Maxwell–Jüttner distribution

In physics, the Maxwell–Jüttner distribution, sometimes called Jüttner–Synge distribution, is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to the Maxwell–Boltzmann distribution, the Maxwell–Jüttner distribution considers a classical ideal gas where the particles are dilute and do not significantly interact with each other. The distinction from Maxwell–Boltzmann's case is that effects of special relativity are taken into account. In the limit of low temperatures

T

{\displaystyle T}

much less than

m

c

2

/

k

B

{\displaystyle mc^{2}/k_{\text{B}}}

(where

m

{\displaystyle m}

is the mass of the kind of particle making up the gas,

c

{\displaystyle c}

is the speed of light and

k

B

{\displaystyle k_{\text{B}}}

is Boltzmann constant), this distribution becomes identical to the Maxwell–Boltzmann distribution.

The distribution can be attributed to Ferencz Jüttner, who derived it in 1911. It has become known as the Maxwell–Jüttner distribution by analogy to the name Maxwell–Boltzmann distribution that is commonly used to refer to Maxwell's or Maxwellian distribution.

Definition

As the gas becomes hotter and

k

B

T

{\displaystyle k_{\text{B}}T}

approaches or exceeds

m

c

2

{\displaystyle mc^{2}}

, the probability distribution for

γ

=

1

/

1

−

v

2

/

c

2

{\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}

in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:

f

(

γ

)

d

γ

=

γ

2

β

(

γ

)

θ

K

2

(

1

θ

)

e

−

γ

/

θ

d

γ

{\displaystyle f(\gamma )\,\mathrm {d} \gamma ={\frac {\gamma ^{2}\,\beta (\gamma )}{\theta \operatorname {K} _{2}\!\left({\frac {1}{\theta }}\right)}}e^{-{\gamma }/{\theta }}\,\mathrm {d} \gamma }

where

β

=

v

c

=

1

−

1

/

γ

2

{\textstyle \beta ={\frac {v}{c}}={\sqrt {1-1/\gamma ^{2}}}}

,

θ

=

k

B

T

m

c

2

{\textstyle \theta ={\frac {k_{\text{B}}T}{mc^{2}}}}

, and

K

2

{\displaystyle \operatorname {K} _{2}}

is the modified Bessel function of the second kind.

Alternatively, this can be written in terms of the momentum as

f

(

p

)

d

3

p

=

1

4

π

θ

K

2

(

1

θ

)

e

−

γ

(

p

)

/

θ

d

3

p

(

m

c

)

3

{\displaystyle f(\mathbf {p} )\,\mathrm {d} ^{3}\mathbf {p} ={\frac {1}{4\pi \theta \operatorname {K} _{2}\!\left({\frac {1}{\theta }}\right)}}e^{-\gamma (p)/\theta }\,{\frac {\mathrm {d} ^{3}\mathbf {p} }{(mc)^{3}}}}

where

γ

(

p

)

=

1

+

(

p

m

c

)

2

{\textstyle \gamma (p)={\sqrt {1+\left({\frac {p}{mc}}\right)^{2}}}}

. If the momentum distribution is symmetric in all 3 dimensions, the distribution can also be written in terms of the magnitude of the momentum, after integrating the 3-dimensional distribution over a sphere:

f

(

p

)

d

p

=

p

2

θ

K

2

(

1

θ

)

e

−

γ

(

p

)

/

θ

d

p

(

m

c

)

3

{\displaystyle f(p)\,\mathrm {d} p={\frac {p^{2}}{\theta \operatorname {K} _{2}\!\left({\frac {1}{\theta }}\right)}}e^{-\gamma (p)/\theta }{\frac {\mathrm {d} p}{(mc)^{3}}}}

The Maxwell–Jüttner equation is covariant, but not manifestly so, and the temperature of the gas does not vary with the gross speed of the gas.

Jüttner distribution graph

A visual representation of the distribution in particle velocities for plasmas at four different temperatures:

Where thermal parameter has been defined as

μ

=

m

c

2

k

B

T

=

1

θ

{\textstyle \mu ={\frac {mc^{2}}{k_{\text{B}}T}}={\frac {1}{\theta }}}

.