velocity distribution is
<math display="block"> f(\mathbf{v}) \equiv \left[\frac{2\pi k_\text{B}T}{m}\right]^{-3/2} \exp\left(-\frac{1}{2}\frac{m\mathbf{v}^2}{k_\text{B}T} \right). </math>

The integral can easily be done by changing to coordinates <math> \mathbf{u} = \mathbf{v}_1-\mathbf{v}_2 </math> and <math display="inline"> \mathbf{U} = \tfrac{1}{2}(\mathbf{v}_1 + \mathbf{v}_2).</math>

==Limitations==
The Maxwell–Boltzmann distribution assumes that the velocities of individual particles are much less than the speed of light, i.e. that <math>T \ll \frac{m c^2}{k_\text{B}}</math>. For electrons, the temperature of electrons must be <math>T_e \ll 5.93 \times 10^9~\mathrm{K}</math>. For distribution of speeds of relativistic particles, see [[Maxwell–Jüttner distribution]].

==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=MaxwellA/><ref name=MaxwellB/><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 | s2cid = 38272381 }}</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. '''Translation available at''': http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf {{Webarchive|url=https://web.archive.org/web/20210305005604/http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf |date=2021-03-05 }}</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 linear in the ratio of the energy of that state to the temperature of the system: there are constants <math>k</math> and <math>C</math> such that, for all <math>i</math>,
<math display="block">-\log \left(\frac{N_i}{N}\right) = \frac{1}{k}\cdot\frac{E_i}{T} + C.</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>{{ cite book | last = Parker | first = Sybil P. | title = McGraw-Hill Encyclopedia of Physics | date = 1993 | publisher = McGraw-Hill | isbn = 978-0-07-051400-3 | edition = 2nd}}</ref>

This relation can be written as an equation by introducing a normalizing factor:
{{NumBlk||<math display="block">
\frac{N_i} N =
\frac{
    \exp\left(-\frac{E_i}{k_\text{B}T}\right)
}{
    \displaystyle \sum_j \exp\left(-\tfrac{E_j}{k_\text{B}T}\right)
}</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,
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].
The denominator in {{EquationNote|1|equation 1}} is 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 display="block">E=\frac{p^2}{2m}</math>|{{EquationRef|2}}}}

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

{{NumBlk||<math display="block">
\frac{N_i}{N} = 
\frac{1}{Z} 
\exp \left(-\frac{p_{i, x}^2 + p_{i, y}^2 + p_{i, z}^2}{2m k_\text{B}T}\right)</math>
|{{EquationRef|3}}}}

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

{{NumBlk||<math display="block">
f_\mathbf{p} (p_x, p_y, p_z) 
\propto
\exp \left(-\frac{p_x^2 + p_y^2 + p_z^2}{2m k_\text{B}T}\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|equation 4}} over all {{mvar|p<sub>x</sub>}}, {{mvar|p<sub>y</sub>}}, and {{mvar|p<sub>z</sub>}} yields a factor of 
<math display="block">\iiint_{-\infty}^{+\infty} \exp\left(-\frac{p_x^2 + p_y^2 + p_z^2}{2m k_\text{B}T}\right) dp_x\, dp_y\, dp_z
= \Bigl[ \sqrt{\pi} \sqrt{2m k_\text{B}T} \Bigr]^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[\frac{1}{2\pi m k_\text{B}T}\right]^{3/2}
\exp\left(-\frac{p_x^2 + p_y^2 + p_z^2}{2m k_\text{B}T}\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>m k_\text{B}T</math>. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with <math display="inline">a = \sqrt{m k_\text{B}T}</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 display="block"> f_E(E) \, dE = f_p(\mathbf p) \, d^3 \mathbf p,</math>|{{EquationRef|7}}}}
where <math>d^3 \mathbf p</math> is the infinitesimal phase-space volume of momenta corresponding to the energy interval {{mvar|dE}}.
Making use of the spherical symmetry of the energy-momentum dispersion relation <math>E = \tfrac{| \mathbf p|^2}{2m},</math> this can be expressed in terms of {{mvar|dE}} as
{{NumBlk|:|<math>  d^3 \mathbf p = 4 \pi \left| \mathbf p \right|^2 \, d{\left|\mathbf p\right|} = 4 \pi m \sqrt{2mE} \ dE.</math>|{{EquationRef|8}}}}
Using then ({{EquationNote|8}}) in ({{EquationNote|7}}), and expressing everything in terms of the energy {{mvar|E}}, we get
<math display="block">\begin{align}
f_E(E) \, dE &= \left[\frac{1}{2\pi m k_\text{B}T}\right]^{3/2} \exp\left(-\frac{E}{k_\text{B}T}\right) 4 \pi m \sqrt{2mE} \ dE \\[1ex]
 &= 2 \sqrt{\frac{E}{\pi}} \, \left[\frac{1}{k_\text{B}T}\right]^{3/2} \exp\left(-\frac{E}{k_\text{B}T}\right) \, dE
\end{align}</math>
and finally

{{Equation box 1 |indent=: |equation=
<math>f_E(E) = 2 \sqrt{\frac{E}{\pi}} \, \left[\frac{1}{k_\text{B}T}\right]^{3/2} \exp\left(-\frac{E}{k_\text{B}T} \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 energy distribution can be written equivalently as a [[gamma distribution]], using a shape parameter, <math>k_\text{shape} = 3/2</math> and a scale parameter, <math>\theta_\text{scale} = k_\text{B}T.</math>

Using the [[equipartition theorem]], given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split <math>f_E(E) \, dE</math> into a set of [[chi-squared distribution]]s, where the energy per degree of freedom, {{mvar|ε}} 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 = [https://books.google.com/books?id=QF6iMewh4KMC&pg=PA434 434] | url = https://books.google.com/books?id=QF6iMewh4KMC}}</ref>
<math display="block">f_\varepsilon(\varepsilon)\,d\varepsilon
= \sqrt{\frac{1}{\pi\varepsilon k_\text{B}T}} ~ \exp\left(-\frac{\varepsilon}{k_\text{B}T}\right)\,d\varepsilon</math>

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.

===Distribution for the velocity vector===

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

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

and using {{math|1='''p''' = ''m'''''v'''}} we get

{{Equation box 1 |indent=: |equation=
<math>
f_\mathbf{v} (v_x, v_y, v_z) =
\biggl[\frac{m}{2\pi k_\text{B}T} \biggr]^{3/2}
\exp\left(-\frac{m\left(v_x^2 + v_y^2 + v_z^2\right)}{2 k_\text{B}T}\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 {{math|[''dv<sub>x</sub>'', ''dv<sub>y</sub>'', ''dv<sub>z</sub>'']}} about velocity {{math|1='''v''' = [''v<sub>x</sub>'', ''v<sub>y</sub>'', ''v<sub>z</sub>'']}} is

<math display="block">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 display="inline">k_\text{B}T / m</math>.
It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity
{{math|[''v<sub>x</sub>'', ''v<sub>y</sub>'', ''v<sub>z</sub>'']}} is the product of the distributions for each of the three directions:
<math display="block">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 display="block">
f_v (v_i) =
\sqrt{\frac{m}{2 \pi k_\text{B}T}}
\exp \left(-\frac{mv_i^2}{2k_\text{B}T}\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 display="inline">\sigma_{v_x} = \sigma_{v_y} = \sigma_{v_z} = \sqrt{k_\text{B}T / 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 covariance <math display="inline">\Sigma_{\mathbf{v}} = \left(\frac{k_\text{B}T}{m}\right)I</math>, where <math>I</math> is the {{nowrap|3 &times; 3}} identity matrix.

A notable property of the distribution for the velocity vector is direction-independence, which means that velocity components are [[normal distribution|normally distributed]] in any selected direction, not only in three base directions <math>x</math>, <math>y</math>, and <math>z</math>.<ref name="ChemTexts0111-1">{{cite journal |last1=Rapp-Kindner |first1=I.|last2=Ősz |first2=K. |last3=Lente |first3=G. |year=2025 |title=The ideal gas law: derivations and intellectual background |journal=ChemTexts |volume=11 |issue=1 |pages=1 |doi=10.1007/s40828-024-00198-9 |doi-access=free }}</ref>

===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 display="block">v = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>
and the [[volume element]] in [[spherical coordinates]]
<math display="block"> 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) =
\sqrt{\frac{2}{\pi}} \, \biggl[\frac{m}{k_\text{B}T}\biggr]^{3/2} v^2 \exp\left(-\frac{mv^2}{2k_\text{B}T}\right).
</math>}}

== In ''n''-dimensional space ==
In {{mvar|n}}-dimensional space, Maxwell–Boltzmann distribution becomes:
<math display="block"> f(\mathbf{v}) ~ d^n\mathbf{v} = \biggl[\frac{m}{2 \pi k_\text{B}T}\biggr]^{n/2} \exp\left(-\frac{m|\mathbf{v}|^2}{2k_\text{B}T}\right) ~d^n\mathbf{v} </math>

Speed distribution becomes:
<math display="block"> f(v) ~ dv = A \exp\left(-\frac{mv^2}{2k_\text{B} T}\right)  v^{n-1} ~ dv </math>
where <math> A </math> is a normalizing constant.

The following integral result is useful:
<math display="block">\begin{align}
\int_{0}^{\infty} v^a \exp\left(-\frac{mv^2}{2k_\text{B} T}\right) dv  
&= \left[\frac{2k_\text{B} T}{m}\right]^\frac{a+1}{2} \int_{0}^{\infty} e^{-x}x^{a/2} \, dx^{1/2} \\[2pt]
&= \left[\frac{2k_\text{B} T}{m}\right]^\frac{a+1}{2} \int_{0}^{\infty} e^{-x}x^{a/2}\frac{x^{-1/2}}{2} \, dx \\[2pt]
&= \left[\frac{2k_\text{B} T}{m}\right]^\frac{a+1}{2} \frac{\Gamma{\left(\frac{a+1}{2}\right)}}{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 display="block"> \langle v \rangle 
= \frac
{\displaystyle \int_{0}^{\infty} v \cdot v^{n-1} \exp\left(-\tfrac{mv^2}{2k_\text{B} T}\right) \, dv}
{\displaystyle \int_{0}^{\infty} v^{n-1} \exp\left(-\tfrac{mv^2}{2k_\text{B} T}\right) \, dv}
= \sqrt{\frac{2k_\text{B} T}{m}} ~~ \frac{\Gamma{\left(\frac{n+1}{2}\right)}}{\Gamma{\left(\frac{n}{2}\right)}}</math>
which is the [[expectation value|mean]] speed itself <math display="inline">v_\mathrm{avg} = \langle v \rangle = \sqrt{\frac{2k_\text{B} T}{m}} \  \frac{\Gamma \left(\frac{n+1}{2}\right)}{\Gamma \left(\frac{n}{2}\right)}.</math>

<math display="block"> \begin{align}
\langle v^2 \rangle 
&= \frac
{\displaystyle\int_{0}^{\infty} v^2 \cdot v^{n-1} \exp\left(-\tfrac{mv^2}{2k_\text{B} T}\right) \, dv}
{\displaystyle\int_{0}^{\infty} v^{n-1} \exp\left(-\tfrac{mv^2}{2k_\text{B}T}\right) \, dv} \\[1ex]
&= \left[\frac{2k_\text{B}T}{m}\right] \frac{\Gamma {\left(\frac{n+2}{2}\right)}}{\Gamma {\left(\frac{n}{2}\right)}} \\[1.2ex]
&= \left[\frac{2k_\text{B}T}{m}\right] \frac{n}{2} = \frac{n k_\text{B}T}{m}
\end{align}</math>
which gives root-mean-square speed  <math display="inline">v_\text{rms} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{n k_\text{B}T}{m}}.</math>

The derivative of speed distribution function:
<math display="block">\frac{df(v)}{dv} = A \exp\left(-\frac{mv^2}{2k_\text{B}T}\right) \biggl[-\frac{mv}{k_\text{B}T} v^{n-1}+(n-1)v^{n-2}\biggr] = 0 </math>

This yields the most probable speed ([[Mode (statistics)|mode]]) <math display="inline">v_\text{p} = \sqrt{\left(n-1\right) k_\text{B}T/m}.</math>

==Extension to real gases==

The derivations show that the validity of the Maxwell–Boltzmann velocity distribution is limited to ideal gases. A generalization of the formula to all gases (ideal and real alike) is known, its derivation starts from the fact that the properties of both ideal and real gases must be independent of the direction. The formula obtained contains <math>pV_\text{m}</math> terms instead of <math>RT</math>, where <math>p</math> is the pressure, <math>V_\text{m}</math> is the molar volume of the gas sample: <ref name="JMathChem631742">{{cite journal |last=Lente |first=G. |year=2025 |title=Direction independence as a key property to derive a particle speed distribution in real gases |journal=Journal of Mathematical Chemistry |volume=63 |doi=10.1007/s10910-025-01742-9 |doi-access=free }}</ref>

<math display="block"> f(v) = \biggl[\frac{M}{2 \pi pV_\text{m}}\biggr]^{{3}/{2}} \, 4\pi v^2 \exp\left(-\frac{Mv^2}{2pV_\text{m}}\right). </math>

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

== Notes ==
{{reflist|group=note}}

==References==

{{reflist}}

==Further reading==

* {{ cite book | last1 = Tipler | first1 = Paul Allen | last2 = Mosca | first2 = Gene | title = Physics for Scientists and Engineers: with Modern Physics | date = 2008 | publisher = W.H. Freeman | isbn = 978-0-7167-8964-2 | edition = 6th |location = New York }}
* {{ cite book | title = Thermodynamics: From Concepts to Applications | last1 = Shavit | first1 = Arthur | last2 = Gutfinger | first2 = Chaim | url = https://www.worldcat.org/title/244177312 | date = 2009 | publisher = CRC Press | isbn = 978-1-4200-7368-3 | edition = 2nd | oclc = 244177312}}
* {{ cite book | last = Ives | first = David J. G. | title = Chemical Thermodynamics | date = 1971 | publisher = Macdonald Technical and Scientific |isbn = 0-356-03736-3 | series = University Chemistry }}
* {{ cite book | last = Nash | first = Leonard K. | title = Elements of Statistical Thermodynamics | date = 1974 | publisher = Addison-Wesley | isbn = 978-0-201-05229-9 | edition = 2nd | series = Principles of Chemistry }}
* {{ cite journal | last1 = Ward | first1 = C. A. | last2 = Fang | first2 = G. | year = 1999 | title = Expression for predicting liquid evaporation flux: Statistical rate theory approach | journal = Physical Review E | volume = 59 | issue = 1 | pages = 429–440 | doi = 10.1103/physreve.59.429 | issn = 1063-651X}}
* {{cite journal | last1 = Rahimi | first1 = P | last2 = Ward | first2 = C.A. | year = 2005 | title = Kinetics of Evaporation: Statistical Rate Theory Approach | journal = International Journal of Thermodynamics | volume = 8 | issue = 9| pages = 1–14}}

==External links==
{{Commonscat|Maxwell–Boltzmann distributions}}
* [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]]