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Maxwell–Boltzmann distribution

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Also known as Maxwell distribution

specific probability distribution function, important in physics

Key facts

Parameters
0"}}'> a > 0 {\displaystyle a>0}
Support
x ∈ ( 0 ; ∞ ) {\displaystyle x\in (0;\infty )}
Pdf
2 π x 2 a 3 exp ⁡ ( − x 2 2 a 2 ) {\displaystyle {\sqrt {\frac {2}{\pi }}}\,{\frac {x^{2}}{a^{3}}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} (where exp is the exponential function )
Cdf
erf ⁡ ( x 2 a ) − 2 π x a exp ⁡ ( − x 2 2 a 2 ) {\displaystyle \operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}\,{\frac {x}{a}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)} (where erf is the error function )
Mean
μ = 2 a 2 π {\displaystyle \mu =2a{\sqrt {\frac {2}{\pi }}}}
Mode
2 a {\displaystyle {\sqrt {2}}a}
Variance
σ 2 = a 2 ( 3 π − 8 ) π {\displaystyle \sigma ^{2}={\frac {a^{2}(3\pi -8)}{\pi }}}
Skewness
γ 1 = 2 2 ( 16 − 5 π ) ( 3 π − 8 ) 3 / 2 ≈ 0.48569 {\displaystyle \gamma _{1}={\frac {2{\sqrt {2}}(16-5\pi )}{(3\pi -8)^{3/2}}}\approx 0.48569}
Excess kurtosis
γ 2 = 4 ( − 96 + 40 π − 3 π 2 ) ( 3 π − 8 ) 2 ≈ 0.10816 {\displaystyle \gamma _{2}={\frac {4(-96+40\pi -3\pi ^{2})}{(3\pi -8)^{2}}}\approx 0.10816}
Entropy
ln ⁡ ( a 2 π ) + γ − 1 2 {\displaystyle \ln \left(a{\sqrt {2\pi }}\right)+\gamma -{\frac {1}{2}}}

via Wikipedia infobox

Wikidata facts

Named after
Ludwig Boltzmann
Image
Maxwell-Boltzmann distribution 1.png
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Commons category
Maxwell–Boltzmann distributions
maintained by WikiProject
WikiProject Mathematics
Sources (2)

via Wikidata · CC0

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Encyclopedic overview

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 speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions 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. 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.

Excerpted from Wikipedia’s “Maxwell–Boltzmann distribution” article, available under the CC BY-SA 4.0 licence.