Maxwell–Jüttner distribution explained

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 much less than mc ^{2 /k} _{B} (where m is the mass of the kind of particle making up the gas, c is the speed of light and k _{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.^[1] 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 _{BT} approaches or exceeds mc ^{2} , the probability distribution for $\backslash gamma=1/\backslash sqrt$ in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:^[2]

$$f(\backslash gamma)\; =\; \backslash frac\; e^$$

where $\backslash beta\; =\; \backslash frac\; =\; \backslash sqrt,$ $\backslash theta=\backslash frac,$ and

\operatorname{K} _{2} is the modified Bessel function of the second kind.

Alternatively, this can be written in terms of the momentum as$$f(\backslash mathbf)\; =\; \backslash frac\; e^$$where $\backslash gamma(p)\; =\; \backslash sqrt$. 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.^[3]

Jüttner distribution graph

A visual representation of the distribution in particle velocities for plasmas at four different temperatures:^[4]

Where thermal parameter has been defined as $\backslash mu\; =\; \backslash frac\; =\; \backslash frac$.

The four general limits are:

• ultrarelativistic temperatures

\mu \ll 1 \iff \theta \gg 1

• relativistic temperatures:

\mu < 1 \iff \theta > 1 ,

• weakly (or mildly) relativistic temperatures:

\mu > 1 \iff \theta < 1 ,

• low temperatures:

\mu \gg 1 \iff \theta \ll 1 ,

Limitations

Some limitations of the Maxwell–Jüttner distributions are shared with the classical ideal gas: neglect of interactions, and neglect of quantum effects. An additional limitation (not important in the classical ideal gas) is that the Maxwell–Jüttner distribution neglects antiparticles.

If particle-antiparticle creation is allowed, then once the thermal energy

k _{BT} is a significant fraction of mc ^{2} , particle-antiparticle creation will occur and begin to increase the number of particles while generating antiparticles (the number of particles is not conserved, but instead the conserved quantity is the difference between particle number and antiparticle number). The resulting thermal distribution will depend on the chemical potential relating to the conserved particle–antiparticle number difference. A further consequence of this is that it becomes necessary to incorporate statistical mechanics for indistinguishable particles, because the occupation probabilities for low kinetic energy states becomes of order unity. For fermions it is necessary to use Fermi–Dirac statistics and the result is analogous to the thermal generation of electron–hole pairs in semiconductors. For bosonic particles, it is necessary to use the Bose–Einstein statistics.^[5]

Perhaps most significantly, the basic

MB distribution has two main issues: it does not extend to particles moving at relativistic speeds, and  it assumes anisotropic temperature (where each DoF does not have the same translational kinetic energy). While the classic Maxwell–Jüttner distribution generalizes for the case of special relativity, it fails to consider the anisotropic description.

Derivation

The Maxwell–Boltzmann (

MB ) distribution \operatorname{pdf} _{MB} describes the velocities u or the kinetic energy $\backslash varepsilon\; =\; \backslash frac\; m\backslash mathbf^2$ of the particles at thermal equilibrium, far from the limit of the speed of light, i.e:

$\backslash theta\; \backslash equiv\; \backslash sqrt,\backslash \; \backslash \; u\; \backslash ll\; c$

Or, in terms of the kinetic energy:

\varepsilon \ll mc ^{2}

where

\theta is the temperature in speed dimensions, called thermal speed, and d denotes the kinetic degrees of freedom of each particle. (Note that the temperature is defined in the fluid’s rest frame, where the bulk speed u _{b} is zero. In the non-relativistic case, this can be shown by using $\backslash varepsilon\; =\; \backslash frac\; m\; (\backslash mathbf-\backslash mathbf\_b)^2$.

The relativistic generalization of Eq. (1a), that is, the Maxwell–Jüttner (

MJ ) distribution, is given by:

where

\beta \equiv { u }/ and \gamma(\beta) = ( 1- \beta ^{2)} ^{-{1} /{ 2 }} . (Note that the inverse of the unitless temperature \theta is the relativistic coldness \zeta , Rezzola and Zanotti, 2013.) This distribution (Eq. 2) can be derived as follows. According to the relativistic formalism for the particle momentum and energy, one has

While the kinetic energy is given by

\varepsilon = E - E _{0} = (\gamma - 1 ) E _{0} . The Boltzmann distribution of a Hamiltonian is \operatorname{pdf} _{MJ(H)} \propto e ^{- (H)/(k _{BT})} . In the absence of a potential energy, H is simply given by the particle energy E , thus:

(Note that

E is the sum of the kinetic \varepsilon and inertial energy $E\_0,\backslash frac\; =\; \backslash frac$). Then, when one includes the d -dimensional density of states:

So that:

Where

d \Omega _{d} denotes the d -dimensional solid angle. For isotropic distributions, one has

or

Then,

d (\gamma\beta) = \gamma (\gamma ^{2-1)} ^{- (1)/(2)} d \gamma = \beta ^{-1} d \gamma so that:

Or:

Now, because

(E)/(k _{BT}) = (\gamma)/(\theta) . Then, one normalises the distribution . One sets

And the angular integration:$$\backslash mathrmp\_1\; \backslash cdots\; \backslash mathrmp\_d=\; B\_d\; p^\; \backslash mathrmp=\; \backslash frac\; B\_d\backslash left(mc\; \backslash right)^d\backslash left(\backslash left(\backslash frac\; \backslash right)^2\; \backslash right)^\; \backslash mathrm\backslash left(\backslash frac\; \backslash right)^2,$$

Where

B _{d} = (2 \pi ^{d} /{ 2)/() } is the surface of the unit d-dimensional sphere. Then, using the identity \gamma ^{2} = \left( (p)/(mc) \right) ^{2} + 1 one has:and

Where one has defined the integral:

The Macdonald function (Modified Bessel function of the II kind) (Abramowitz and Stegun, 1972, p.376) is defined by:

So that, by setting

n = (d +1)/(2) , z = (1)/(\theta) one obtains:

Hence,

Or

The inverse of the normalization constant gives the partition function

Z \equiv (1)/(N) :

Therefore, the normalized distribution is:

Or one may derive the normalised distribution in terms of:

Note that

\theta can be shown to coincide with the thermodynamic definition of temperature.

Also useful is the expression of the distribution in the velocity space.^[6] Given that

(d (\beta\gamma))/(d \beta) = \gamma ^{3} , one has:

Hence

Take

d = 3 (the “classic case” in our world):

And

Note that when the

MB distribution clearly deviates from the MJ distribution of the same temperature and dimensionality, one can misinterpret and deduce a different MB distribution that will give a good approximation to the MJ distribution. This new MB distribution can be either:

• a convected

MB distribution, that is, an MB distribution with the same dimensionality, but with different temperature T _{MB} and bulk speed u _{b} (or bulk energy $E\_b\backslash equiv\; \backslash frac\; m\backslash left(\backslash mathbf\; +\; \backslash mathbf\_b\backslash right)^2$)

• an

MB distribution with the same bulk speed, but with different temperature T _{MB} and degrees of freedom d _{MB} . These two types of approximations are illustrated.

Other properties

The

MJ probability density function is given by:

$$\backslash operatorname\_\backslash text(\backslash gamma)\; =\; \backslash frac\; \backslash gamma^2\; \backslash ,\backslash beta(\backslash gamma)e^$$

This means that a relativistic non-quantum particle with parameter

\theta has a probability of \operatorname{pdf} _{MJ(\gamma)d\gamma} of having its Lorentz factor in the interval [\gamma,\gamma + d \gamma] .

The

MJ cumulative distribution function is given by:

$$\backslash operatorname\_\backslash text(\backslash gamma)\; =\; \backslash frac\; \backslash int\_1^\backslash gamma\; ^2\; \backslash sqrt\; \backslash ,\; e^\; \backslash mathrm\backslash gamma\text{'}$$

That has a series expansion at

\gamma = 1 :

$$\backslash operatorname\_\backslash text(\backslash gamma)=\; \backslash frac\; \backslash frac\; ^3\; +\; \backslash frac\; \backslash frac\; ^5\; +\; \backslash mathcal\backslash left(^7\; \backslash right)$$

By definition

\lim _{\gamma\toinfty} \operatorname{cdf} _{MJ(\gamma)} = 1 , regardless of the parameter \theta .

To find the average speed,

\langle v\rangle _{MJ} , one must compute $\backslash int\_1^\backslash infty\; \backslash operatorname\_\backslash text(\backslash gamma)\; \backslash ,\; v(\backslash gamma)\; \backslash ,\backslash mathrm\backslash gamma$, where $v(\backslash gamma)\; =\; c\backslash sqrt$ is the speed in terms of its Lorentz factor. The integral simplifies to the closed- form expression:

$$\backslash langle\; v\backslash rangle\_\backslash text\; =\; 2c\; \backslash frac$$

This closed formula for

\langle v\rangle _{MJ} has a series expansion at \theta = 0 :

$$\backslash frac\backslash langle\; v\backslash rangle\_\backslash text\; =\; \backslash sqrt\backslash sqrt\; -\backslash frac^3\; +\; \backslash mathcal\backslash left(^5\backslash right)$$

Or substituting the definition for the parameter

\theta :$$\backslash langle\; v\backslash rangle\_\backslash text\; =\; \backslash sqrt\; -\backslash frac\; \backslash frac^3\; +\; \backslash cdots$$

Where the first term of the expansion, which is independently of

c , corresponds to the average speed in the Maxwell–Boltzmann distribution, \langle v\rangle _{MB} = \sqrt{ (8)/(\pi) (k _{BT})/(m) } , whilst the following are relativistic corrections.

This closed formula for

\langle v\rangle _{MJ} has a series expansion at \theta = inf ty :

$$\backslash frac\backslash langle\; v\backslash rangle\_\backslash text\; =\; 1\; -\; \backslash frac\backslash frac\; +\; \backslash mathcal\backslash left(\backslash frac\backslash right)$$

Or substituting the definition for the parameter

\theta :

$$\backslash langle\; v\backslash rangle\_\backslash text\; =\; c\; -\backslash frac\; c^5\; \backslash frac\; +\; \backslash cdots$$

Where it follows that

c is an upper limit to the particle's speed, something only present in a relativistic context, and not in the Maxwell–Boltzmann distribution.

References

1. Jüttner . F. . Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie . 10.1002/andp.19113390503 . Annalen der Physik . 339 . 5 . 856–882 . 1911 . 1911AnP...339..856J .
2. Book: Synge, J.L . 1957 . The Relativistic Gas . Series in physics . . 57003567.
3. Chacon-Acosta. Guillermo . Dagdug. Leonardo. Morales-Tecotl. Hugo A.. 2009 . On the Manifestly Covariant Jüttner Distribution and Equipartition Theorem . Physical Review E. 81 . 2 Pt 1 . 021126 . 0910.1625. 2010PhRvE..81b1126C . 10.1103/PhysRevE.81.021126 . 20365549 . 39195896 .
4. Lazar. M.. Stockem. A.. Schlickeiser. R.. 2010-12-03. Towards a Relativistically Correct Characterization of Counterstreaming Plasmas. I. Distribution Functions. The Open Plasma Physics Journal. 3. 1.
5. See first few paragraphs in https://arxiv.org/abs/hep-th/9604039v1 for extended discussion.
6. Dunkel . Jörn . Talkner . Peter . Hänggi . Peter . 2007-05-22 . Relative entropy, Haar measures and relativistic canonical velocity distributions . New Journal of Physics . 9 . 5 . 144 . 10.1088/1367-2630/9/5/144 . 15896453 . 1367-2630. cond-mat/0610045 . 2007NJPh....9..144D .