342. Thermal equilibrium in special relativity
ASTRO, COND-MAT — By David Cubero on April 8, 2009 at 3:48 pm
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David Cubero is professor at the Department of Applied Physics of the University of Sevilla. Dmitry.
Special relativity, despite being more than a hundred years old, still shows an intriguing capacity to surprise us in very fundamental issues, such as thermal equililbrium. In this post, we will review a recent controversy about the proper velocity distribution of dilute gases at thermal equilibrium.
In 1911, just six years after Albert Einstein had presented his theory of special relativity to the world, Ferencz Juettner formulated [1] a generalization to relativistic particles of the celebrated formula that describes the one-particle momentum distribution of a gas in equilibrium in the non-relativistic limit, namely, the Maxwell-Boltzmann distribution:
where 
, 
is the Botlzmann constant, 
the temperature, 
a normalization constant and 
the particle energy. In the non-relativistic case, this energy is simply given by the kinetic energy of the particle: 
. Using a maximum entropy derivation, Juettner proposed to replace this non-relativistic expression with its relativistic counterpart:
The corresponding momentum distribution, i.e. Eq. (1) with (2), is called since then the Juettner distribution. The velocity distribution 
can be obtained by transforming to the particle velocity using the relativistic formula of the momemtum 
, where 
.
Despite lacking a rigorous microscopic derivation due to the intrinsic difficulties introduced by special relativity (and more specifically due to the difficulty of formulating a relativistically consistent Hamiltonian mechanics of interacting particles), the Juettner distribution had been widely accepted among theorists for most of the 20th century. Starting from the 1980s, a number of theorists began to question the validity of the Juettner distribution, based on “manifestly covariant” theories [2], or the lack of explicit compatibility with relativity of the original maximum-entropy principle used by Juettner [3]. Using a maximum-entropy principle in combination with Lorentz symmetry, the following “modified” Juettner distribution was proposed [3]:
thus differing in a 
prefactor.
Evidently, identifying the correct relativistic equilibrium distribution is essential for the proper interpretation of experiments in high energy and astro-physics. This controversy was resolved in a simple one-dimensional system in Ref. [4] by resorting to computer simulations. Generally, fully relativistic simulations are very hard to carry out because interactions at a distance are very difficult to implement in special relativity, ussually requiring the introduction of fields. This difficulty was avoided in Ref. [4] by considering a two-component gas made of impenetrable point-particles with elastic point-like binary collisions in 1D. The results were conclusive, favoring the Juettner distribution as the only distribution able to account for the common equilibrium of the two-component gas.
One-particle velocity distributions using lab time (Juettner) and proper-time (Modified Juettner) parameterizations of a 1D two-component gas with 5000 light particles of mass 
and 5000 heavy particles of mass 
.
Later on, it was found that the modified Juettner distribution could be generated from a Monte Carlo simulation in which collisions are not consistent with the laws of Special Relativity. More specifically, the modified Juettner distribution could be obtained with a uniform random pairing technique, which is known to violate the relativistic invariance of the number of collisions in a space-time element [5].
This would have settled the controversy, had not it been for the fact that the modified Juettner function actually represents a physical distribution, as it was realized first in Ref. [6] from the analysis of relativistic Brownian processes. In our recent manuscript [7], using similar fully-relativistic molecular dynamics simulations of a two-component gas as in [4], we demonstrate that the modified Juettner distribution is obtained as the stationary one-particle distribution when a proper-time parameterization is used instead of the laboratory time, both when ensemble and time averages are carried out. The proper-time paramatrization is the natural mathematical choice when one tries to extend the theory of stochastic processes to general relativity. But also, it plays a crucial role in the description of particle creation/annihilation processes. These processes are a consequence of the equivalence of mass and energy, and thus, one of the baffling features attributable to the theory of Special Relativity. If we want to address questions like, for example, what is the typical energy distribution at the end of a particle’s life-time? Then we are compelled to consider a proper-time parameterization, which yields the modified Juettner distribution.
Last but not last, our work also reveals a close connection between time parameters and entropy in special relativity, thus, providing a better understanding of Juettner’s original derivation.
Some literature
[1] F. Juettner, Ann. Phys. (Leipzig) 34, 856 (1911).
[2] L. P. Horwitz, W. C. Schieve, and C. Piron, Ann. Phys. (N.Y.) 137, 306 (1981); L. P. Horwitz, S. Shashoua, and W. C. Schieve, Physica (Amsterdam) 161A, 300 (1989).
[3] J. Dunkel, P. Talkner, and P. Hanggi, New J. Phys. 9, 144 (2007).
[5] F. Peano, M. Marti, and L. O. Silva, Phys. Rev. E 79, 025701(R) (2009).
[6] J. Dunkel, P. Hanggi, and S. Weber, Phys. Rev. E 79, 010101(R) (2009).
8 Comments
I apologize but I find this “mystery” completely silly. The Juettner distribution is clearly the correct one. It has always been and one can check that it is correct by a 3-line calculation (plus some words that I expand unnecessarily, for pedagogical purposes) – a calculation that students taking the 1st lecture on relativistic QM should be able to reproduce as a homework exercise.
Take a 1-particle Hilbert space (or subspace of the full Hilbert space, with N=1) and work in a box, to be certain about the measure issues. It is useful to do this problem directly at the quantum level because the number of states – the volume on phase space – is normalized properly and automatically. Proper classical physics always emerges as the limit of quantum mechanics, and that’s the case of relativistic statistical mechanics, too.
The density matrix must be exp(-beta H) where H=p_0. This is a completely general formula valid for any system in quantum mechanics, as I will discuss later. The thermal density matrix must commute with H, and so on. Consider “p” in the lattice derived from the box, and you see that the normal derivation holds, except that one must use the correct E(p) dispersion relation.
There is a simple way to see that exp(-beta H)/H would be an inconsistent thermal density matrix for one species. Consider two species that are very weakly interacting. Put them to the same temperature. Because of the weak interactions, the thermal matrix must (almost exactly) factorize. But if the total density matrix were exp(-beta H1)/H1 (otimes) exp(-beta H2)/H2, then it wouldn’t commute with the total Hamiltonian H1+H2+small interactions, so the matrix couldn’t be in equilibrium. The only way to write down the correct density matrix that factorizes properly for pairs of clusters is the simple and universal exp(-beta H_total) without any denominator.
Also, it makes no sense to “deform” the thermal equilibrium by replacing the laboratory time by the proper time because one cannot define any equilibrium with respect to a “temperature computed relatively to the proper time”. What would it even mean?
If one needs to know something about the distribution of particles in an ensemble before they decay, he must simply solve the full problem where the number of particles can change. If the initial state is given by a temperature, it still uses the original Juettner distribution, although it must be combined with a subsequent complicated calculation reflecting the decay rate of various particles.
But the modified Juettner distribution would only be justified if one could actually define a new kind of equilibrium based on the “proper time temperature”, and not just find a calculation of something else where a formula resembling the modified Juettner distribution appears.
I am sure that the modified equilibrium could appear at various places – e.g. in Schwinger parameterization – but the temperature would have a different interpretation than a normal temperature.
Dear Lubos,
first of all, I agree with (almost) all your remarks concerning the correctness of the Juettner distribution and its derivation from relativistic quantum mechanics. There is, however, a somewhat “hidden” assumption in your line of reasoning which, I think, deserves to be emphasized and/or clarified:
The Hamiltonian “H” you are referring to in your comment is directly linked to a global (e.g., inertial or laboratory) coordinate-time “t” (i.e., “H” describes the evolution of the quantum system with respect to “t”). Hence, when adopting this type of time-parameterization, one obtains the Juettner function as “t”-stationary distribution.
The purpose of our paper [7] is to illustrate by means of a (very) simple example how concepts like “stationarity”, “entropy”, etc. are affected if one chooses – instead of “t” – other time parameters as, e.g., the proper-time “tau”, to quantify the evolution of a subsystem (here, a single particle). With regard to your example, the main conclusion could be summarized as follows:
A (nonlinear) change of the time parameter leads to a modified Hamiltonian and thus to a modified stationary distribution. Qualitatively, this is, perhaps, not too surprising but I think that it is worthwhile to also quantitatively understand the resulting modifications. Another “side-result” of our study is that a Lorentz-invariant time-parameter requires a Lorentz-invariant reference measure (reference probability density) in the associated entropy definition; this establishes a nice link between time-parameters and (Shannon) entropy.
Last but not least, let me briefly address why we think it is interesting to look into these questions:
(i) While the discussion in [7] refers to a simple equilibrium model, more relevant applications concern non-equilibrium scenarios. A quite useful, semi-phenomenological way of describing such phenomena is by stochastic processes or, (almost) equivalently, Fokker-Planck equations in phase space (see, e.g., [6]). If adopting such an approach in a relativistic setting, one faces similar problems concerning the choice of an appropriate evolution parameter (personally, I think, that simplified stochastic models that take into account annihilation processes are most naturally constructed in terms of proper-time rather than coordinate-time).
The results in [7] show that the different, time-parameter-dependent “stationary” distributions obtained from relativistic stochastic processes [6] are consistent with those obtained from simple deterministic gas simulations.
(ii) In special relativity, it is still unproblematic to unambiguously define global time parameters “t” – in general relativity, this is usually not the case anymore and proper-time may be a “better” parameter to start with.
Therefore, in my opinion, it is useful to have a profound understanding of how distributions that are based on different time-parameters are related to each other.
Dear Jorn,
if I understand you correctly, what you are talking about is rather trivial thing.
Suppose you indeed focus on the dynamics of a single particle in the gas. Its motion is Brownian, with correlation properties of the white noise determined by the temperature of the gas (FDT). If you choose different time parametrization, the form of the Fokker-Planck equation will be different, and of course the asymptotic
form of the distribution P – solution of the F.-P. equation will really depend on what 
is.
As far as I understand from the Lubos’ comment, he wants to study thermodynamic properties of the gas as a whole, so of course one has to choose the world time for that.
So, basically, before discussing the ambiguity you are talking about maybe it is good to explain what kinds of physical problems you want to deal with.
Cheers,
Dmitry.
Dear Dmitry & Lubos,
thanks for your comments. I basically agree with most of the things you are saying but probably have a slightly different perception of what should be considered “trivial”.
To Dmitry:
1) As I already tried to state in my previous comment (maybe not clearly enough), I agree that formulating the question and guessing the qualitative answer is trivial. But I am not so sure, whether working out quantitative consequences in a mathematically rigorous way, as done in [6], really is that trivial [please, see also "4)" below] – matter of taste I guess.
2) For applications and future directions please see (i) and (ii) of the previous comment.
To Lubos:
3) I agree that “non-local” physics (such as, e.g., thermodynamics) is determined by symmetries. In a globally Poincare-invariant theory this includes invariance under time translations and, hence, energy conservation. However, I am not sure that space-like hyperplanes “t=const” are a good choice with regard to extensions of, e.g., TD to GR (my view on this issue is pretty much summarized in http://arxiv.org/abs/0902.4651)
4) With regard to your “derivation” of the modified Juettner function: I agree that your simple argument captures the physical essence of transforming distributions based on different time parameters – but, given the fact that the transformation from coordinate-time to proper-time is trajectory-dependent, it seems to me more like a (very well motivated) guess rather than a satisfactory proof. Therefore, I prefer the mathematically rigorous derivation in [6], which also gives you the transformation behavior of various kinetic coefficients. But this is, perhaps, again just a matter of taste.
5) Generally, I think that your arguments work perfectly fine in SR, but they might run into difficulties in GR, where global time translation invariance and things like that become “fishy”. Personally, I would not exclude the possibility that including proper-time in QM or QFT could be a fruitful idea (unfortunately, I am aware of only a very limited number of useful papers dealing with proper-time in relativistic QM/QFT).
I think this is, perhaps, a good time to stop “defending” our work
Everybody is happily invited to read the papers and to ignore them if deemed too trivial … but perhaps one or two readers find at least some minor parts of the discussion worth reading.
Thanks again to both of you for your input,
Jorn
Dear Jorn,
Sorry if you felt offended, it was never my intention to criticize your work, the word “trivial” was really directed to Lubos
Thank you very much for your explanations anyway.
My impression is that using proper time in relativistic quantum mechanics is the essence of first quantized QFT. As long as you deal with free theory, it is easy to show that first and second quantizations are equivalent (which is done for example in Polyakov’s “Gauge fields and strings” in a nice way). If you want to consider an interacting field theory, then first quantized picture becomes a horrible mess. For example, how to describe the process of particle creation and annihilation? In a sense, this is the reason why perturbative string theory is also a mess and people fight for years to progress from lower order to the next higher order of the perturbation theory – that’s because second quantization of string theory (or proper string field theory in other words) is not yet invented.
Cheers and thanks again,
Dmitry.
P.S. I would add string theory at finite temperature to the list of applications and future directions
Dear Jorn, thanks for your answer. I think that you are being clear and I understand what you’re saying. But it still seems to me that you are neglecting they key difference between the “laboratory time” and “proper time”.
What’s the difference? The difference is that physics is actually symmetric under translations by a time interval as measured in the lab frame. On the other hand, there is no symmetry under “proper time translations”, is there?
Equivalently, using Noether’s map, only the normal Hamiltonian (or other generators of symmetries) is conserved, which means that I can require the quantity to be conserved, and require that the thermal density matrix commutes with it.
I think that none of these operations can be done with the “proper time translations”. How do you write the “proper time Hamiltonian” and what properties does it have? So let me say, I completely endorse your observation that if the proper time were used instead of the normal time, things formally become the modified distribution.
By the way, there’s a trivial derivation of the formula in this case. The extra 1/E is therebecause the exponential is essentially multiplied by the Lorentz-invariant phase space, a standard thing in field theory. It’s clear that if time is replaced by a Lorentz-invariant quantity, proper time, the old-fashioned 3D phase space is not allowed and must be replaced by the Lorentz-invariant phase space, with the extra 1/2E factor (the overall 2 is a convention), multiplying the now-Lorentz-invariant exponential (because the things in the exponent are defined to be Lorentz-invariant, too).
But this replacement is formal and the modified density matrix probably doesn’t have any important applications or ways to prepare the ensemble.
Let me tell you an example: Hagedorn density. The density of states in perturbative string theory is such that the partition sum, measured in spacetime with respect to lab time, diverges above some (Hagedorn) temperature. But if one used a different Hamiltonian, e.g. the worldsheet one – linked to some kind of proper time – the partition sum would never diverge.
It’s because the density of states goes like exp(sqrt(N).C) where N is the excitation level, which grows more slowly than exp(C’N) for any C’. So the properties are completely different. Does this counting mean that there is no special behavior near the Hagedorn temperature? Of course, it doesn’t: there is an important transition over there. But one must use the physical, spacetime temperature.
David (also thanks), indeed, the action at a distance is not allowed in relativity and a proper treatment dictates that one should start with fields. But this argument has nothing to do with the validity of my complaints, and it doesn’t allow you to circumvent any of the problems.
You can always imagine that you e.g. integrate out all the fields and get a particular, delayed interaction between the charged particles. There will exist a lag but this will play no role for the qualitative conclusions. For example, if you confine all the particles into a box, which is generally OK, the lag will always be bounded, and it can’t possibly influence the (absent) 1/E prefactor in the distribution at high energies.
Alternatively, I can work with the fields – with quantum field theory – from the beginning to the end, as particle physicists usually do. Still, it is possible to derive the thermal distribution from field theory. Field theory contains n-particle states so all the things I need are there. And even if you start with an interacting field theory and you want to determine the representation of states containing largely decoupled particles, you will confirm the original Juettner distribution.
My derivation did that, after all. It didn’t make any assumption about not having fields. You are fooling yourself if you say otherwise. If there is an argument that XY is wrong and you invent a possible “loophole”, you should also have a responsibility to check whether the loophole actually changes the result or not. If it doesn’t, the loophole is fictitious and your argument that it is relevant is wrong. If you had done this procedure, you would find out that the “loophole”, if one admits its existence in the first place, is inconsequential.
You say that you don’t know why you would have to define the “proper time temperature” to justify the modified distribution. Is it a real question? You have to define the “proper time temperature” because it appears, in the form of beta=1/T, in this very distribution, doesn’t it? It’s some parameter and one must know what it means and how it can be in principle measured for the distribution – otherwise the distribution is just a physically meaningless function of ill-defined parameters.
The corresponding “proper” beta=1/T in the normal distribution can be measured by thermometers.
It’s very important and very physical, and unless you invent a thermometer that measure the beta in your modified Juettner distribution, I would argue that the distribution is unphysical – it is as random a distribution as any other function you can write down.
Well, the flaw I see in your argument is that you are using Hamiltonians. The usual interaction potentials of Quantum Mechanics are not strictly allowed in Special Relativity because they assume an instantaneous interaction at a distance. You need to introduce fields, and that complicates things a lot. As far as I know, a rigorous derivation of the Juettner distribution is still lacking.
Also, I don’t see the problem with the example you mention about a gas with decaying particles. If the particle decaying time is much longer than the equilibration time you can still use the Juettner distribution to account for the particle’s energy in the box at a random time, and the modified Juettner for its energy just before collapsing. I don’t see why one has to define a “proper time temperature” to justify the modified Juettner distribution. In a more complex setup the calculation would be much harder, but still one would have to use a proper-time parameterization.
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