Handbook of Differential Entropy

116 Handbook of Differential Entropy

4.16 Maxwell Distribution

The Maxwell, or Maxwell-Boltzmann distribution, is named after James Clerk

Maxwell and Ludwig Boltzmann whose work laid the foundation for the kinetic

theory of gases. They proved that, in ideal gases with insignificant quantum

effects, particle speeds and also particle energies are described by the Maxwell-

Boltzmann distribution. This distribution is exactly the Chi distribution with

3 degrees of freedom with σ =

p

3/2α.

0 1 2 3 4 5

0

0.5

1

1.5

2

x

p(x)

α = 2

α = 1

α = 0.5

FIGURE 4.31

Probability density functions for the Maxwell distribution.

Probability density: p(x) =

(

4

√

π

x

2

e

−x

2

/(α

2

)

α

3

: 0 ≤ x < ∞

0 :