General Statement and Questions
I am trying to figure out the proper way to model a velocity/momentum distribution function that is correct in the relativistic limit. I would like to determine/know two things:
- Is there an analytical form for an anisotropic relativistic momentum distribution (i.e., the relativistic analog of the bi-Maxwellian distribution)?
- What does temperature (i.e., kinetic temperature) mean in the relativistic limit?
- Temperature cannot be a Lorentz invariant, can it?
- It certainly cannot be invariant if the average particle thermal energies correspond to relativistic thermal velocities, correct?
- So how can just a simple scalar temperature be a proper normalization factor as a Lagrange multiplier in the Maxwell-Jüttner distribution, for instance?
- Extra: Is there an appropriate relativistic version of the $\kappa$ distribution (see this arXiv PDF for reference, or e-print number 1003.3532 if you do not trust links)?
Background
I am aware of the Maxwell-Jüttner distribution of particle species $\alpha$, given by:
$$
f_{\alpha} \left( p \right) = \Lambda \ exp \left[ – \Theta_{0} \ m c^{2} \ \gamma \left( p \right) – \sum_{i=1}^{3} \Theta_{i} \ c \ p^{i} \right]
$$
where $\Lambda$, $\Theta_{0}$, and $\Theta_{i}$ are Lagrange multipliers, $p$ is the relativistic momentum, and $\gamma\left( p \right)$ is the Lorentz factor. The $\Theta_{\nu}$ terms are 4-vector components with units of inverse energy.
In the isotropic limit, one can set all $\Theta_{i}$ $\rightarrow$ 0. This leads to the canonical form of the isotropic relativistic momentum distribution function is given by:
$$
f_{\alpha} \left[ \gamma \left( p \right) \right] = \Lambda e^{ – \Theta_{0} \ m c^{2} \ \gamma \left( p \right) }
$$
where $\Theta_{0}$ was shown1 to be the inverse of a temperature.
The Issue
The definition of $\Lambda$, however, has led to multiple results, as stated by Treumann et al. [2011]:
the correct (non-angular-dependent part of the) relativistic thermal-equilibrium distribution should become the modified-Jüttner distribution. (The ordinary Maxwell-Jüttner distribution function was derived by F. Jüttner, 1911, who obtained it imposing translational invariance in momentum space only.)
An attempt4 was made to derive $\Lambda$ by imposing Lorentz invariance on only momentum space, ignoring the spatial coordinates of the volume integral. However, Treumann et al. [2011] note that:
This is either not justified at all or it is argued that the particles are all confined to a fixed box which is unaffected by the Lorentz transformation and invariance. However, the momentum and configuration space volume elements the product of which forms the phase-space volume element, are not independent, as we have demonstrated above. Even in this case of a fixed outer box, the particle's proper spaces experience linear Lorentz contractions when seen from the stationary frame of the observer, i.e., from the box-frame perspective. The consequence is that the extra proper Lorentz factor $\gamma\left( p \right)$ in the phase-space volume element cancels thereby guaranteeing and restoring Lorentz invariance…
They go on to show that the correct Lagrange multipliers are:
$$
\Theta_{0} = \frac{ 1 }{ T } \\
\Lambda = \frac{ N^{0} }{ 4 \pi \ m^{2} T^{2} } \left[ 3 K_{2}\left( \frac{ m c^{2} }{ T } \right) + \frac{ m c^{2} }{ T } K_{1}\left( \frac{ m c^{2} }{ T } \right) \right]^{-1}
$$
where $N^{0}$ is the scalar part of the particle current density 4-vector (i.e., number density), $K_{i}(x)$ is the second order modified Bessel function, and $T$ is a scalar temperature. Notice there is an additional term (i.e., $K_{1}(x)$) in the normalization factor $\Lambda$, which is why they called this the modified Maxwell-Jüttner distribution. This accounts for Lorentz invariance in the phase-space element, not just momentum-space.
What I am looking for…
Regardless of its accuracy, the distribution function in Treumann et al. [2011] still only assumes an isotropic distribution and I am still a bit confused how the temperature is just a scalar. In plasma physics, it is more appropriate to think of it as kind of a pseudotensor derived from the pressure tensor or 2nd moment of the distribution function. So am I supposed to interpret relativistic temperatures through the energy-momentum tensor or something else? See more details about velocity moments here: https://physics.stackexchange.com/a/218643/59023.
In many situations, plasmas can be described as either a bi-Maxwellian or bi-kappa [e.g., Livadiotis, 2015] velocity distribution functions. The bi-Maxwellian is given by:
$$
f\left( v_{\parallel}, v_{\perp} \right) = \frac{ 1 }{ \pi^{3/2} \ V_{T \parallel} \ V_{T \perp}^{2} } \ exp\left[ – \left( \frac{ v_{\parallel} – v_{o, \parallel} }{ V_{T \parallel} } \right)^{2} – \left( \frac{ v_{\perp} – v_{o, \perp} }{ V_{T \perp} } \right)^{2} \right]
$$
where $\parallel$($\perp$) refer to directions parallel(perpendicular) with respect to a quasi-static magnetic field, $\mathbf{B}_{o}$, $V_{T_{j}}$ is the $j^{th}$ thermal speed (actually the most probable speed), and $v_{o, j}$ is the $j^{th}$ component of the bulk drift velocity of the distribution (i.e., from the 1st velocity moment).
The bi-kappa distribution function is given by:
$$
f\left( v_{\parallel}, v_{\perp} \right) = A \left[ 1 + \left( \frac{ v_{\parallel} – v_{o, \parallel} }{ \sqrt{ \kappa – 3/2 } \ \theta_{\parallel} } \right)^{2} + \left( \frac{ v_{\perp} – v_{o, \perp} }{ \sqrt{ \kappa – 3/2 } \ \theta_{\perp} } \right)^{2} \right]^{- \left( \kappa + 1 \right) }
$$
where the amplitude is given by:
$$
A = \left( \frac{ \Gamma\left( \kappa + 1 \right) }{ \left( \pi \left( \kappa – 3/2 \right) \right)^{3/2} \ \theta_{\parallel} \ \theta_{\perp}^{2} \ \Gamma\left( \kappa – 1/2 \right) } \right)
$$
and where $\theta_{j}$ is the $j^{th}$ thermal speed (also the most probable speed), $\Gamma(x)$ is the complete gamma function and we can show that the average temperature is just given by:
$$
T = \frac{ 1 }{ 3 } \left( T_{\parallel} + 2 \ T_{\perp} \right)
$$
if we assume a gyrotropic distribution (i.e., shows symmetry about $\mathbf{B}_{o}$ so that the two perpendicular components of a diagonalized pressure tensor are equal).
In summary, I would prefer a relativistically consistent bi-kappa distribution but would be very happy with the bi-Maxwellian version as well.
Update
After several conversations with the R. Treumann, he and his colleague decided to look into an anisotropic Maxwell-Jüttner distribution. I also referred him to this page and he decided to try and remain consistent with the original Maxwell-Jüttner distribution normalization to avoid further confusion.
His new results can be found in the arXiv paper with e-print number 1512.04015.
Summary of Results
One of the interesting things noted by Treumann and Baumjohann is that one cannot simply take the expression for energy and split the momentum into parallel and perpendicular terms as has occasionally been done in the past. Part of the issue is that the normalizations factors, i.e., temperature-like quantities, are not relativistically invariant. The temperature in this case is more akin to a pseudotensor than a scalar (Note: I use pseudotensor very lightly/carelessly here).
They use the Dirac tensor, from the Klein-Gordon approach, to define the energies. They treat the pressure as a proper tensor, with an assumed inverse, to define what they call the temperature tensor.
Unfortunately, the equation cannot be reduced analytically, but it is useful none-the-less given the alternative is to assume the unrealistic case of an isotropic velocity distribution in a relativistic plasma.
References
- Israel, W. "Relativistic kinetic theory of a simple gas," J. Math. Phys. 4, 1163-1181, doi:10.1063/1.1704047, 1963.
- Treumann, R.A., R. Nakamura, and W. Baumjohann "Relativistic transformation of phase-space distributions," Ann. Geophys. 29, 1259-1265, doi:10.5194/angeo-29-1259-2011, 2011.
- Jüttner, F. "Das Maxwellsche Gesetz der Geschwindigkeitsverteilung in der Relativtheorie," Ann. Phys. 339, 856-882, doi:10.1002/andp.19113390503, 1911.
- Dunkel, J., P. Talkner, and P. Hänggi "Relative entropy, Haar measures and relativistic canonical velocity distributions," New J. Phys. 9, 144-157, doi:10.1088/1367-2630/9/5/144, 2007.
- Livadiotis, G. "Introduction to special section on Origins and Properties of Kappa Distributions: Statistical Background and Properties of Kappa Distributions in Space Plasmas," J. Geophys. Res. Space Physics 120, 1607-1619, doi:10.1002/2014JA020825, 2015.
Best Answer
I think you make it sound much more mysterious than it is. The relativistic distribution function is $$ f_p = \frac{1}{(2\pi)^3}\exp(-(\mu-u\cdot p)/T)\, $$ where $u_\alpha$ is the 4-velocity of the fluid, $p_\alpha$ is the 4-momentum of the particle, $T$ is temperature, and $\mu$ is the chemical potential. This is sometimes called the Juttner distribution, and it has obvious generalizations to Bose-Einstein and Fermi-Dirac statistics.
This expression is obviously Lorentz-invariant ($u\cdot p$ is a scalar, and so are $\mu$ and $T$), it reduces to Boltzmann in the rest frame of the fluid, and it is Gallilean invariant for slowly moving fluids.
The funny Bessel functions appear if you try to determine the fugacity $e^{\mu/T}$ in terms of the density, $n=N_0$ with $$ N_\mu = \int \frac{d^3p}{p^0} p_\mu f_p\, , $$ because now your normalization factor contains integrals over $\exp(-\sqrt{p^2+m^2}/T)$
Additional Remarks: After some prodding by the OP I looked at the Treumann et al. paper (it is available on the arxive, http://arxiv.org/abs/1105.2120). I initially thought that this was just a needlessly complicated rederivation of well-known results, but this is not the case. The paper is just wrong. (Frankly, it is not a good sign if a paper that points out a major flaw in relativistic kinetic theory is published in the physics section of the arxive, and a geophysics journal.)
The paper starts out o.k., observing that since $d^3xd^3p$ is Lorentz invariant, $f_p$ must be a scalar. However, they write down a distribution function which is clearly not a scalar unless I assume that $T$ is the zeroth component of a vector. Even if that could be arranged, the result is nor right because it is not Galilean invariant for small velocities. He then writes down a non-covariant expression for $N_\mu$. If $f_p$ is a scalar, $d^3p p_\mu f_p$ is not a vector, and as a result his particle density $N_0$ does not transform as a density (I wrote the correct expression above). Since $N_0$ is wrong, the normalization $\Lambda$ is also wrong.
More Remarks: After more prodding, an attempt to answer the original questions:
1) The original Boltzmann distribution is anisotropic if the fluid velocity $\vec{u}$ is not zero (the distribution is isotropic in the fluid rest frame), but presumably you are looking for something more general. The distribution function of a viscous fluid is anisotropic already in the rest frame of the fluid, $f_p=f_p^0 +\delta f_p$ with $$\delta f_p \sim \eta\sigma_{\alpha\beta}p^\alpha p^\beta,$$ where $\sigma_{\alpha\beta}$ is the relativistic strain tensor. Even more generally, we can parameterize the distribution function in terms of the currents and the stresses. This is the relativistic version of Grad's 13 moment method, see, for example, http://arxiv.org/abs/1301.2912. People have also written down models for highly anisotropic distribution functions, see for example http://arxiv.org/abs/1007.0130 .
2) The temperature is a scalar. It is related to the kinetic energy per particle in the rest frame. The kinetic energy density is the 00 component of a tensor, and transforms accordingly.