Equilibrium Distribution Function of a Relativistic Dilute Perfect Gas F. Debbasch 14th September 2007 Abstract An alternative to the J¨ uttner distribution has been recently proposed by several authors. The litterature on the topic is reviewed critically. It is found that the J¨ uttner distribution is correct and that the alternative distribution contradicts quantum field theory, statistical physics and continuum mechanics.

1

Introduction

F. J¨ uttner was the first to derive the equilibrium distribution function of a special relativistic dilute perfect gas [16]. This result has become one of the cornerstones of relativistic statistical physics and it also has been used in countless applications [7, 1, 4]. Recently, various authors (see [20, 19, 9, 10] and references included therein) have nevertheless chalanged the correctness of J¨ uttner distribution and proposed an alternative equilibrium distribution for special relativistic dilute perfect gas. Some of the authors [20] who introduced the alternative distribution even challenge the validity of standard Bose-Einstein (BE) and Fermi-Dirac (FD) distributions in the relativistic realm. The aim of this article is to review critically all arguments in favor of both the J¨ uttner and the alternative distribution. Our main conclusion is that the J¨ uttner distribution is correct and that the alternative one is not. Indeed, we recall that the J¨ uttner distribution can be derived from quantum field theory as the dilute gas limit of the BE and FD distributions and that it is also consistent with all principles of statistical physics and continuum mechanics; on the contrary, the alternative distribution and its related modified BE and FD distributions are invalidated by observations of black body radiation and contradict Gibbs’ statistics, quantum field theory and some basic principles of continuum mechanics.

2 2.1

Why the Ju ¨ ttner distribution is correct I: Non quantum arguments General framework

Kinetic theory describes a perfect gas through the so-called one-particle distribution function in phase-space. 1

Let R be an arbitary Lorentz frame R, with associated coordinates (tR , xR , pR ), the natural reference measure to be used in one particle phase-space is dmR = d3 xR d3 pR /h3 . The statistical law obeyed in R by the independent particles of the gaz can be caracterized by its density f R with respect to dmR and this density is what physicists call the one-particle distribution function (or, more cursively, one-particle distribution). The one-particle distribution f R is thus defined by X R 3 R R R f R (tR , xR , pR ) =< δ 3 (xR − xR (1) i (t ))δ (p − pi (t )) > i

where the sum extends over all particles in the system and angular brackets denote statistical averaging. It can be proven by an explicit computation [7, 8] that this definition actually makes f R independent of R and that the repartition of particles in phase-space is thus caracterized in an invariant manner by a single, Lorentz frame-independent distribution f ; this is true for both in and out of equilibrium situations; general consequences are critically examined in [2]. Suppose now that the local macroscopic state of the gaz is described by a conserved particle current J and a stress-energy tensor T . These fields can be evaluated from f through the following relations [14]: Z pµ f (x, p), (2) J µ (x) = dS mc M Z pµ ν µν T (x) = dS p f (x, p), (3) mc M where M is the mass-shell, defined by p2 = pµ pµ = m2 c2 and dS is the scalar measure associated to the metric gM induced by the space-time metric g on M. Given a Lorentz frame R, one has dS = (mc/h3 )d3 pR /pR 0 . Equation (2) then transcribes into the familiar definitions: Z d 3 pR R R ρ (x ) = f (xR , pR ) (4) 3 M h and JR (xR ) =

Z M

d3 pR pR f (x, p), h 3 pR 0

(5)

where ρR and JR are respectively the particle density and 3-particle current in 3-D physical space. As for equation (3), it transcribes into the usual statistical definitions for the energy density, pressure and stresses of a perfect gaz. Because of definition (1) for f R , the spatial volume element to be used in conjunction with ρR and JR is d3 xR ; this element is represented in 4-dimensional language by the 4-surface element dΣµ with components (d3 xR , 0, 0, 0) in R[14]. Let F be an arbitrary distribution in phase space. The conditional entropy of f with respect to F is associated to a current Sf /F defined by:   Z pµ f (x, p) Sfµ/F (x) = − dS f (x, p) ln . (6) mc F (x, p) M The Boltzmann-Gibbs entropy of f is then simply the conditional entropy of f with respect to the uniform distribution F = 1. Note that this last distribution cannot be converted into a measure because both the mass-shell and the Lorentz dependent flat 3-D physical space are not compact. 2

2.2

Gibbs statistics leads to the Ju ¨ ttner distribution

Applying Gibbs statistics to single particles is a standard way to obtain the usual Maxwell distribution describing the equilibrium states of non relativistic perfect gases [17]. Gibbs principle can also be applied to the relativistic framework presented above. Let Re be the global Lorentz rest frame of gas in equilibrium. By Gibbs principle, the equilibrium one particle distribution

function of a relativistic perfect gas coincides with exp µ − ε(pRe ) /(kB T ) , where µ is the chemical potential and ε(pRe ) is the energy of the particle in Re . This distribution is precisely the distribution proposed by J¨ uttner in 1911. The same distribution can also be obtained by extremizing the Boltzmann-Gibbs entropy ((6) with h = 1) of the gas [14].

3

Why the Ju ¨ ttner distribution is correct II: Quantum arguments

Quantum relativistic statistical Physics is best discussed within the framework of Quantum Field Theory (QFT). Indeed, out of equilibrium dynamics and relaxation towards equilibrium are driven by interactions and QFT is at the moment the sole theoretical descrition of quantum interactions consistent with the principles of both special and general relativity. Special relativistic perfect gases in equilibrium correspond to a two-fold idealization: (i) first, that there is a Lorentz frame in which system has reached a substantially time-independent state at finite temperature (ii) that the system is dilute enough to make interactions, in the equilibrium state, negligible. This section is made of two separate subsections. The first one reviews the quantum theory of a free scalar field at vanishing temperature. The major motivation behind this first subsection is to present how macroscopic fields like a charge current J and a stress-energy tensor T appear in the quantum framework. The first subsection also sets the tone for the following one, which reviews quantum field theory at non vanishing temperature. This second subsection discusses how the Bose-Einstein (BE) and Fermi-Dirac (FD) distributions are derived by applying Gibbs’ statistics to quantum field theory, and how the J¨ uttner distribution appears as a limit case of both the BE and FD ones.

3.1 3.1.1

Quantum field theory at vanishing temperature In a single Lorentz frame with spatially periodic boundary conditions

As its very name indicates, Quantum Field Theory is a quantum theory of fields, not of particles and the particle concept is well-known to have only a limited usefulness, even in flat space-time [5, 22]. Let us restrict for the moment the discussion to a free field in flat Minkovski space-time. The classical dynamics of this field is covariantly described by an action S. The simplest and historical way to quantize the field is to use the so-called canonical quantization scheme [15, 6], whose standard implementation makes it mandatory to derive an Hamiltonian from the action. This necessitates the introduction of a time. Among the physically reasonable options, the most practical one [15, 6] is to choose as time the time-coordinate tR of an arbitrary inertial or Lorentz frame R. This choice 3

makes sense physically because the time-coordinate of a Lorentz frame is an observable quantity; this is also the most practical choice because the matrix formed out of the components of the Minkovski metric in an inertial coordinate basis is identical to the unit matrix. Note that there is no way to associate a proper time to a classical or a quantum field. The concept of proper time is intrinsically linked to the particle concept and particles are not the fundamental objects of QFT. Choosing a ‘proper time’ to derive the Hamiltonian of a field from its action is therefore out of the question. Choosing the time-coordinate tR of a certain Lorentz frame R as time, one can derive from the action S a Hamiltonian H R , together with a 3-vector PR ; these represent what an observer at rest in R would interpret as the total energy and total 3-momentum of the field. It is a standard result of field theory [3] that (H R , PR ) are the components in R of a 4-vector P ; thus, if R0 is another 0 0 Lorentz frame, the quantities (H R , PR ) can be derived from (H R , PR ) by the standard Lorentz transformation linking R to R0 and P represents the total energy-momentum of the field. Note that P is an intrinsic quantity which depends only on the field and the Minkovski metric, and not on any Lorentz frame. Canonical quantization can be performed using H R as Hamiltonian [15, 6]. To make the discussion as precise as possible, consider the simplest case of a free real scalar field. The scalar field can be Fourier analyzed spatially in R and the Fourier mode associated to each wave vector kR is a harmonic oscillator p R 2 4 of frequency ω(k ) = m c /~2 + kR · kR , where m is the mass of the field. The free scalar field quantized in R is thus a collection of quantized harmonic oscillators [11, 15]. It is important at this stage to specify the set KR of values allowed to kR . The set KR depends on the boundary conditions used for the field. The simplest ones [11, 15] impose spatial periodicity of the field in R; these are sometimes refered to by saying that the field is enclosed in a rigid box at rest in R; let Lx , Ly , Lz be the length of the sides of the box, as measured in R. One then has KR = {kR = (2πqx /Lx , 2πqy /Ly , 2πqz /Lz ) , (qx , qy , qz ) ∈ N3 }; in particular, KR is then discrete. The energy eigenstates of the quantum field in R decompose as direct products of the energy eigenstates of each separate oscillators. Since a quantized harmonic oscillator of frequency ω has eigenstates [11] indexed by the set of positive integers, the eigenstate n having the energy εn = (n + 1/2)~ω, the eigenstates of the field quantized in R are completely determined by a set of integers {nR (kR ), P kR ∈ KR }. The value of H R corresponding to an eigenstate R R R R R is simply H = kR ∈KR n (k )~ω(kP). Note that the eigenstates of H R R R R R are eigenstates of P too, since P = kR ∈KR n (k )~k . These states will hereafter simply be called the eigenstates of the field in R. The particle content of the theory is recovered by interpreting, for each kR ∈ KR , the integer nR (kR ) associated to a certain eigenstate as the number of particles with energy ~ω(kR ) and momentum ~kR that an observer at rest in R detects when the field is inPthat eigenstate. The total number of particles for observers at rest in R is kR ∈KR nR (kR ). Note that their is no constraint on nR (kR ); in particular, this number can be superior to unity; this means that the particles corresponding to a real scalar field are bosons and a real scalar field is therefore qualified as bosonic. As well known, fields of integer spins are bosonic and fields of half-integer spins are fermionic (i.e. for such fields, the number of particles per state cannot exceed unity). 4

3.1.2

Comparing different Lorentz frames

The periodic boundary conditions used in the preceding section can be relaxed by formally letting Lx , Ly and Lz tend to infinity. The set KR is then simply R3 and this limit will be called from now on the continuous limit. As the lengths Lx , Ly and Lz tend to infinity, all discrete summations over kR are better and better approximated by integrals against the measure V R d3 pR /h3 defined in the R one-particle momentum space MR = R3 (here, V R is the physical 3-volume of the system in R). The corrresponding measure in the R one-particle phase-space P R is dmR = d3 xR d3 pR /h3 . The particle and momentum spatial densities in R then read: Z d3 p R NR = n (p) (7) VR h IR3 and

PR = VR

Z IR3

d3 p p nR (p). h

(8)

It is a standard result [15, 13, 22] of QFT that, for any two Lorentz-frame R and R0 , there is in the continuous limit a one-to-one correspondance between the eigenstates of the field in R and the eigenstates of the field in R0 . This correspondance is expressed in particle language by saying that a particle with 0 3-momentum ~kR in R corresponds to a particle with 3-momentum ~kR in 0 R0 , the link between ~kR and ~kR being best expressed by stating that the 4 0 quantities (~kµR ) = (~ω(kR ), ~kR ) are connected to the 4 quantities (~kR µ0 ) = 0

R0

(~ω(kR ), ~k0 ) by the standard Lorentz transformation connecting R and R0 . The 4 quantities (~kµR ) are then interpreted as the 4 components in R of a particle 4-momentum p = ~k, defined independently of any Lorentz frame. Momentum space M is also Lorentz frame independent and continuous, and is called the mass-shell. Remark that the number nR (k), now denoted by n(p), is also R-independent. As well-known [15, 7, 14], one can construct an invariant, Lorentz frame independent volume measure dS on the mass-shell; this measure is the volume associated to the metric induced on the mass-shell by the flat Minkovski spacetime metric; given an arbitrary Lorentz frame R, a standard calculation leads to dS = (mc/h3 )d3 pR /pR 0 and equation (7) becomes: NR = VR

Z dS M

pR 0 n(p). mc

Thus suggests to also consider the 3-vector Z N R VR pR = dS n(p). R V mc M

(9)

(10)

The four quantities (N R /V R , N R VR /V R ) are the four components in the Rcoordinate basis of a 4-vector J which represents the particle-current in spacetime. Equations (9) and (10) can be combined into: Z pµ Jµ = dS n(p) (11) mc M

5

Similarly, the stress-energy tensor T of the field reads: Z pµ ν µν T = p n(p) dS mc M

(12)

Thus, the QFT obtained from classical field theory by canonical quantization is, in the continuous limit, Lorentz invariant [15]; this is so even though the construction of a Hamiltonian for both classical and quantum fields depends on a choice of Lorentz frame. Note that the necessity of choosing a time is a general feature of all canonical approaches in Physics and that these approaches are nevertheless perfectly covariant. For example, there exist an hamiltonian formulation of general relativity [21]. The construction of the Hamiltonian itself can only be carried out after the space-time has been foliated into a family of time-dependent 3D space-like submanifolds, i.e. after a time-coordinate has been chosen in space-time, but the field equations, being those of general relativity, do not depend on the choice of foliation. Remark finally that path integral quantization [15, 21, 22] also requires the introduction of a foliation of the spacetime by a family of 3-D space-like submanifolds; path-integral formulations of QFT are nevetheless covariant.

3.2

Quantum field theory at finite temperature. Canonical and grand canonical treatments

Consider now a free real scalar field in equilibrium at finite temperature in Minkovski space1 . The physical notion of equilibrium entails that there exist a Lorentz frame Re where the quantities which characterize the dynamics of the system (be they the states of the system or the observables in Heisenberg representation) do not depend on the time-coordinate te . On the other hand, a system at thermodynamical equilibrium is described by a scalar density operator (i.e. a density operator common to all inertial observers) ρe and it is a standard result of statistical physics [11, 17] that, up to an additive constant, the logarithm of this density operator depends linearly on the additive integrals of motion only. Generically, the only such integral available is the 4-momentum P of the field. The fact that ρe is a scalar then entails that the equilibrium is fully characterized by a 4-vector β such that ln ρe = cβµ P µ + C, where C is a normalization constant. The quantity β 2 is interpreted [14] as the squared 2 2 inverse temperature 1/kB T and the unit 4-vector uµ = β µ /((β.β)1/2 ) as the 4-velocity of the Lorentz frame Re . It is then staightforward to evaluate the thermal averages of all physically interesting quantities. One finds that the thermal equilibrium of a free real scalar field in Minkovski space-time can be described in the particle language by stating [11, 18] that the average number of particles with 4-momentum p is given by the usual Bose-Einstein distribution: n ¯ BE (p) =

1 exp (ε(pRe )/(kB T ))

−1

,

(13)

where ε(pRe ) is the energy of the particle in Re . As well known, this distribution is actually recovered from finite temperature quantum field theory for all particles of integer spin or helicity. Particles of half-integer spin on the other-hand 1 All

following developments are carried out in the continuous limit

6

obey the Fermi-Dirac distribution: n ¯ F D (p) =

1 exp (ε(pRe )/(kB T ))

+1

.

(14)


High dilutions correspond to exp ε(pRe )/(kB T )  1 for all pRe ; in that case, the Bose-Einstein and the Fermi-Dirac distributions are both well approximated by the simple Boltzman-J¨ uttner distribution:
n ¯ BJ (p) = exp −ε(pRe )/(kB T ) . (15) Suppose now a conserved charge is associated to the field under consideration. The total charge Q then enters [17] the equilibrium density operator ρe . The resulting equilibrium particle distributions can be recovered from (13) and (14) by simply ‘replacing’ the 4-momentum p by q = p − µu, where µ represents the chemical potential of the particle system [11]. The Bose-Einstein and FermiDirac distributions then read: n ¯ µBE (p) = n ¯ µF D (p) =

1 , − µ)/(kB T )) − 1

(16)

1 exp ((ε(pRe ) − µ)/(kB T )) + 1

(17)

exp ((ε(pRe )

and the corresponing Boltzmann-J¨ uttner, a.k.a. J¨ uttner distribution, is:
n ¯ µBJ (p) = exp (µ − ε(pRe ))/(kB T ) .

(18)

Remark that the BE and FD statistics can also be derived directly from maximum entropy (or microcanonical) arguments applied to quantum fields or, when the notion of particle is relevant, to systems of N bosons or fermions [17]. As explained above in section 3.1.2, the particle current J is space-time is given in terms of n ¯ by Z pµ Jµ = dS n ¯ (p) (19) mc M and the stress-energy tensor T by: Z µν T =

dS

M

pµ ν p n ¯ (p). mc

(20)

The equilibrium Boltzmann-Gibbs entropy current is given by (see Section 2.1): Z pµ µ SBG (x) = − dS n ¯ (p) ln (¯ n(p)) . (21) mc M To sum up: a single scalar function can be used by all Lorentz observers as equilibrium one-particle distribution. This result, which actually extends beyond equilibrium situations, is perhaps the most important result of relativistic statistical physics and was derived here as a direct consequence of quantum field theory. As mentioned above in Section 2, a completely non quantum proof of the same result exists and can be found in [7, 8].

7

4 4.1

Why the alternative distribution is not correct Experimental evidence against the alternative distribution

As explained in the previous Section, the J¨ uttner distribution is simply the dilute gaz limit of both the BE and the FD distributions. Thus, stating that the J¨ uttner distribution is wrong in the relativistic domain is tantamount to stating that both the BE and the FD distributions are also wrong in this domain. But the validity of the BD distribution for relativistic particles has been confirmed experimentaly. For example, all standard calculations pertaining to black body radiation [17] are developed by applying the BE distribution to photons and the results of these calculations have all been firmly confirmed by numerous experiments. The results of these experiments thus de facto invalidate the alternative distribution.

4.2

Theoretical arguments in favor of the alternative distribution I: Universal scalar time parameter arguments

This approach is perhaps best illustrated by Wm. C. Schieve’s recent article (see [20] and references included therein). A relativistic continuous medium is modeled there as a gaz of N interacting particles and a universal scalar time is used to parametrize the evolution of the state of a system. The first conceptual objection against the whole approach is that no such time is accessible to observations and that physical theories should try as much as possible to avoid introducing objects which cannot be related, at least indirectly, to experiments. The real and apparently only reason to develop the approach presented in [20] is a historical one, namely the desire to treat relativistic interactions in the framework of action-at-a-distance theories. These theories are today widely considered unrealistic; indeed, not only does the theoretical framework used in [20] allow the (non quantum) particles to wander off their mass-shells, but action-at-a-distance theories do not seem to permit a theoretical treatment of the particle creation/anihilation phenomenon, which is naturally an experimental fact. It is well known that quantum field theory has been notably built to permit precisely a theoretical treatment of particle creations/anihilation and it is the best current model of matter, confirmed by countless experiments. Actually, quantum field theory is at the moment the sole theoretical description of quantum interactions consistent with the principles of relativity. As reviewed in Section 3.1.1, relativistic quantum field theory predicts that the particles associated to fields of integer spins are bosons and that the particles associated to fields of half-integer spins are fermions. Maximum entropy (microcanonical) arguments as well as canonical and grand canonical treatments then imply (see Section 3..2) that bosons and fermions obey respectively the BE and the FD statistics. As stated explicitely in [20], the universal time/action-at-a-distance approach favored by Wm. C. Schieve leads to different distributions for particles of integer and half-integer spins; let us call these distributions the modified BE and FD distributions. The difference between these modified distributions and

8

the standard ones increases with the temperature and explicit computations have been done [20] in the limit where the ratio θ of the temperature to the rest mass of the particles tend to infinity. The problem with these new statistics is that the standard BE statistics derived from finite temperature quantum field theory has been experimentally confirmed in the limit where the ratio θ tends to infinity. Indeed, all standard computations concerning black body radiation are carried out with the BE distribution and are naturally confirmed by countless experiments. To sum up: the universal time approach has no theroretical justification, except the desire to treat relativistic interactions in the framework of action-ata-distance theories. These theories are today widly considered obsolete because they allow (non quantum) particles to wander off their mass-shells but do not seem to permit a theoretical approach to the particle creation/anihilation phenomenon. Moreover, the one-particle distributions predicted by the universal time approach differ from the standard BE and FD distributions predicted by QFT and it is well-known that the applicability of the standard BE distribution in the ultra-relativistic regime is confirmed by all experiments carried out so far on black body radiation. Thus, the universal time approach developped in [20] is not only theoretically inferior to any approach based on quantum field theory, but it is also invalidated by countless experiments.

4.3

Theoretical arguments in favor of the alternative distribution II: Lehmann’s equilibrium statistical mechanics

E. Lehmann’s article [19] shares with the litterature discussed above the desire to work outside of a quantum field theoretical framework and retains a single universal time to parametrize the evolution of a N -particle system; but [19] is sufficiently different from all other articles to warrant a separate discussion. Lehmann actually considers two possible choices of time-parameter. The first one, which he calls semi-covariant, is to retain as parameter the time-coordinate tR of an arbitrary Lorentz frame R. This choice is the one commonly made (see Section 2.1 above); when used properly, it naturally leads to the usual BE and FD distributions and thus, in the dilute gaz limit, to the J¨ uttner distribution (see Sections 2.2 and 3.2 of the present article). The other choice, which Lehmann calls covariant and apparently favors, is to retain as parameter the proper time of the particles constituting the system. There are two distinct arguments which speak against this second choice. The first problem with the so-called covariant choice of time is simply that it cannnot be extended to develop a statistical treatment of quantum fields (see above, Section 3). The second problem is an experimental one: it is simply impossible to have experimental access to any physical quantity describing the macroscopic state of a gaz at fixed common value of the proper-times of all particles constituting the gaz. Indeed, only quantities taken at fixed value of the time-coordinate tR of some Lorentz frame R can be measured in real (as opposed to Gedanken) experiments. This reflects into the theoretical framework of continuum mechanics: given an arbitrary Lorentz frame R, densities of intensive variables in continuum mechanics are spatial densities at fixed time-coordinate in R and continuum mechanics does not even know of the proper-times of the particles constituting the system.

9

The choice of time-parameter Lehmann’s calls covariant is thus experimentaly irrealistic and any statistical mechanics based on this choice can be related, neither to observations nor to standard realistic continuum mechanics. But there is another difficulty with the work presented in [19]. Most calculations of the article are presented using what the author claims to be canonical ensembles of particles distributed in a continuous classical phase-space. For each choice of time, the author introduces a certain reference measure in phase space and defines these ensembles by densities of the classical form K exp(−βµ .pµ ) with respect to the introduced reference measure (here and below, K is a normalization constant). However, since the approach developed in [19] does not start from QFT, [19] has to introduce the reference measures by hand. Let R be an arbitray Lorentz frame, with inertial coordinates (tR , xR ). For both choices of time, the reference measure retained by [19] in momentum space is the standard scalar measure dS = (mc/h3 )d3 pR /pR 0 introduced in Section2.1 above. The reference measure in physical 3-D space depends on the choice of R time. For the so-called semi-covariant choice, [19] retains dVsc = d3 xR as refR 3 3 R 3 R erence measure in physical space, which leads to dµsc = (mc/h )d x d p /pR 0 as reference measure in phase-space; for the so-called covariant choice, [19] retains dVcR = mcd3 xR /pR 0 as reference measure in physical space, which leads 2 3 3 R 3 R 2 to dµR = (mc) /h d x d p /(pR c 0 ) as reference measure in phase-space. The crucial remark is that both reference measures differ from the reference measure dmR used in Section 2.1 above and derived in Section 3.1 by considering R R 2 R R R the continuous limit of QFT; indeed, dµR sc = dm /p0 and dµc = dm /(p0 ) . Thus, describing equilibium by assigning a density of the form K exp(−βµ .pµ ) R with respect to either dµR sc or dµc actually contradicts Gibbs’ statistics. Indeed, the canonical ensemble treatment assigns a weight of the form K exp(−βµ .pµ ) to each quantum state of a particle, and it is a standard result derived from the continuous limit of QFT (see Section 3.2 above) that, in any Lorentz frame R, a one-particle quantum state ‘occupies’ the volume dmR = d3 xR d3 pR , and R not dµR sc or dµc . The whole statistical treatment presented in [19] is therefore invalid. Naturaly, the correct canonical treatment is well known to deliver the J¨ uttner distribution (see Section 2.2). Lehmann also proposes a grand canonical treatment, but it suffers from the same inadequacies as the so-called canonical one.

4.4

4.4.1

Theoretical arguments in favor of the alternative distribution III: Collision simulations. Entropy arguments Collision simulations

Recently, P. H¨ anggi and J. Dunkel performed simultations of 1-D collisions between two types of particles and interpreted their numerical findings as demonstration that the J¨ uttner distribution is not correct and should be replaced by the alternative distribution [9]. The asumptions used in [9] are very general and do seem to leave ‘very little freedom for modifications such that one could hope to recover the standard J¨ uttner. distribution ’ [9]. We will now show that the numerical findings of [9], when properly interpreted, actually confirm the validity of the J¨ uttner distribution. Let us first review rapidly the argumentation presented in [9]. The main 10

starting point of this argumentation is (eq. (8) in [9]): Z Z Z dXg(X)ΦX (X) = dY dZg(X(Y, Z))ΦY (Y )ΦZ (Z),

(22)

where Y , Z are two independently distributed random variables, X(Y, Z) is ‘derived’ random variable, ΦX dX, ΦY dY , ΦZ dZ are the associated probability measures and g an arbitrary function. The authors then show that the validity of (22) for all g implies: Z ∂Z ΦX (X) = dY | | ΦY (Y )ΦZ (Z(Y, X)). (23) ∂X Consider now a collision between two particles of momenta p and P , and let pˆ and Pˆ be momenta of the particles after the collision2 . As well known, it is possible, for 1-D collisions, to obtain explicit expressions of both pˆ and Pˆ in terms of p and P ; following [9], we will denote these expressions, somewhat abusively, by pˆ(p, P ) and Pˆ (p, P ). These can be partially inversed into two functions P (p, Pˆ ) and p(P, pˆ) which will be of use later on. The authors of [9] apply (23) to Y = p, Z = P , X = Pˆ and to Y = P , Z = p and X = pˆ. At equilibrium, the law obeyed by Pˆ must be indentical to the law obeyed by P and the law obeyed by pˆ must be identical to the one obeyed by p. This leads to the following relations: Z ∂P ˆ | φ(p)Φ(P (p, Pˆ )), (24) Φ(P ) = dp | ∂ Pˆ Z ∂p φ(ˆ p) = dP | | Φ(P )φ(p(P, pˆ)), (25) ∂ pˆ where φ(p)dp is the law obeyed by p and Φ(P )dP is the law obeyed by P . Stating these relations is equivalent (see equations (22) and (23)) to stating that, for all functions g and h, Z Z Z dPˆ g(Pˆ )Φ(Pˆ ) = dp dP g(Pˆ (p, P ))φ(p)Φ(P ) (26) and

Z

Z dˆ ph(ˆ p)φ(ˆ p) =

Z dP

dph(ˆ p(P, p))Φ(P )φ(p).

(27)

Numerical simulations presented in [9] indicate that equations (26) and (27) are solved by 1 exp(−βε(p)), (28) φ(p) = a ε 1 Φ(P ) = A exp(−βε(P )), (29) ε where a and A are two normalization constants, β is the inverse temperature and ε(p) is the energy in R of a particle with momentum component p in R. Note that ε(p) = p0 , the zeroth component of the 2-momentum associated to p. follow in this section the notations of [9]; the four quantities p, P , pˆ and Pˆ here represent 1-D spatial momentum components in an arbitrary but fixed Lorentz frame R. The measure normalization factor mc/h3 is also put to unity. 2 We

11

The distributions (28) and (29) are thus identical with one another and differ from the J¨ uttner distribution by a multiplicative factor 1/ε; this completes the evidence presented in [9] against the J¨ uttner distribution. We now recall that the one-particle distribution function f is a Lorentz scalar (see Sections 2 and 3). In equilibrium and the absence of force-field, f depends only on the momentum p of the particle and coincides with the Lorentz invariant function n ¯ (p) introduced in Section 3.2. We will now show that φ(p) and Φ(P ) used in [9] are not Lorentz invariant and, thus, do not coincide with the scalar distribution functions governing the equilibrium of the colliding particles. Consider equations (26) and (27). These equations are not Lorentz invariant if one supposes the functions φ and Φ to be Lorentz scalars. Suppose indeed φ and Φ to be Lorentz scalars and also choose as test quantities g and h two Lorentz scalars; transforming (26) and (27) to another Lorentz frame R0 then leads to: Z Z Z ˆ P0 0 0 ˆ 0 0 0 0 0 0 0 p0 0 P0 ˆ 0 0 ˆ 0 0 ˆ 0 dp dP g (P (p , P ))φ (p )Φ (P ) (30) d P g ( P )Φ ( P ) = 0 0 ˆ p0 P00 P0 and Z

pˆ0 0 0 0 0 0 dˆ p h (ˆ p )φ (ˆ p)= pˆ00

Z

P0 0 dP P00

Z

p0 0 0 0 0 0 dp h (ˆ p (P , p ))Φ0 (P 0 )φ0 (p0 ), p00

(31)

with g 0 (Pˆ 0 ) = g(Pˆ ), h0 (Pˆ 0 ) = h(Pˆ ), Φ0 (P 0 ) = Φ(P ), Φ0 (Pˆ 0 ) = Φ(Pˆ ), φ0 (p0 ) = φ(p) and φ0 (ˆ p0 ) = φ(ˆ p); comparing (30) with (26) and (31) with (27) makes the non invariance of both (26) and (27) manifest. Thus, φ and Φ are not Lorentz scalars; since the relativistic one-particle distribution is necessarily a Lorentz scalar (see paragraph containing equation (24) above), one concludes from the above calculation that the functions φ and Φ are not the one-particle distributions of the colliding particles. The easiest way to understand how φ and Φ are related to Lorentz invariant distribution functions is to introduce the scalar measure dS in momentum-space. Given an arbitrary Lorentz-frame R, one can write dSp = dpR /pR 0 and replace (26) and (27) by the Lorentz-invariant equations Z Z Z ˜ Φ(P ˜ Pˆ ) = dSp dSP g(Pˆ (p, P ))φ(p) ˜ ) dSPˆ g(Pˆ )Φ( (32) and

Z

˜ p) = dSpˆg(ˆ p)φ(ˆ

Z

Z dSP

˜ ˜ )φ(p). dSp g(ˆ p(P, p))Φ(P

(33)

˜ denote Lorentz-invariant one-particle distributions. Comparing where φ˜ and Φ (26)/(27) with (32)/(33) leads to ˜ φ(p) = p0 φ(p)

(34)

˜ Φ(p) = P0 Φ(p)

(35)

and The numerical results presented in [9] led the authors to (28) and (29), which transcribe into ˜ φ(p) = a exp(−βε(p)) (36) 12

and ˜ ) = A exp(−βε(P )) Φ(P

(37)

˜ are J¨ Thus, φ˜ and Φ uttner distributions and the results presented in [9], when properly interpreted, do confirm that the correct, Lorentz-invariant one-particle distribution of dilute relativistic gaz is the J¨ uttner distribution. 4.4.2

Entropy arguments

Let us now discuss another recent attempt [10] to justify the alternative distribution through a maximum entropy argument. Following [10], we will only deal, for simplicity reasons, with a flat space-time of (1 + 1) dimension, the extension to higher dimensions offering no real difficulty. The notations used in the present section are inspired by, but slightly different from the notations used in [10]. Let dµ and dν be two measures on R; the authors define what they call the relative entropy S[dµ | dν] by:   Z dµ (38) S[dµ | dν] = − dµ ln dν IR or, equivalently, Z S[dµ | dν] = −

dνfdµ|dν ln fdµ|dν ,

(39)

IR

where fdµ|dν is the density of dµ with respect to dν. Let R be an arbitrary inertial Galilean (i.e. non relativistic) frame and interpret R as the space of 1-D spatial momentum components p in R. Choose for dν the usual Lebesgue measure dp on R and search for the densities fdµ|dν which maximize the entropy S[dµ | dν] under the conditions Z dpfdµ|dν (p) = N (40) IR

and

Z dpfdµ|dν (p)ε(p) = E

(41)

IR

where ε(p) = p2 /2m is the energy of a free Galilean particle in R and N and E are two positive constants. It is well-known that one thus obtains as density the usual Maxwelian distribution which caracterizes Galilean thermal equilibrium. A first possible but apparently naive extension to Special relativity, is to conserve definition (39) for relative entropies, keep dµ identical the Lebesgue measure in p-space, and p still impose (40) and (41) but with the relativistic expression ε(p) = mc2 1 + p2 /(m2 c2 ) for the energy ε of a free particle. As recalled in [10], this choice leads to the J¨ uttner distribution. The authors then recall another well-known fact (see Section 2.1 above) i.e. that the Lebesgue measure dp in momentum-space is not Lorentz invariant, but that dS = dp/p0 is. They therefore argue that a proper relativistic treatment should use definition (39) of relative entropies with dν = dS and, accordingly, change dp into dS in the constraints (40) and (41) (keeping naturally p ε(p) = mc2 1 + p2 /(m2 c2 )). The extremalization calculation then delivers the alternative distribution as equilibrium distribution. 13

Let us now show that this second approach is not correct and that the first, apparently naive form of the constraints is both not naive and correct. Any conservative quantity is represented by a 4-current in space-time. For systems in equilibrium in flat space-time (and in the absence of external fields acting on the particles), all currents are actually position-independent. Choose for R be the global rest-frame of the system in equilibrium. The one-particle distribution is then the number n ¯ (p) introduced in Section 3.2, where p now stands for the spatial momentum component in R. The function n ¯ is necessarily even in p, dS = dp/p0 and the only non-vanishing components (in R) of the µ 3 three currents j µ , T µν and SBG given by equations (19), (20) and (21) read: Z j0 = n ¯ (p)dp, (42) R

T 00 =

Z

p0 n ¯ (p)dp =

IR

T

Z ε(p)¯ n(p)dp,

(43)

IR

11

Z = IR

and 0 SBG =

p i pi n ¯ (p)dp p0

(44)

n ¯ ln n ¯ dp.

(45)

Z IR

Let us recall the physical interpretion of these components. The quantity j 0 represents the particle spatial density in R and, as such, can be integrated against dx to deliver the total (scalar) number of particles N in the system. The energy spatial density (in R) is T 00 ; integrating T 00 against dx leads to the total energy E = N ε of the system in R. The components T 11 represent the 0 is the Boltzmann-Gibbs entropy spatial density in pressure of the gaz and SBG R; its integral over dx delivers the total (scalar) entropy N σ of the system. Equations (42), (43) and (45) prove that the correct measure to use in (39), (40) and (41) is the Lebesgue measure dp, and not the Lorentz invariant measure dS = dp/p0 . This is so because all extensive quantities, even scalar ones, are geometrically represented by a 4-current, and never by scalars (see Sections 2 and 3); for example, the particle number is represented by a 4-vector field j µ which is obtained by integrating pµ against the invariant (scalar) measure n ¯ dp/p0 ; the current j µ is thus the first, and not the zeroth moment of the distribution n ¯ . The particle spatial density N/V is the zeroth component of j µ and is therefore obtained by integrating p0 , and not 1, against the measure n ¯ dp/p0 . The factors p0 and p0 cancel and one obtains (42). This relation may not seem covariant if one imports from Galilean statistical physics the idea that the particle spatial density N/V is the zeroth moment of n ¯ . But Lorentzian geometry imposes that any scalar extensive quantity be represented by a 4current, and the particle spatial density is consequently a first moment of n ¯, which makes (40) covariant. As noted in [10], the extremization calculation with dν = dp does deliver the J¨ uttner distribution as equilibrium distribution of a relativistic dilute perfect gaz. This is naturally nothing less than the wellknown result (see Sections 2.2 and 3.2), which states that a microcanonical treatment of relativistic dilute prefect gaz leads to the J¨ uttner distribution as equilibrium one-particle distribution. 3 BG

stands for Boltzman-Gibbs

14

5

Discussion

We have recalled the various standard derivations of the J¨ uttner distribution and discussed all arguments presented recently in favor of a new, alternative, oneparticle distribution for special relativistic dilute perfect gaz. The arguments in favor of this new distribution fall into three categories. The first category entails arguments developed in the framework of action-at-a-distance theories. These arguments suffer from all the shortcommings of such theories; in particular, particles are allowed off their mass-shells and the proposed treatment cannot be extended to relativistic quantum fields. The alternative distribution is also derived from new, modified BE and FD statistics. The modified BE statistics is however invalidated by all existing experiments on black body radiation. The recent work of E. Lehmann constitutes a category unto itself and endavours to build a covariant treatment of relativistic statitical physics based on a covariant extension of Gibbs ensembles. The problems with this work are threefold. First, the treatment presented in [19] cannot be extended to relativistic quantum fields. Second, the various densities introduced by the author are not observable and cannot be put in correspondence with densities in relativistic continuum mechanics (and Maxwell Electromagnetism). Third, [19] uses wrong measures in one-particle phase-space and thus, construct ensembles which actually violate Gibbs’ statistics. The third category regroups refs. [9] and [10]. The first work [9] offers numerical simulations of 1-D relativistic collisions between non quantum particles; the authors deduce the new, alternative distribution from the results of these simulations through a theoretical interpretation involving a choice of measure in momentum-space. We have proven that the choice of measure retained in [9] leads to non-covariant equations and is therefore inadequate. We have also shown that the numerical simulations presented in [9], when interpreted through a proper choice of measure in momentum-space, do indeed confirm the validity of J¨ uttner distribution. The second work [10] relies on a maximum entropy argument. The argument is non quantum and necessitates a choice of measure in momentum-space. The authors envisage two possible choices. The first one leads to J¨ uttner distribution but is rejected by the authors on the ground that it is not covariant; as for the second choice, it leads to the alternative to J¨ uttner distribution. We have shown that the second and retained choice of measure in [10] is not compatible with the geometrical status of various fields - particle and entropy 4-current, stress-energy tensor- entering the local caracterization of a relativistic gaz. This choice is therefore not covariant and, therefore, inadequate. We have also proven that, contrary to appearances, the first choice of measure envisaged in [10] is perfectly covariant and that the whole argument developed by the authors turns out to be identical to the standard microcanonical justification of the J¨ uttner distribution. As a conclusion, let us recall that quantum field theory can be developed in curved [5, 12, 22] as well as in flat space-time, at both vanishing and nonvanishing temperatures. When quantum efffects are discarded, the theory leads to standard general relativistic statistical physics [7, 14], including standard general relativistic kinetic theory. These theories thus stand on a very firm background. If any future experiment or theory were to contradict traditional relativistic statistical physics, be it in flat or in curved space-time, it would 15

de facto contradict quantum field theory and thus, the best currently available model of matter, confirmed by countless experiments. The present author considers the event of such a dramatic development as highly unlikely.

References [1] A. Anile and Y. Choquet-Bruhat, editors. Relativistic Fluid Dynamics, volume 1385 of Lecture Notes in Mathematics, Berlin, 1987. Springer-Verlag. [2] R. Balescu. Relativistic Statistical Thermodynamics. Physica, 40:309–338, 1968. [3] A.O. Barut. Electrodynamics and classical theory of fields and particles. The Macmillan Company, 1964. [4] J. Bernstein. Kinetic theory in the expanding universe. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1988. [5] N.D. Birrel and P.C.W. Davies. Quantum fields in curved space. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1982. [6] N.N. Bogoliubov and D.V. Shirkov. Quantum Fields. jamin/Cummings Publishing Company, inc., Reading, 1983.

The Ben-

[7] S.R. de Groot, W.A. van Leeuwen, and CH.G. van Weert. Relativistic Kinetic Theory. Principles and Applications. North-Holland, Amsterdam, 1978. [8] F. Debbasch, J.P. Rivet, and W.A. van Leeuwen. Invariance of the relativistic one-particle distribution function. Physica A, 301:181, 2001. [9] J. Dunkel and P. H¨ anggi. One-dimensional nonrelativistic and relativistic Brownian motions: A microscopic collision model. Phys. A, 374:559–572, 2007. [10] J. Dunkel, P. Talkner, and P. H¨anggi. Haar measures, relative entropy and the relativistic canonical velocity distribution. arXiv:cond-math/0610045 v2, 2006. [11] R.P. Feynman. Statistical Mechanics. A set of Lectures. Addison-Wesley Publishing Company, inc., Reading, 1972. [12] S.A. Fulling. Aspects of Quantum Field Theory in Curved Space-Time, volume 17 of London Mathematical Society Student Texts. Cambridge University Press, 1989. [13] R. Haag. Local Quantum Physics. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1992. [14] W. Israel. Covariant fluid mechanics and thermodynamics: An introduction. In A. Anile and Y. Choquet-Bruhat, editors, Relativistic Fluid Dynamics, volume 1385 of Lecture Notes in Mathematics, Berlin, 1987. Springer-Verlag. 16

[15] C. Itzikson and J.B. Zuber. Quantum Field Theory. Mc Graw and Hill, New York, 1980. [16] F. J¨ uttner. Ann. Phys. (Leipzig), 34:856, 1911. [17] L.D. Landau and E.M. Lifshitz. Statistical Physics, volume 5 of Course of Theoretical Physics. Pergamon Press, first edition, 1959. [18] M. Le Bellac. Thermal Field Theory. Cambridge University Press, Cambridge, 1996. [19] E. Lehmann. Covariant Equilibrium Statistical Mechanics. J. Math. Phys., 47:023303, 2006. [20] W.C. Schieve. Covariant Relativistic Statistical Mechanics of Many Particles. Found. of Phys., 35(8):1359–1381, 2005. [21] R.M. Wald. General Relativity. The University of Chicago Press, Chicago, 1984. [22] R.M. Wald. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. The University of Chicago Press, Chicago, 1994.

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