Maxwell–Jüttner distribution
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.
is the mass of the kind of particle making up the gas,
The distribution can be attributed to Ferencz Jüttner, who derived it in 1911.
in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:[2]
Alternatively, this can be written in terms of the momentum as
[3] A visual representation of the distribution in particle velocities for plasmas at four different temperatures:[4] Where thermal parameter has been defined as
The four general limits are: 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
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.
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).
[clarification needed] While the classic Maxwell–Jüttner distribution generalizes for the case of special relativity, it fails to consider the anisotropic description.
of the particles at thermal equilibrium, far from the limit of the speed of light, i.e.:
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
(Note that the inverse of the unitless temperature
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
one obtains: Hence, Or The inverse of the normalization constant gives the partition function
can be shown to coincide with the thermodynamic definition of temperature.
Also useful is the expression of the distribution in the velocity space.
distribution of the same temperature and dimensionality, one can misinterpret and deduce a different
distribution that will give a good approximation to the
This means that a relativistic non-quantum particle with parameter
The integral simplifies to the closed- form expression:
, corresponds to the average speed in the Maxwell–Boltzmann distribution,
is an upper limit to the particle's speed, something only present in a relativistic context, and not in the Maxwell–Boltzmann distribution.
This article incorporates text by George Livadiotis available under the CC BY 3.0 license.
