Maxwell Speed Distribution

Averages

There are three candidates for what is called the "average" value of the speed of the Maxwell speed distribution.

Firstly, by finding the maximum of the MSD (by differentiating, setting the derivative equal to zero and solving for the speed), we can determine the most probable speed. Calling this v_mp, we find that:

Second, we can find the mean value of v from the MSD. Calling this :

Third and finally, we can find the root mean square of the speed by finding the expected value of v2. (Alternatively, and much simpler, we can solve it by using the equipartition theorem.) Calling this v_rms:

Notice that

These are three different ways of defining the average velocity, and they are not numerically the same. It is important to be clear about which quantity is being used.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g.

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Bose–Einstein
Burr
chi-squared
chi
Coxian
Dagum
Davis
Erlang
exponential
F
Fermi–Dirac
folded normal
Fréchet
Gamma
generalized inverse Gaussian
half-logistic
half-normal
Hotelling's T-squared
hyper-exponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell speed
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete: Ewens; multinomial; Dirichlet-multinomial; negative multinomial

Continuous: Dirichlet; Generalized Dirichlet; multivariate normal; Multivariate stable; multivariate Student; normal-scaled inverse gamma; normal-gamma

Matrix-valued: inverse matrix gamma; inverse-Wishart; matrix normal; matrix t; matrix gamma; normal-inverse-Wishart; normal-Wishart; Wishart

Directional

Univariate (circular) directional: Circular uniform; univariate von Mises; wrapped normal; wrapped Cauchy; wrapped exponential; wrapped Lévy

Bivariate (spherical): Kent
Bivariate (toroidal): bivariate von Mises

Multivariate: von Mises–Fisher; Bingham

Degenerate and singular

Degenerate: discrete degenerate; Dirac delta function

Singular: Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Maxwell Speed Distribution - Averages