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Maxwell–Boltzmann distribution

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(Redirected fromRoot-mean-square speed)

Not to be confused withMaxwell–Boltzmann statistics.

Specific probability distribution function, important in physics

This article is about particle energy levels and velocities. For system energy states, seeBoltzmann distribution.

Maxwell–Boltzmann distribution
Probability density function
Cumulative distribution function
Parameters	$a>0$
Support	$x\in (0;\infty )$
PDF	${\sqrt {\frac {2}{\pi }}}\,{\frac {x^{2}}{a^{3}}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)$ (whereexp is theexponential function)
CDF	$\operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}\,{\frac {x}{a}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)$ (whereerf is theerror function)
Mean	$\mu =2a{\sqrt {\frac {2}{\pi }}}$
Mode	${\sqrt {2}}a$
Variance	$\sigma ^{2}={\frac {a^{2}(3\pi -8)}{\pi }}$
Skewness	$\gamma _{1}={\frac {2{\sqrt {2}}(16-5\pi )}{(3\pi -8)^{3/2}}}$
Excess kurtosis	$\gamma _{2}={\frac {4(-96+40\pi -3\pi ^{2})}{(3\pi -8)^{2}}}$
Entropy	$\ln \left(a{\sqrt {2\pi }}\right)+\gamma -{\frac {1}{2}}$

Inphysics (in particular instatistical mechanics), theMaxwell–Boltzmann distribution, orMaxwell(ian) distribution, is a particularprobability distribution named afterJames Clerk Maxwell andLudwig Boltzmann.

It was first defined and used for describing particlespeeds inidealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very briefcollisions 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 ormolecules), and the system of particles is assumed to have reachedthermodynamic equilibrium.^[1] The energies of such particles follow what is known asMaxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies withkinetic energy.

Mathematically, the Maxwell–Boltzmann distribution is thechi distribution with threedegrees of freedom (the components of thevelocity vector inEuclidean space), with ascale parameter measuring speeds in units proportional to the square root of $T/m$ (the ratio of temperature and particle mass).^[2]

The Maxwell–Boltzmann distribution is a result of thekinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, includingpressure anddiffusion.^[3] The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (themagnitude 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 classicalideal gas, which is an idealization of real gases. In real gases, there are various effects (e.g.,van der Waals interactions,vortical flow,relativistic speed limits, and quantumexchange interactions) that can make their speed distribution different from the Maxwell–Boltzmann form. However,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 idealplasmas, which are ionized gases of sufficiently low density.^[4]

The distribution was first derived by Maxwell in 1860 onheuristic grounds.^[5]^[6] 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 in the phase space, with the constraint ofconservation of average energy $\langle H\rangle =E;$
Canonical ensemble.

Distribution function

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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 spaced³v, centered on a velocity vector $\mathbf {v}$ of magnitude $v {\displaystyle v}$ , is given by $f(\mathbf {v} )~d^{3}\mathbf {v} ={\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{{3}/{2}}\,\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)~d^{3}\mathbf {v} ,$ where:

m is the particle mass;
k_B is theBoltzmann constant;
T isthermodynamic temperature;
$f(\mathbf {v} )$ is a probability distribution function, properly normalized so that ${\textstyle \int f(\mathbf {v} )\,d^{3}\mathbf {v} }$ over all velocities is unity.

The speed probability density functions of the speeds of a fewnoble gases at a temperature of 298.15 K (25 °C). They-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 $d^{3}\mathbf {v} =dv_{x}\,dv_{y}\,dv_{z}$ , for velocities in a standard Cartesian coordinate system, or as $d^{3}\mathbf {v} =v^{2}\,dv\,d\Omega$ in a standard spherical coordinate system, where $d\Omega =\sin {v_{\theta }}~dv_{\phi }~dv_{\theta }$ is an element of solid angle and ${\textstyle v^{2}=|\mathbf {v} |^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}$ .

The Maxwellian distribution function for particles moving in only one direction, if this direction isx, is $f(v_{x})~dv_{x}={\sqrt {\frac {m}{2\pi k_{\text{B}}T}}}\,\exp \left(-{\frac {mv_{x}^{2}}{2k_{\text{B}}T}}\right)~dv_{x},$ which can be obtained by integrating the three-dimensional form given above overv_y andv_z.

Recognizing the symmetry of $f(v)$ , one can integrate over solid angle and write a probability distribution of speeds as the function^[7]

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

Thisprobability density function gives the probability, per unit speed, of finding the particle with a speed nearv. This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter ${\textstyle a={\sqrt {k_{\text{B}}T/m}}\,.}$ The Maxwell–Boltzmann distribution is equivalent to thechi distribution with three degrees of freedom andscale parameter ${\textstyle a={\sqrt {k_{\text{B}}T/m}}\,.}$

The simplestordinary differential equation satisfied by the distribution is: ${\begin{aligned}0&=k_{\text{B}}Tvf'(v)+f(v)\left(mv^{2}-2k_{\text{B}}T\right),\\[4pt]f(1)&={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{k_{\text{B}}T}}{\biggr ]}^{3/2}\exp \left(-{\frac {m}{2k_{\text{B}}T}}\right);\end{aligned}}$

or inunitless presentation: ${\begin{aligned}0&=a^{2}xf'(x)+\left(x^{2}-2a^{2}\right)f(x),\\[4pt]f(1)&={\frac {1}{a^{3}}}{\sqrt {\frac {2}{\pi }}}\exp \left(-{\frac {1}{2a^{2}}}\right).\end{aligned}}$ With theDarwin–Fowler method of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result.

Simulation of a 2D gas relaxing towards a Maxwell–Boltzmann speed distribution

Relaxation to the 2D Maxwell–Boltzmann distribution

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For particles confined to move in a plane, the speed distribution is given by

$P(s<|\mathbf {v} |<s{+}ds)={\frac {ms}{k_{\text{B}}T}}\exp \left(-{\frac {ms^{2}}{2k_{\text{B}}T}}\right)ds$

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 theBoltzmann equation. The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is amolecular dynamics (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle. They interact viaperfectly 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

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Solar Atmosphere Maxwell–Boltzmann Distribution. — The Maxwell–Boltzmann distribution corresponding to the solar atmosphere. Particle masses are oneproton mass,m_p =1.67×10⁻²⁷ kg ≈1 Da, and the temperature is the effective temperature of theSun's photosphere,T = 5800 K. ${\tilde {V}}$ , ${\bar {V}}$ , andV_rms mark the most probable, mean, and root mean square velocities, respectively. Their values are ${\tilde {V}}$ ≈9.79 km/s, ${\bar {V}}$ ≈11.05 km/s, andV_rms ≈12.00 km/s.

{\displaystyle {\tilde {V}}} — The Maxwell–Boltzmann distribution corresponding to the solar atmosphere. Particle masses are oneproton mass,m_p =1.67×10⁻²⁷ kg ≈1 Da, and the temperature is the effective temperature of theSun's photosphere,T = 5800 K. ${\tilde {V}}$ , ${\bar {V}}$ , andV_rms mark the most probable, mean, and root mean square velocities, respectively. Their values are ${\tilde {V}}$ ≈9.79 km/s, ${\bar {V}}$ ≈11.05 km/s, andV_rms ≈12.00 km/s.

Themean speed $\langle v\rangle$ , most probable speed (mode)v_p, and root-mean-square speed ${\textstyle {\sqrt {\langle v^{2}\rangle }}}$ can be obtained from properties of the Maxwell distribution.

This works well for nearlyideal,monatomic gases likehelium, but also formolecular gases like diatomicoxygen. This is because despite the largerheat capacity (larger internal energy at the same temperature) due to their larger number ofdegrees of freedom, theirtranslational kinetic energy (and thus their speed) is unchanged.^[8]

The most probable speed,v_p, is the speed most likely to be possessed by any molecule (of the same massm) in the system and corresponds to the maximum value or themode off(v). To find it, we calculate thederivative⁠ ${\tfrac {df}{dv}},$ ⁠ set it to zero and solve forv: ${\frac {df(v)}{dv}}=-8\pi {\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{3/2}\,v\,\left[{\frac {mv^{2}}{2k_{\text{B}}T}}-1\right]\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)=0$ with the solution: ${\frac {mv_{\text{p}}^{2}}{2k_{\text{B}}T}}=1;\quad v_{\text{p}}={\sqrt {\frac {2k_{\text{B}}T}{m}}}={\sqrt {\frac {2RT}{M}}}$ where:
- R is thegas constant;
- M is molar mass of the substance, and thus may be calculated as a product of particle mass,m, andAvogadro constant,N_A: $M=mN_{\mathrm {A} }.$
For diatomic nitrogen (N₂, the primary component ofair)^{[note 1]} atroom temperature (300 K), this gives
$v_{\text{p}}\approx {\sqrt {\frac {2\cdot 8.31\,\mathrm {J{\cdot }{mol}^{-1}K^{-1}} \ 300\,\mathrm {K} }{0.028\,\mathrm {{kg}{\cdot }{mol}^{-1}} }}}\approx 422\,\mathrm {m/s} .$
The mean speed is theexpected value of the speed distribution, setting ${\textstyle b={\frac {1}{2a^{2}}}={\frac {m}{2k_{\text{B}}T}}}$ : ${\begin{aligned}\langle v\rangle &=\int _{0}^{\infty }v\,f(v)\,dv\\[1ex]&=4\pi \left[{\frac {b}{\pi }}\right]^{3/2}\int _{0}^{\infty }v^{3}e^{-bv^{2}}dv=4\pi \left[{\frac {b}{\pi }}\right]^{3/2}{\frac {1}{2b^{2}}}\\[1.4ex]&={\sqrt {\frac {4}{\pi b}}}={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}={\sqrt {\frac {8RT}{\pi M}}}={\frac {2}{\sqrt {\pi }}}v_{\text{p}}\end{aligned}}$
The mean square speed $\langle v^{2}\rangle$ is the second-orderraw moment of the speed distribution. The "root mean square speed" $v_{\text{rms}}$ is the square root of the mean square speed, corresponding to the speed of a particle with averagekinetic energy, setting ${\textstyle b={\frac {1}{2a^{2}}}={\frac {m}{2k_{\text{B}}T}}}$ : ${\begin{aligned}v_{\text{rms}}&={\sqrt {\langle v^{2}\rangle }}=\left[\int _{0}^{\infty }v^{2}\,f(v)\,dv\right]^{1/2}\\[1ex]&=\left[4\pi \left({\frac {b}{\pi }}\right)^{3/2}\int _{0}^{\infty }v^{4}e^{-bv^{2}}dv\right]^{1/2}\\[1ex]&=\left[4\pi \left({\frac {b}{\pi }}\right)^{3/2}{\frac {3}{8}}\left({\frac {\pi }{b^{5}}}\right)^{1/2}\right]^{1/2}={\sqrt {\frac {3}{2b}}}\\[1ex]&={\sqrt {\frac {3k_{\text{B}}T}{m}}}={\sqrt {\frac {3RT}{M}}}={\sqrt {\frac {3}{2}}}v_{\text{p}}\end{aligned}}$

In summary, the typical speeds are related as follows: $v_{\text{p}}\approx 88.6\%\ \langle v\rangle <\langle v\rangle <108.5\%\ \langle v\rangle \approx v_{\text{rms}}.$

The root mean square speed is directly related to thespeed of soundc in the gas, by $c={\sqrt {\frac {\gamma }{3}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{3f}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{2f}}}\ v_{\text{p}},$ where ${\textstyle \gamma =1+{\frac {2}{f}}}$ is theadiabatic index,f is the number ofdegrees of freedom of the individual gas molecule. For the example above, diatomic nitrogen (approximatingair) at300 K, $f=5$ ^{[note 2]} and $c={\sqrt {\frac {7}{15}}}v_{\mathrm {rms} }\approx 68\%\ v_{\mathrm {rms} }\approx 84\%\ v_{\text{p}}\approx 353\ \mathrm {m/s} ,$ the true value for air can be approximated by using the average molar weight ofair (29 g/mol), yielding347 m/s at300 K (corrections for variablehumidity are of the order of 0.1% to 0.6%).

The average relative velocity ${\begin{aligned}v_{\text{rel}}\equiv \langle |\mathbf {v} _{1}-\mathbf {v} _{2}|\rangle &=\int \!d^{3}\mathbf {v} _{1}\,d^{3}\mathbf {v} _{2}\left|\mathbf {v} _{1}-\mathbf {v} _{2}\right|f(\mathbf {v} _{1})f(\mathbf {v} _{2})\\[2pt]&={\frac {4}{\sqrt {\pi }}}{\sqrt {\frac {k_{\text{B}}T}{m}}}={\sqrt {2}}\langle v\rangle \end{aligned}}$ where the three-dimensional velocity distribution is $f(\mathbf {v} )\equiv \left[{\frac {2\pi k_{\text{B}}T}{m}}\right]^{-3/2}\exp \left(-{\frac {1}{2}}{\frac {m\mathbf {v} ^{2}}{k_{\text{B}}T}}\right).$

The integral can easily be done by changing to coordinates $\mathbf {u} =\mathbf {v} _{1}-\mathbf {v} _{2}$ and ${\textstyle \mathbf {U} ={\tfrac {1}{2}}(\mathbf {v} _{1}+\mathbf {v} _{2}).}$

Limitations

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The Maxwell–Boltzmann distribution assumes that the velocities of individual particles are much less than the speed of light, i.e. that $T\ll {\frac {mc^{2}}{k_{\text{B}}}}$ . For electrons, the temperature of electrons must be $T_{e}\ll 5.93\times 10^{9}~\mathrm {K}$ .

Derivation and related distributions

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Maxwell–Boltzmann statistics

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Main articles:Maxwell–Boltzmann statistics § Derivations, andBoltzmann distribution

The original derivation in 1860 byJames Clerk Maxwell was an argument based on molecular collisions of theKinetic 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.^[5]^[6]^[9] After Maxwell,Ludwig Boltzmann in 1872^[10] also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (seeH-theorem). He later (1877)^[11] derived the distribution again under the framework ofstatistical thermodynamics. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known asMaxwell–Boltzmann statistics (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particlemicrostate. 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 $k {\displaystyle k}$ and $C {\displaystyle C}$ such that, for all $i {\displaystyle i}$ , $-\log \left({\frac {N_{i}}{N}}\right)={\frac {1}{k}}\cdot {\frac {E_{i}}{T}}+C.$ 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.^[1]^[12]

This relation can be written as an equation by introducing a normalizing factor:

{\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)}}

where:

N_i is the expected number of particles in the single-particle microstatei,
N is the total number of particles in the system,
E_i is the energy of microstatei,
the sum over indexj takes into account all microstates,
T is the equilibrium temperature of the system,
k_B is theBoltzmann constant.

The denominator inequation 1 is a normalizing factor so that the ratios $N_{i}:N$ add up to unity — in other words it is a kind ofpartition function (for the single-particle system, not the usual partition function of the entire system).

Because velocity and speed are related to energy, Equation (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

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The potential energy is taken to be zero, so that all energy is in the form of kinetic energy.The relationship betweenkinetic energy and momentum for massive non-relativistic particles is

E={\frac {p^{2}}{2m}}

wherep² is the square of the momentum vectorp = [p_x,p_y,p_z]. We may therefore rewrite Equation (1) as:

{\frac {N_{i}}{N}}={\frac {1}{Z}}\exp \left(-{\frac {p_{i,x}^{2}+p_{i,y}^{2}+p_{i,z}^{2}}{2mk_{\text{B}}T}}\right)

where:

Z is thepartition function, corresponding to the denominator inequation 1;
m is the molecular mass of the gas;
T is the thermodynamic temperature;
k_B is theBoltzmann constant.

This distribution ofN_i :N isproportional to theprobability density functionf_p for finding a molecule with these values of momentum components, so:

f_{\mathbf {p} }(p_{x},p_{y},p_{z})\propto \exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mk_{\text{B}}T}}\right)

Thenormalizing constant can be determined by recognizing that the probability of a molecule havingsome momentum must be 1.Integrating the exponential inequation 4 over allp_x,p_y, andp_z yields a factor of $\iiint _{-\infty }^{+\infty }\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mk_{\text{B}}T}}\right)dp_{x}\,dp_{y}\,dp_{z}={\Bigl [}{\sqrt {\pi }}{\sqrt {2mk_{\text{B}}T}}{\Bigr ]}^{3}$

So that the normalized distribution function is:

$f_{\mathbf {p} }(p_{x},p_{y},p_{z})=\left[{\frac {1}{2\pi mk_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mk_{\text{B}}T}}\right)$ (6)

The distribution is seen to be the product of three independentnormally distributed variables $p_{x}$ , $p_{y}$ , and $p_{z}$ , with variance $mk_{\text{B}}T$ . Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with ${\textstyle a={\sqrt {mk_{\text{B}}T}}}$ . The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using theH-theorem at equilibrium within theKinetic theory of gases framework.

Distribution for the energy

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The energy distribution is found imposing

f_{E}(E)\,dE=f_{p}(\mathbf {p} )\,d^{3}\mathbf {p} ,

where $d^{3}\mathbf {p}$ is the infinitesimal phase-space volume of momenta corresponding to the energy intervaldE.Making use of the spherical symmetry of the energy-momentum dispersion relation $E={\tfrac {|\mathbf {p} |^{2}}{2m}},$ this can be expressed in terms ofdE as

d^{3}\mathbf {p} =4\pi |\mathbf {p} |^{2}d|\mathbf {p} |=4\pi m{\sqrt {2mE}}\ dE.

Using then (8) in (7), and expressing everything in terms of the energyE, we get ${\begin{aligned}f_{E}(E)dE&=\left[{\frac {1}{2\pi mk_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)4\pi m{\sqrt {2mE}}\ dE\\[1ex]&=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{k_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)\,dE\end{aligned}}$ and finally

$f_{E}(E)=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{k_{\text{B}}T}}\right]^{3/2}\exp \left(-{\frac {E}{k_{\text{B}}T}}\right)$ (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 agamma distribution, using a shape parameter, $k_{\text{shape}}=3/2$ and a scale parameter, $\theta _{\text{scale}}=k_{\text{B}}T.$

Using theequipartition theorem, given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split $f_{E}(E)dE$ into a set ofchi-squared distributions, where the energy per degree of freedom,ε is distributed as a chi-squared distribution with one degree of freedom,^[13] $f_{\varepsilon }(\varepsilon )\,d\varepsilon ={\sqrt {\frac {1}{\pi \varepsilon k_{\text{B}}T}}}~\exp \left(-{\frac {\varepsilon }{k_{\text{B}}T}}\right)\,d\varepsilon$

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 thespecific heat of a gas.

Distribution for the velocity vector

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Recognizing that the velocity probability densityf_v is proportional to the momentum probability density function by

$f_{\mathbf {v} }d^{3}\mathbf {v} =f_{\mathbf {p} }\left({\frac {dp}{dv}}\right)^{3}d^{3}\mathbf {v}$

and usingp =mv we get

$f_{\mathbf {v} }(v_{x},v_{y},v_{z})={\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{3/2}\exp \left(-{\frac {m\left(v_{x}^{2}+v_{y}^{2}+v_{z}^{2}\right)}{2k_{\text{B}}T}}\right)$

which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element[dv_x,dv_y,dv_z] about velocityv = [v_x,v_y,v_z] is

$f_{\mathbf {v} }{\left(v_{x},v_{y},v_{z}\right)}\,dv_{x}\,dv_{y}\,dv_{z}.$

Like the momentum, this distribution is seen to be the product of three independentnormally distributed variables $v_{x}$ , $v_{y}$ , and $v_{z}$ , but with variance ${\textstyle k_{\text{B}}T/m}$ .It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity[v_x,v_y,v_z] is the product of the distributions for each of the three directions: $f_{\mathbf {v} }{\left(v_{x},v_{y},v_{z}\right)}=f_{v}(v_{x})f_{v}(v_{y})f_{v}(v_{z})$ where the distribution for a single direction is $f_{v}(v_{i})={\sqrt {\frac {m}{2\pi k_{\text{B}}T}}}\exp \left(-{\frac {mv_{i}^{2}}{2k_{\text{B}}T}}\right).$

Each component of the velocity vector has anormal distribution with mean $\mu _{v_{x}}=\mu _{v_{y}}=\mu _{v_{z}}=0$ and standard deviation ${\textstyle \sigma _{v_{x}}=\sigma _{v_{y}}=\sigma _{v_{z}}={\sqrt {k_{\text{B}}T/m}}}$ , so the vector has a 3-dimensional normal distribution, a particular kind ofmultivariate normal distribution, with mean $\mu _{\mathbf {v} }=\mathbf {0}$ and covariance ${\textstyle \Sigma _{\mathbf {v} }=\left({\frac {k_{\text{B}}T}{m}}\right)I}$ , where $I {\displaystyle I}$ is the3 × 3 identity matrix.

Distribution for the speed

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The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is $v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}$ and thevolume element inspherical coordinates $dv_{x}\,dv_{y}\,dv_{z}=v^{2}\sin \theta \,dv\,d\theta \,d\phi =v^{2}\,dv\,d\Omega$ where $\phi$ and $\theta$ are thespherical coordinate angles of the velocity vector.Integration of the probability density function of the velocity over the solid angles $d\Omega$ yields an additional factor of $4\pi$ .The speed distribution with substitution of the speed for the sum of the squares of the vector components:

$f(v)={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{k_{\text{B}}T}}{\biggr ]}^{3/2}v^{2}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right).$

Inn-dimensional space

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Inn-dimensional space, Maxwell–Boltzmann distribution becomes: $f(\mathbf {v} )~d^{n}\mathbf {v} ={\biggl [}{\frac {m}{2\pi k_{\text{B}}T}}{\biggr ]}^{n/2}\exp \left(-{\frac {m|\mathbf {v} |^{2}}{2k_{\text{B}}T}}\right)~d^{n}\mathbf {v}$

Speed distribution becomes: $f(v)~dv=A\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)v^{n-1}~dv$ where $A {\displaystyle A}$ is a normalizing constant.

The following integral result is useful: ${\begin{aligned}\int _{0}^{\infty }v^{a}\exp \left(-{\frac {mv^{2}}{2k_{\text{B}}T}}\right)dv&=\left[{\frac {2k_{\text{B}}T}{m}}\right]^{\frac {a+1}{2}}\int _{0}^{\infty }e^{-x}x^{a/2}\,dx^{1/2}\\[2pt]&=\left[{\frac {2k_{\text{B}}T}{m}}\right]^{\frac {a+1}{2}}\int _{0}^{\infty }e^{-x}x^{a/2}{\frac {x^{-1/2}}{2}}\,dx\\[2pt]&=\left[{\frac {2k_{\text{B}}T}{m}}\right]^{\frac {a+1}{2}}{\frac {\Gamma {\left({\frac {a+1}{2}}\right)}}{2}}\end{aligned}}$ where $\Gamma (z)$ is theGamma function. This result can be used to calculate themoments of speed distribution function: $\langle v\rangle ={\frac {\displaystyle \int _{0}^{\infty }v\cdot v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2k_{\text{B}}T}}\right)\,dv}{\displaystyle \int _{0}^{\infty }v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2k_{\text{B}}T}}\right)\,dv}}={\sqrt {\frac {2k_{\text{B}}T}{m}}}~~{\frac {\Gamma {\left({\frac {n+1}{2}}\right)}}{\Gamma {\left({\frac {n}{2}}\right)}}}$ which is themean speed itself ${\textstyle v_{\mathrm {avg} }=\langle v\rangle ={\sqrt {\frac {2k_{\text{B}}T}{m}}}\ {\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\Gamma \left({\frac {n}{2}}\right)}}.}$

${\begin{aligned}\langle v^{2}\rangle &={\frac {\displaystyle \int _{0}^{\infty }v^{2}\cdot v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2k_{\text{B}}T}}\right)\,dv}{\displaystyle \int _{0}^{\infty }v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2k_{\text{B}}T}}\right)\,dv}}\\[1ex]&=\left[{\frac {2k_{\text{B}}T}{m}}\right]{\frac {\Gamma {\left({\frac {n+2}{2}}\right)}}{\Gamma {\left({\frac {n}{2}}\right)}}}\\[1.2ex]&=\left[{\frac {2k_{\text{B}}T}{m}}\right]{\frac {n}{2}}={\frac {nk_{\text{B}}T}{m}}\end{aligned}}$ which gives root-mean-square speed ${\textstyle v_{\text{rms}}={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {nk_{\text{B}}T}{m}}}.}$

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

This yields the most probable speed (mode) ${\textstyle v_{\text{p}}={\sqrt {\left(n-1\right)k_{\text{B}}T/m}}.}$

Notes

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^The calculation is unaffected by the nitrogen being diatomic. Despite the largerheat capacity (larger internal energy at the same temperature) of diatomic gases relative to monatomic gases, due to their larger number ofdegrees of freedom, ${\frac {3RT}{M_{\text{m}}}}$ is still the meantranslational kinetic energy. Nitrogen being diatomic only affects the value of the molar massM =28 g/mol.See e.g. K. Prakashan,Engineering Physics (2001),2.278.
^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.

References

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^^a ^bMandl, Franz (2008).Statistical Physics. Manchester Physics (2nd ed.). Chichester: John Wiley & Sons.ISBN 978-0471915331.
^Young, Hugh D.; Friedman, Roger A.; Ford, Albert Lewis; Sears, Francis Weston; Zemansky, Mark Waldo (2008).Sears and Zemansky's University Physics: With Modern Physics (12th ed.). San Francisco: Pearson, Addison-Wesley.ISBN 978-0-321-50130-1.
^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.)
^N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press, Inc., 1986, among many other texts on basic plasma physics
^^a ^bMaxwell, 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.[1]
^^a ^bMaxwell, 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.[2]
^Müller-Kirsten, H. J. W. (2013). "2".Basics of Statistical Physics (2nd ed.).World Scientific.ISBN 978-981-4449-53-3.OCLC 822895930.
^Serway, Raymond A.; Faughn, Jerry S. & Vuille, Chris (2011).College Physics, Volume 1 (9th ed.). Cengage Learning. p. 352.ISBN 9780840068484.
^Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium".Studies in History and Philosophy of Modern Physics.57:53–65.arXiv:1702.01411.Bibcode:2017SHPMP..57...53G.doi:10.1016/j.shpsb.2017.01.001.S2CID 38272381.
^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.
^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 inWissenschaftliche Abhandlungen, Vol. II, pp. 164–223, Leipzig: Barth, 1909.Translation available at:http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf Archived 2021-03-05 at theWayback Machine
^Parker, Sybil P. (1993).McGraw-Hill Encyclopedia of Physics (2nd ed.). McGraw-Hill.ISBN 978-0-07-051400-3.
^Laurendeau, Normand M. (2005).Statistical Thermodynamics: Fundamentals and Applications. Cambridge University Press. p. 434.ISBN 0-521-84635-8.

External links

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Wikimedia Commons has media related toMaxwell–Boltzmann distributions.

"The Maxwell Speed Distribution" from The Wolfram Demonstrations Project atMathworld

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling'sT-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher'sz Kaniadakis κ-Gaussian Gaussianq Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson'sS_U Landau Laplace Asymmetric Logistic Noncentralt Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student'st Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakisκ-exponential Kaniadakisκ-Gamma Kaniadakisκ-Weibull Kaniadakisκ-Logistic Kaniadakisκ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda