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 much less than, 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 approaches or exceeds, the probability distribution for in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:
where and is the modified Bessel function of the second kind.
Alternatively, this can be written in terms of the momentum as
where. 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
The four general limits are:
  • ultrarelativistic temperatures
  • relativistic temperatures:
  • weakly relativistic temperatures:
  • low temperatures:

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 is that the Maxwell–Jüttner distribution neglects antiparticles.
If particle-antiparticle creation is allowed, then once the thermal energy is a significant fraction of, particle-antiparticle creation will occur and begin to increase the number of particles while generating antiparticles. 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.
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. While the classic Maxwell–Jüttner distribution generalizes for the case of special relativity, it fails to consider the anisotropic description.

Derivation

The Maxwell–Boltzmann distribution describes the velocities or the kinetic energy of the particles at thermal equilibrium, far from the limit of the speed of light, i.e.:
where the low-energy limit can be expressed as or equivalently, Here, is the expression of temperature as a speed, called thermal speed, and denotes the kinetic degrees of freedom of each particle. distribution, is given by:
where and This distribution 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. The Boltzmann distribution of a Hamiltonian is In the absence of a potential energy, is simply given by the particle energy, thus:
is defined by:
So that, by setting one obtains:
Hence,
Or
The inverse of the normalization constant gives the partition function
Therefore, the normalized distribution is:
Or one may derive the normalised distribution in terms of:
Note that can be shown to coincide with the thermodynamic definition of temperature.
Also useful is the expression of the distribution in the velocity space. Given that one has:
Hence
Take :
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 and bulk speed
  • an MB distribution with the same bulk speed, but with different temperature and degrees of freedom. These two types of approximations are illustrated.

Other properties

The MJ probability density function is given by:
This means that a relativistic non-quantum particle with parameter has a probability of of having its Lorentz factor in the interval

Cumulative distribution function

The MJ cumulative distribution function is given by:
That has a series expansion at
By definition regardless of the parameter

Average speed

To find the average speed, one must compute, where is the speed in terms of its Lorentz factor.
The integral simplifies to the closed-form expression:
This closed formula for has a series expansion at
Or substituting the definition for the parameter
Where the first term of the expansion, which is independently of, corresponds to the average speed in the Maxwell–Boltzmann distribution,, whilst the following are relativistic corrections.
This closed formula for has a series expansion at
Or substituting the definition for the parameter
Where it follows that is an upper limit to the particle's speed, something only present in a relativistic context, and not in the Maxwell–Boltzmann distribution.