Maxwell–Jüttner distribution

In physics, the Maxwell–Jüttner distribution is the distribution of speeds of particles in a hypothetical gas of relativistic particles. Similar to Maxwell's 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's case is that effects of special relativity are taken into account. In the limit of low temperatures ${\displaystyle T}$ much less than ${\displaystyle mc^{2}/k}$ (where ${\displaystyle m}$ is the mass of the kind of particle making up the gas, ${\displaystyle c}$ is the speed of light and ${\displaystyle k}$ is Boltzmann's constant), this distribution becomes identical to the Maxwell–Boltzmann distribution.

The distribution can be attributed to , 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 distribution.

The distribution function[]

Maxwell–Jüttner distribution over Lorentz factor (relativistic Maxwellian), for a gas at different temperatures. Speed is represented in terms of the Lorentz factor.

As the gas becomes hotter and ${\displaystyle kT}$ approaches or exceeds ${\displaystyle mc^{2}}$, the probability distribution for ${\textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$ in this relativistic Maxwellian gas is given by the Maxwell–Jüttner distribution:^[2]

${\displaystyle f(\gamma )={\frac {\gamma ^{2}\beta }{\theta K_{2}(1/\theta )}}\exp \left(-{\frac {\gamma }{\theta }}\right)}$

where ${\displaystyle \beta ={\frac {v}{c}}={\sqrt {1-1/\gamma ^{2}}},}$ ${\displaystyle \theta ={\frac {kT}{mc^{2}}},}$ and ${\displaystyle K_{2}}$ is the modified Bessel function of the second kind.

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

${\displaystyle f(\mathbf {p} )={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}\exp \left(-{\frac {\gamma (p)}{\theta }}\right)}$

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

Juttner Distribution Graph^[4][]

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

Where we have defined the thermal parameter ${\displaystyle \mu =mc^{2}/k_{b}T}$.

The four general limits are:

• ultrarelativistic temperatures μ << 1
• relativistic temperatures: μ < 1,
• weakly (or mildly) relativistic temperatures: μ > 1 ,
• low temperatures: μ >> 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 ${\displaystyle kT}$ is a significant fraction of ${\displaystyle mc^{2}}$, 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 hasn’t the same translational kinetic energy). While the classic Maxwell-Juttner distribution generalizes for the case of special relativity, it fails to consider the anisotropic description.

Derivation of MJ Distribution[]

The Maxwell–Boltzmann (MB) distribution P_M describes the velocities u or the kinetic energy ${\displaystyle \epsilon ={\frac {1}{2}}m\mathbf {u} ^{2}}$ of the particles at thermal equilibrium, far from the limit of the speed of light, i.e:

${\displaystyle P_{\text{MB}}\left(\mathbf {p;\theta } \right)\mathbf {=(\pi } m^{\mathbf {2} }\mathbf {\theta } ^{\mathbf {2} }\mathbf {)} ^{\mathbf {-} {\frac {\mathbf {1} }{\mathbf {2} }}\mathbf {d} }\exp \left(-{\frac {{\frac {1}{2m}}\mathbf {p} ^{2}}{k_{b}T}}\right).\ \ \ \ (1a)}$

${\displaystyle \theta \equiv {\sqrt {\frac {2k_{b}T}{m}}},\ \ u\ll c}$

Or, in terms of the kinetic energy:

${\displaystyle P_{mb}(\epsilon ;T)={\frac {(k_{b}T)^{-1/2d}}{\Gamma ({\frac {1}{2}}d)}}exp(-{\frac {\epsilon }{K_{b}T}})\epsilon ^{{\frac {1}{2}}d-1}(1b)}$

${\displaystyle \epsilon \ll mc^{2}}$

where θ 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 ε = ${\displaystyle {\frac {1}{2}}m(\mathbf {u-} \mathbf {u} _{\mathbf {b} })^{2}}$ .

The relativistic generalization of Eq. (1a), that is, the Maxwell–Jüttner (MJ) distribution, is given by:

${\displaystyle P_{MJ}(\gamma )\propto \gamma ^{2}\beta (\gamma )\bullet e^{\frac {-\gamma }{\Theta }},\Theta \equiv {\frac {k_{b}T}{E_{0}}},\ E_{0}=mc^{2}\ \ \ \ (2)}$

where β ≡ ${\displaystyle {\frac {\mathbf {u} }{\mathbf {c} }}}$ and γ (β) ≡ (1-${\displaystyle \beta ^{2})^{-{\frac {1}{2}}}\ }$ . (Note that the inverse of the unitless temperature ${\displaystyle \Theta }$ is the relativistic coldness ζ , Rezzola and Zanotti, 2013.) This distribution (Eq. 2) can be derived as follows. According to the relativistic formalism for the particle momentum and energy, we have

${\displaystyle \mathbf {p=} mc\bullet \gamma \left(\beta \right)\bullet \mathbf {\beta } ,\ E\left(\beta \right)=\gamma (\beta )\bullet E_{0}\ \ \ \ \ \ \ \ \ (3)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

While the kinetic energy is given by ${\displaystyle \epsilon =E-E_{0}=(\gamma -1)\bullet E_{0}}$. The Boltzmann distibuition of a Hamiltonian is P_mj(H) ${\displaystyle \propto \exp {\left\lbrack -{\frac {H}{K_{b}T}}\right\rbrack .}}$ In the absence of a potential energy, H is simply given by the particle energy E, thus:

${\displaystyle P_{\text{MJ}}\left(E\right)\operatorname {\propto exp} \left(-{\frac {E}{k_{b}T}}\right)\propto \exp \left(-{\frac {\gamma }{\Theta }}\right)\ \ \ \ \ (4a)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

(Note that E is the sum of the kinetic ε and inertial energy ${\displaystyle E_{0},{\frac {\epsilon }{k_{b}T}}={\frac {\gamma -\ 1}{\Theta }}}$). Then, when we include the d-dimensional density of states:

${\displaystyle P_{\text{MJ}}\left(\gamma \right)\propto p\left(\gamma \right)^{d-1}{\frac {{\text{dp}}\left(\gamma \right)}{\text{dγ}}}\bullet \exp \left(-{\frac {\gamma }{\Theta }}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4b)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

So that:

${\displaystyle {\int _{}^{}{P_{\text{MJ}}\left(\mathbf {p} \right)dp_{1}\ldots .dp_{d}\propto }\int _{}^{}{\exp \left\lbrack -{\frac {E\left(\mathbf {p} \right)}{k_{b}T}}\right\rbrack dp_{1}\ldots .dp_{d}}}}$

${\displaystyle {=\int _{}^{}{\exp \left\lbrack -{\frac {E\left({\text{γ.}}\Omega _{d}\right)}{k_{b}T}}\right\rbrack d\Omega _{d}p^{d-1}{\text{dp}}}}}$

${\displaystyle {=\int _{\Omega _{d}}^{}{\exp \left\lbrack -{\frac {E\left({\text{γ.}}\Omega _{d}\right)}{k_{b}T}}\right\rbrack \bullet \left\lbrack p\left(\gamma \right)^{d-1}{\frac {{\text{dp}}\left(\gamma \right)}{\text{dγ}}}\right\rbrack d\Omega _{d}{\text{dγ}}}}}$

Where ${\displaystyle d\Omega _{d}\ }$denotes the d-dimensional solid angle. For isotropic distributions, we have

${\displaystyle \int P_{MJ}(p)dp_{1}....dp_{d}\propto \int exp[-{\frac {E(p)}{K_{b}T}}]\times \ \ \ \ \ \ (5a)}$

${\displaystyle [p(\gamma )^{d-1}{\frac {dp(\gamma )}{d\gamma }}]d\Omega _{d}d\gamma \equiv \int _{\Omega _{d}}d\Omega _{d}\bullet \int P_{MJ}(\gamma )d\gamma ,}$

or

${\displaystyle P_{mj}(\gamma )\propto exp[-{\frac {E(\gamma )}{k_{b}T}}]\bullet p(\gamma )^{d-1}{\frac {dp(\gamma )}{d\gamma }}\ \ \ \ \ \ (5b)}$

Then, ${\displaystyle d\left({\text{γβ}}\right)=\gamma \left(\gamma ^{2}-1\right)^{-{\frac {1}{2}}}d\gamma =\beta ^{-1}{\text{dγ}}}$ so that:

${\displaystyle {p\left(\gamma \right)^{d-1}{\frac {{\text{dp}}\left(\gamma \right)}{\text{dγ}}}=\left({\text{mc}}\right)^{d}\left({\text{γβ}}\right)^{d-1}{\frac {d\left({\text{γβ}}\right)}{\text{dγ}}}}\ \ \ \ \ \ \ (6)}$

${\displaystyle {=\left({\text{mc}}\right)^{d}\gamma ^{d-1}\beta ^{d-2},}}$

Or:

${\displaystyle P_{MJ}(\gamma )\propto \gamma ^{d-1}\beta ^{d-2},e^{-{\frac {\gamma }{\Theta }}}\propto \gamma (\gamma ^{2}-1)^{{\frac {d}{2}}-1}\bullet e^{-{\frac {\gamma }{\Theta }}}\ \ \ \ \ (7)\ }$

Now, because ${\displaystyle {\frac {E}{k_{b}T}}={\frac {\gamma }{\Theta }}}$. Then, we normalize the distribution Eq. (7). We set

${\displaystyle P_{MJ}(p,\Theta )dp_{1}dp_{2}....dp_{d}\ \ \ \ \ \ (8)}$

${\displaystyle =N\bullet e^{\frac {-\gamma (p)}{\Theta }}dp_{1}dp_{2}....dp_{d}}$

And the angular integration:

${\displaystyle {dp_{1}dp_{2}\ldots dp_{d}=B_{d}p^{d-1}{\text{dp}}}}$

${\displaystyle {={\frac {1}{2}}b_{d}\left({\text{mc}}\right)^{d}\left\lbrack \left({\frac {p}{\text{mc}}}\right)^{2}\right\rbrack ^{{\frac {d}{2}}-1}d\left({\frac {p}{\text{mc}}}\right)^{2},}}$

Where ${\displaystyle B_{d}=2\pi ^{\frac {d}{2}}/\Gamma \left({\frac {d}{2}}\right)}$ is the surface of the unit d-dimensional sphere. Then, using the identity ${\displaystyle \gamma ^{2}=\left({\frac {p}{\text{mc}}}\right)^{2}+1}$ we have:

${\displaystyle P_{\text{MJ}}(\mathbf {p} }$;${\displaystyle \mathbf {\Theta } \ )dp_{1}dp_{2}\ldots dp_{d}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

${\displaystyle =N\bullet {\frac {1}{2}}B_{d}\left({\text{mc}}\right)^{d}\bullet e^{-{\frac {\gamma }{\Theta }}}\left(\gamma ^{2}-1\right)^{{\frac {d}{2}}-1\ }d\left(\gamma ^{2}-1\right).}$

and

${\displaystyle {1=\int _{-\infty }^{\infty }{P_{\text{MJ}}\left(\mathbf {p;} \Theta \right)dp_{1}dp_{2}\ldots \ldots dp_{d}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)}}$${\displaystyle {=N\bullet {\frac {1}{2}}B_{d}\left({\text{mc}}\right)^{d}\bullet \int _{0}^{\infty }{e^{-{\frac {\gamma }{\Theta }}}\left(\gamma ^{2}-1\right)^{{\frac {d}{2}}-1}}d\left(\gamma ^{2}-1\right)}}$

${\displaystyle {=N\bullet {\frac {1}{2}}B_{d}\left({\frac {d}{2}}\right)^{-1}\left({\text{mc}}\right)^{d}\Theta ^{-1}\bullet \int _{1}^{\infty }{e^{-{\frac {\gamma }{\Theta }}}\left(\gamma ^{2}-1\right)^{\frac {d}{2}}}{\text{dγ}}}}$

${\displaystyle {=N\bullet {\frac {1}{2}}B_{d}\left({\frac {d}{2}}\right)^{-1}\left({\text{mc}}\right)^{d}\Theta ^{-1}\bullet I_{d},}}$

Where we have defined the integral:

${\displaystyle I_{d}\equiv \int _{1}^{\infty }{e^{-{\frac {\gamma }{\Theta }}}\left(\gamma ^{2}-1\right)^{\frac {d}{2}}}d\gamma .\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }$

The Macdonald function (Avramovitz and Stegun, 1972, p.376) is defined by:

${\displaystyle K_{n}(z)\equiv {\frac {\pi ^{\frac {1}{2}}({\frac {1}{2}}z)^{n}}{\Gamma (n+{\frac {1}{2}})}}\int _{1}^{\infty }e^{-zy}(\gamma ^{2}-1)^{n-{\frac {1}{2}}}d\gamma \ \ \ \ \ \ (12)}$

So that, by setting ${\displaystyle n={\frac {d+1}{2}},\ z={\frac {1}{\Theta }}}$ we obtain:

${\displaystyle I_{d}=\Gamma ({\frac {d}{2}}+1)\pi ^{-{\frac {1}{2}}}K_{\frac {d+1}{2}}({\frac {1}{\Theta }})(2\Theta )^{\frac {d+1}{2}}\ \ \ \ \ (13)}$

Hence,

${\displaystyle {N^{-1}={\frac {\pi ^{\frac {d}{2}}}{\Gamma \left({\frac {d}{2}}\right)}}\left({\frac {d}{2}}\right)^{-1}\Gamma \left({\frac {d}{2}}+1\right)\pi ^{-{\frac {1}{2}}}K_{\frac {d+1}{2}}\left({\frac {1}{\Theta }}\right)\left(2\Theta \right)^{\frac {d+1}{2}}\backslash n}{=\pi ^{\frac {d-1}{2}}2^{-{\frac {d+1}{2}}}\left({\text{mc}}\right)^{d}\bullet \Theta ^{\frac {d-1}{2}}K_{\frac {d+1}{2}}\left({\frac {1}{\Theta }}\right),\ \ \ \ \ \ \ \ \ \ (14a)\ \ \ \ \ \ \ \ \ }}$

Or

${\displaystyle N=\ \pi ^{\frac {1-d}{2}}2^{-{\frac {d+1}{2}}}\left({\text{mc}}\right)^{-d}\bullet \Theta ^{\frac {1-d}{2}}K_{\frac {d+1}{2}}\left({\frac {1}{\Theta }}\right)^{-1}\ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(14b\right)}$

The inverse of the normalization constant gives the partition function ${\displaystyle Z\equiv {\frac {1}{N}}:}$

${\displaystyle Z=\pi ^{\frac {d-1}{2}}2^{-{\frac {d+1}{2}}}\left({\text{mc}}\right)^{d}\bullet \Theta ^{\frac {d-1}{2}}K_{\frac {d+1}{2}}\left({\frac {1}{\Theta }}\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(14c\right)}$

Therefore, the normalized distribution is:

${\displaystyle P_{MJ}(P;\Theta )dp_{1}dp_{2}...dp_{d}\ \ \ \ \ (15a)}$

${\displaystyle =\pi ^{\frac {1-d}{2}}2^{-{\frac {d+1}{2}}}(mc)^{-d}\bullet \Theta ^{\frac {1-d}{2}}K_{\frac {d+1}{2}}({\frac {1}{\Theta }})^{-1}\times exp[-{\frac {\gamma (p)}{\Theta }}]dp_{1}dp_{2}....dp_{d}}$

Or we may derive the normalized distribution in terms of:

${\displaystyle P_{mj}(\gamma ;\Theta )d\gamma \ \ \ \ \ \ \ (15b)}$

${\displaystyle ={\frac {\pi ^{\frac {1}{2}}2^{\frac {1-d}{2}}}{\Gamma ({\frac {d}{2}})}}K_{\frac {d+1}{2}}({\frac {1}{\Theta }})^{-1}\Theta ^{\frac {1-d}{2}}e^{-{\frac {\gamma }{\Theta }}}(\gamma ^{2}-1)^{d/2-1}\gamma d\gamma }$

Note that ${\displaystyle \Theta }$ can be shown to coincide with the thermodynamic definition of temperature.

Also useful is the expression of the distribution in the velocity space (Dunkel et al., 2007). Given that ${\displaystyle {\frac {d\left({\text{βγ}}\right)}{\text{dβ}}}=\gamma ^{3}}$, we have:

${\displaystyle {dp_{1}dp_{2}\ldots dp_{d}=p^{d-1}{\text{dpd}}\Omega _{d}}{=\left({\text{mc}}\right)^{d}\gamma ^{d-1}\beta ^{d-1}{\frac {d\left({\text{βγ}}\right)}{\text{dβ}}}{\text{dβd}}\Omega _{b}}{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}$

${\displaystyle {=\left({\text{mc}}\right)^{d}\gamma ^{d+2}\beta ^{d-1}{\text{dβd}}\Omega _{b}}{=\left({\text{mc}}\right)^{d}\gamma ^{d+2}d\beta _{1}\ldots ..d\beta _{b}}}$

Hence

${\displaystyle {P_{\text{MJ}}\left(\mathbf {\beta } ;\Theta \right)d\beta _{1}d\beta _{2}\ldots .d\beta _{d}}\ \ \ \ \ (15c)}$

${\displaystyle {=\pi ^{\frac {1-d}{2}}2^{-{\frac {d+1}{2}}}\bullet \Theta ^{{\frac {1-d}{2}}\ }K_{\frac {d+1}{2}}\left({\frac {1}{\Theta }}\right)^{-1}}{\times \exp {\left\lbrack \ -{\frac {\gamma \left(\beta \right)}{\Theta }}\right\rbrack \gamma \left(\beta \right)^{d+2}d\beta _{1}d\beta _{2}\ldots d\beta _{d}}}}$

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

${\displaystyle P_{MJ}(p;\Theta )dp_{1}dp_{2}.....dp_{d}\ \ \ \ \ \ (16a)}$

${\displaystyle ={\frac {1}{4\pi }}\left({\text{mc}}\right)^{-3}\bullet {\frac {1}{\Theta }}K_{2}\left({\frac {1}{\Theta }}\right)^{-1}\bullet e^{-{\frac {\gamma (\mathbf {p)} \ }{\Theta }}}dp_{1}dp_{2}dp_{3}}$

And

${\displaystyle {P_{\text{MJ}}\left(\gamma ;\Theta \right)d\gamma =}\ \ \ \ \ (16b)}$

${\displaystyle {{\frac {1}{\Theta }}K_{2}\left({\frac {1}{\Theta }}\right)^{-1}\bullet e^{-{\frac {\gamma }{\Theta }}}\left(\gamma ^{2}-1\right)^{\frac {1}{2}}{\text{γdγ}}}}$

${\displaystyle {P_{\text{MJ}}\left(\mathbf {\beta ;} \Theta \right)d\beta _{1}d\beta _{2}d\beta _{3}=\ \ \ \ \ (16c)}}$

${\displaystyle {\frac {4}{\pi }}\bullet {\frac {1}{\Theta }}K_{2}\left({\frac {1}{\Theta }}\right)^{-1}\bullet \exp \left\lbrack -{\frac {\gamma \left(\beta \right)}{\Theta }}\ \right\rbrack \gamma \left(\beta \right)^{5}d\beta _{1}d\beta _{2}d\beta _{3}}$

Note that when the MB distribution clearly deviates from the MJ distribution of the same temperature and dimensionality, one can misinterpret nd deduce a different MB distribution that will give a good approximation to the MJ distribution. This new MB distribution can be either (i) 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${\displaystyle E_{b}\equiv {\frac {1}{2}}mu+u_{B}^{2}}$), or (ii) 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.

References[]

 This article incorporates text by George Livadiotis available under the CC BY 3.0 license.