Notes on thermodynamics in special relativity
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Notes on thermodynamics in special relativity
M. Przanowski ∗ and J. Tosiek † Institute of PhysicsTechnical University of Lodzul. Wolczanska 219, 90-924 LodzPoland
June 3, 2018
Abstract
Foundations of thermodynamics in special theory of relativity are considered. We argue thatfrom the phenomenological point of view the correct relativistic transformations of heat and absolutetemperature are given by the formulas proposed by H. Ott, H. Arzeli`es and C. Møller. It is shownthat the same transformation rules can be also found from the relativistic Gibbs distribution for idealgas. This distribution has been recently verified by the computer simulations. Phenomenological andstatistical thermometers in relativistic thermodynamics are analyzed.
Keywords : relativistic thermodynamics, the relativistic Gibbs distribution.
PACS numbers: 05.70.-a, 03.30.+p
The present paper is an extended and improved version of the previous work of one of us (MP) [1] andis devoted to the foundations of relativistic thermodynamics in special theory of relativity. Referring thereader to the next section for details we note here that there is a long standing problem concerning therelativistic transformation rules for heat and temperature. First, A. Einstein [2], M. Planck [3], K. vonMosengeil [4], W. Pauli [5], M. von Laue [6] and many others found transformation formulas (2.18) and(2.28). In 1952 A. Einstein changed his primary opinion in favour of new formulas (2.17) and (2.27) [7]which have been also found later on by H. Ott [8], H. Arzeli`es [9] and C. Møller [10]–[12]. (It is worthwhileto point out that in the first edition of Møller’s book in 1952 the ‘classical’ transformation rules (2.18)and (2.28) were considered to be correct and in the second edition from 1972 [12] the things have beenchanged and the formulas (2.17) and (2.27) have been given as the correct ones). Note that perhaps L.D. Landau and E. M. Lifshitz were the first authors who mentioned the transformation formula for T equivalent to (2.27) in the Russian edition of their famous monograph on statistical physics in 1951 (seeEq. (27.4) in [13]).But it is not the all story. P. T. Landsberg and his collaborators in some works [14]–[17] argue thatheat and temperature are Lorentz invariants i.e. (2.19) and (2.29) hold true and then in some other works[18]–[20] they claim that there does not exist any universal relativistic transformation of temperature.Recently this point of view has been strongly supported in [21], [22]. The idea that heat and temperatureare Lorentz invariants has been also analyzed by N. G. van Kampen [23].The above very brief review of development of opinions on the transformation rules for heat andtemperature in relativistic thermodynamics does not sound optimistically. It seems that the eventualrelativistic thermodynamics depends very much on accepted conventions and that there does not existany natural system of conventions which leads to a correct construction of relativistic thermodynamics.However, in our opinion it is not so. A strong confirmation of this opinion we have found in distinguishedpapers by C. Møller [10]–[11] and, recently in a nice work of M. Requardt [24]. In the mentioned worksit is argued that there exists a natural definition of thermodynamic work and then, by the first law of ∗ E-mail address:przan@fis.cinvestav.mx † E-mail address:[email protected] statistical thermometer . In fact one can find many ‘natural’ statisticalthermometers. One of them can be constructed with the use of the generalized principle of equipartitionof energy (5.12).However, if one searches for some parameter which independently of the reference frame characterizessystems being in thermal equilibrium then this parameter is certainly the empirical temperature T and this is what has been pointed out in [36]. From this point of view the natural statistical thermometeris the one defined by (5.16) (see also [36]).We can also define a statistical thermometer which measures the absolute temperature T of the idealgas at rest (see (5.16)) which is equivalent to the empirical temperature or another one which measuresthe absolute temperature T (see (5.15)). Many other possibilities are also acceptable (e.g. (5.17)).In conclusion we find that there are two notions of relativistic temperature. One of them follows fromthe zeroth law of relativistic thermodynamics and this is the empirical temperature , which is a rela-tivistic scalar and which can be identified with the absolute temperature T of a thermodynamic systemat rest. It seems that the analogous point of view concerning T is represented by A. Staruszkiewicz, whocalls T a ’relative scalar’, since it is measured in the proper frame of a thermodynamic system [42].The second one is a consequence of the second law of relativistic thermodynamics and this is the absolute temperature T of the transformation law (2.27). Although in nonrelativistic thermodynamicsone also meets these two notions of temperature, this is only relativistic thermodynamics which showsexplicitly the deep difference between them.We use the Einstein summation convention. The symbol p denotes either a momentum or a pressurebut from the context its meaning is always clear. The subindex ‘0’ corresponds to the proper frame K of the thermodynamic system. 2 Phenomenological relativistic thermodynamics
Consider a continuous medium in thermodynamic equilibrium. We choose a bounded domain of themedium of the volume V with respect to the proper inertial frame K . We assume also that the mediummoves with a constant velocity ~v with respect to the laboratory inertial frame K. The space axes of K and K are assumed to be mutually parallel. The spacetime metric is η ij = η ij = diag( − , − , − , +) . Then the coefficients Λ jk , j, k = 1 , , , x j = Λ jk x k are given by theformulasΛ µν = δ µν − v µ v ν v (cid:0) − γ ( v ) (cid:1) , Λ µ = v µ c γ ( v ) , Λ µ = v µ c γ ( v ) , Λ = γ ( v ) , µ, ν = 1 , , , (2.1)where γ ( v ) := q − v c . The energy–momentum tensor T jk for a continuous medium can be written in the form [12], [43], [44] T jk = ε u j u k + τ jk = T kj , (2.2)where ε is the energy density in the proper inertial frame K , u j = (cid:0) v µ c γ ( v ) , γ ( v ) (cid:1) is the 4–vector ofvelocity of K relative to K, and τ jk is determined by the stress tensor τ µν as follows τ jk = Λ jl Λ km τ lm , τ j = τ j = 0 ⇒ τ jk = Λ jµ Λ kν τ µν . (2.3)We put τ j = τ j = 0 because for a system in the rest the stream of strength τ µ = 0. Moreover, T = ε so that τ = 0.Inserting (2.1) into (2.3) and then (2.2) one quickly finds the total energy E and momentum P µ E = Z V T dx dx dx = γ ( v ) (cid:18) E + v ν v ρ c T νρ (cid:19) , (2.4a) P µ = 1 c Z V T µ dx dx dx = v µ c γ ( v ) E + v ν c (cid:20)(cid:0) γ ( v ) − (cid:1) v µ v ρ v − η µρ (cid:21) T νρ (2.4b)where T νρ := R V τ νρ dx dx dx . As yet we have not used the conservation law ∂T jk ∂x k = 0 . Since our thermodynamic system is inequilibrium this law in the proper frame K takes the form ∂τ µν ∂x ν = 0 . (2.5)With the use of (2.5) one gets [43] T µν = Z V τ µν dx dx dx = Z V ∂ (cid:0) τ µρ x ν (cid:1) ∂x ρ dx dx dx = − I ∂V τ µ ρ x ν n ρ d Σ , (2.6)where n ρ is the ρ ’s component of the unit outward normal vector to the boundary ∂V of V and d Σ isthe surface element of ∂V .By symmetrization of (2.6) we finally get T µν = − I ∂V (cid:16) τ µρ x ν + τ νρ x µ (cid:17) n ρ d Σ . (2.7)In particular in the case of ideal liquid one has T jk = ( p + ε ) u j u k − p η jk , τ µν = − p η µν , T µν = − p V η µν (2.8)3here p stands for the pressure. Tensor T jk in (2.8) can be rewritten in the form of (2.2) as follows T jk = ǫ u j u k + p ( u j u k − η jk )(compare [43], [44]).Substituting (2.8) into (2.4a) and (2.4b) we obtain E = γ ( v ) (cid:18) E + v c p V (cid:19) , (2.9a) P µ = γ ( v ) (cid:16) E + p V (cid:17) v µ c . (2.9b)Remembering that pressure is a Lorentz invariant i.e. p = p and the volume V = V γ ( v ) we can rewrite(2.9a) as follows E + pV = γ ( v ) (cid:0) E + p V (cid:1) . (2.10)From (2.9b) and (2.10) with (2.1) one concludes that (cid:16) P , P , P , E + pVc (cid:17) constitutes a 4–vector. Thiscrucial point has been considered by many authors [5], [10], [11], [24], [26], [27], [45], [46].We assume that the first law of thermodynamics for our thermodynamic system in its proper frame K has the usual form dE = δQ + δL , (2.11)where δQ is the amount of heat entering the system and δL denotes the thermodynamic work doneover the system. For reversible processes one has δQ = T dS , δL = − Z ∂V τ µν n ν dx µ d Σ (2.12)with T being the absolute temperature and S the entropy of the system. For example, in the case ofideal liquid, from (2.8) and (2.12) we obtain δL = p Z ∂V η µν n ν dx µ d Σ = − p dV . (2.13)Differentiating (2.4a), (2.4b) and applying (2.11) one gets the first law of thermodynamics in the inertialframe K dE = γ ( v ) dQ + γ ( v ) (cid:18) δL + v µ v ρ c d T µρ (cid:19) , (2.14a) dP µ = γ ( v ) v µ c δQ + γ ( v ) v µ c δL + v ν c (cid:20)(cid:0) γ ( v ) − (cid:1) v µ v ρ v − η µρ (cid:21) d T νρ . (2.14b)The notion ‘a reversible adiabatic process’ should be independent of the choice of a system of frames.Consequently we claim that: δQ = 0 if and only if for any inertial frame K the heat δQ suppliedto the thermodynamic system vanishes i.e. δQ = 0. Therefore (2.14a) can be rewritten in the form dE = δQ + δLδQ = γ ( v ) δQ , δL = γ ( v ) (cid:18) δL + v µ v ρ c d T µρ (cid:19) (2.15)with δQ standing for the heat supplied to the system and δL denoting the thermodynamic work performedon the system with respect to the frame K. Employing also (2.14b) one can express the first law ofrelativistic thermodynamics in a 4–D form with δQ standing for the heat supplied to the system and δL denoting the thermodynamic work performed on the system with respect to the frame K. Employing also(2.14b) one can express the first law of relativistic thermodynamics in a 4–form dP j = δQ j + δL j , P j = (cid:18) P , P , P , Ec (cid:19) = (cid:18) ~P , Ec (cid:19) , Q j = δQ c u j , δL µ = γ ( v ) v µ c δL + v ν c (cid:20)(cid:0) γ ( v ) − (cid:1) v µ v ρ v − η µρ (cid:21) d T νρ , δL = δLc . (2.16)Thus we arrive at the 4–vector of heat δQ j and the 4–object of relativistic thermodynamic work δL j . The work δL j is not a 4–vector. The spatial part γ ( v ) δQ c ~v of the 4–vector of heat represents the changeof momentum d ~P caused by the transfer of heat. The 4th component of δQ j is proportional (with thecoefficient c ) to the change of energy following from the heat transfer.Formulas (2.15) or (2.16) lead to the relativistic transformation of heat found by H. Ott [8] andindependently by H. Arzeli`es [9] and C. Møller [10], [11] δQ = γ ( v ) δQ (2.17)(see also [12], [24]).This transformation rule differs dramatically from the one given by A. Einstein [2], M. Planck [3], K.von Mosengeil [4], W . Pauli [5] and M. von Laue [6] δQ (Planck) = 1 γ ( v ) δQ . (2.18)(We must note that, as has been pointed out by Ch. Liu [7] after his studies of the letters betweenEinstein and Laue, A. Einstein in 1952 changed his opinion on the validity of (2.18) in favour of (2.17)).If we mention also that by some authors the heat supplied to be the system is considered to be aLorentz invariant (see P. T. Landsberg and his collaborators [14]–[17] and N. G. van Kampen [23]) δQ (L) = δQ (2.19)then one obtains variety of approaches to the problem of relativistic transformation of heat. However, inour opinion according to (2.14a) the most natural transformation rule is given by Eq. (2.17). Ourconviction comes from the fact, that the infinitesimal heat δQ and the infinitesimal work δL must be ofthe form (2.15).The next question is: what about the relativistic transformation of temperature? Before we considerthis question we write down the first law of relativistic thermodynamics (2.14a), (2.14b) in the case ofideal liquid. Inserting (2.8) and (2.13) into (2.14a) and (2.14b) one gets dE = γ ( v ) δQ − pdV + (cid:0) γ ( v ) (cid:1) v c V dp, (2.20a) d ~P = γ ( v ) ~vc δQ + (cid:0) γ ( v ) (cid:1) ~vc V dp. (2.20b)Hence δL = − pdV + (cid:0) γ ( v ) (cid:1) v c V dp , δ~L = (cid:0) γ ( v ) (cid:1) ~vc V dp (2.21)(see [10], [11], [12], [24]).Consider now the relativistic transformation of temperature. In nonrelativistic phenomenologicalthermodynamics we meet two main notions of temperature. The first one is the empirical temperature introduced by the zeroth law of thermodynamics as a number determining the equivalence class of allthermodynamic systems being in thermal equilibrium. We assume that the zeroth law holds true also inrelativistic thermodynamics. Therefore the empirical temperature in relativistic thermodynamicsis a Lorentz invariant . We will identify it with the absolute temperature T in the proper frame. Thesecond notion, the absolute temperature , is introduced by the second law of thermodynamics as theunique (up to a constant factor) integrating factor of the Pfaffian form δQ depending only on theempirical temperature. This leads to the Clausius equality I δQ T = 0 (2.22)5or every cyclic reversible process. We assume that the second law of thermodynamics is in force inrelativistic thermodynamics in any inertial frame. Consequently, the absolute temperature of a thermo-dynamic system moving with a constant velocity ~v with respect to the frame K should be some smoothfunction of the form T = T ( T , v ) , lim v → T ( T , v ) = T (2.23)and the Clausius equality in K reads I δQT = 0 . (2.24)Substituting (2.17) and (2.23) into (2.24) we obtain I δQ (cid:0) γ ( v ) (cid:1) − T ( T , v ) = 0 . (2.25)Comparing (2.25) with (2.22) one arrives at the conclusion that (cid:0) γ ( v ) (cid:1) − T ( T , v ) = bT (2.26)where b is some constant. Taking the limit of Eq. (2.26) for v → b = 1so finally T = γ ( v ) T . (2.27)This is the relativistic transformation of absolute temperature consistent with (2.17) and the second lawof relativistic thermodynamics for reversible processes. This transformation rule is dramatically differentfrom the transformation law [2]–[6] T (Planck) = (cid:0) γ ( v ) (cid:1) − T (2.28)following from (2.18) or the transformation rule [14]–[17] T (L) = T (2.29)consistent with (2.19).In the next section we are going to give some justification for (2.17) and (2.27).Note also that by (2.12), (2.17) and (2.27) one gets δQ = T dS (2.30)which means that according to Planck [3] entropy is a Lorentz invariant S = S (2.31)(see also [10], [11], [27]). In 1911 F. J¨uttner [25] proposed a relativistic counterpart of the famous Maxwell distribution for ideal gasin its proper inertial frame K . The J¨uttner distribution had been accepted for many years [5], [26]–[30],but from the 80’s some authors have raised their objections to this distribution [31]–[35].The similar objections appear when the J¨uttner distribution is generalized to the case when a vesselcontaining gas moves with a constant velocity ~v with respect to the laboratory frame K. Hence, thenatural question arises if the J¨uttner formula and its generalization to the moving thermodynamic system(ideal gas) are correct. Nowadays we are not able to find any experimental solution of this question.Nevertheless, a partial answer comes from the recent outstanding works [36]–[41] where some computer6imulations have been presented. In Refs. [36], [37] the computer simulation has been performed for a1–D ideal gas being a two-component mixture of ideal gases. The numerical results show the perfectagreement with both: the 1–D J¨uttner distribution in the proper frame K and its generalization to themoving vessel. In [38]–[40] a 2–D ideal gas is considered and the numerical results of simulations areagain in the perfect agreement with the 2–D J¨uttner distribution in the proper frame K . Moreover,in [39], [40] the numerical results are shown to be in perfect agreement with the generalization of theJ¨uttner distribution for the moving vessel.In [41] some theoretical considerations show the advantage of the J¨uttner distribution over otherrelativistic distribution functions and the 3–D Monte Carlo simulations confirm the J¨uttner distribution.If so then it is an easy matter to get from these 1–particle distributions the corresponding relativisticGibbs distributions. Finally, the relativistic Gibbs distribution for an ideal gas contained in the vesselmoving with a constant velocity ~v and 4–velocity u j = (cid:0) γ ( v ) ~vc , γ ( v ) (cid:1) reads dw = 1(2 π ~ ) N N ! Z exp (cid:8) − βcu j P j (cid:9) d N p d N q (3.1)where N stands for the number of particles, β = kT with k being the Boltzmann constant, P j = (cid:16) ~ P , E c (cid:17) is the total momentum of the gas, d N p d N q is the phase space volume element and Z denotes thepartition function Z = V N (2 π ~ ) N N ! Z R N exp (cid:8) − βcu j P j (cid:9) d N p. (3.2)Remembering that the measure d p transforms as follows [27] d p = γ ( v ) ~v · ~p c p ~p + m c ! d p (3.3)one gets Z = V N (2 π ~ ) N N ! (cid:18)Z R exp (cid:8) − βcu j p j (cid:9) d p (cid:19) N == V N (2 π ~ ) N N ! Z R exp (cid:26) − βc q ~p + m c (cid:27) " γ ( v ) ~v · ~p c p ~p + m c ! d p ! N == V N (2 π ~ ) N N ! (cid:18)Z R exp (cid:26) − βc q ~p + m c (cid:27) d p (cid:19) N = Z (3.4)where, as in Sec. 2, the subindex ‘0’ corresponds to the proper frame K . Therefore, the partitionfunction Z is a Lorentz invariant [27]. Then the entropy of the gas S reads S = − k D ln (cid:18) Z exp (cid:8) − βcu j P j (cid:9)(cid:19) E = k (cid:0) ln Z + βcu j (cid:10) P j (cid:11)(cid:1) (3.5)with (cid:10) · (cid:11) denoting the expected value (average) with respect to the Gibbs distribution (3.1).Analogous calculations as those done by R. K. Pathria [26], [27] show that the gas pressure is a Lorentzinvariant i.e. p = p exactly as in the case of continuous medium from Sec. 2 and also that the followingformulas, which are closely related to (2.9a), (2.9b), hold true (cid:10) E (cid:11) = γ ( v ) (cid:18)(cid:10) E (cid:11) + v c p V (cid:19) , (3.6a) (cid:10) ~ P (cid:11) = γ ( v ) (cid:18)(cid:10) E (cid:11) + p V (cid:19) ~vc (3.6b)7see also [46]). Hence, as before, (cid:18)(cid:10) ~ P (cid:11) , (cid:10) E (cid:11) + pVc (cid:19) i.e. the average total momentum (cid:10) ~ P (cid:11) and the enthalpyof the gas divided by c namely (cid:10) E (cid:11) + pVc constitute a 4–vector.Employing (3.6a) and (3.6b) one quickly finds u j (cid:10) ~ P j (cid:11) = (cid:10) E (cid:11) c . (3.7)Thus, although (cid:10) ~ P j (cid:11) is not a 4–vector, the ‘scalar product’ u j (cid:10) ~ P j (cid:11) is a Lorentz invariant.Inserting (3.4) and (3.7) into (3.5) we obtain the result S = k (cid:0) ln Z + β (cid:10) E (cid:11) (cid:1) = S (3.8)which confirms the Planck formula (2.31).Differentiating (3.6a) and (3.6b) and performing the same calculations as in Sec. 2 one arrives at thefirst law of relativistic thermodynamics for ideal gas in the form (2.20a) and (2.20b) with E := (cid:10) E (cid:11) , ~ P := (cid:10) ~ P (cid:11) . (3.9)Hence, again we are led to the relativistic transformations of heat (2.17) and absolute temperature (2.27).However, we are going to give a slightly deeper insight into the problem by applying some statisticalconsiderations which are well known in nonrelativistic statistical physics and which lead from the Gibbsdistributions to the first law of thermodynamics [13].To this end we rewrite the relativistic Gibbs distribution (3.1) in a quantum form w n = 1 Z exp (cid:8) − βcu j P jn (cid:9) , n = 1 , , . . . (3.10)where the subindex ‘ n ’ denotes a quantum state. One assumes that the total 4–momentum eigenvalues P jn depend on some external thermodynamic parameters λ , . . . , λ s . Differentiating the formula ∞ X n =1 w n = 1 (3.11)with respect to T , λ , . . . , λ s , and performing some simple manipulations we get u j d (cid:10) P j (cid:11) = 1 kβc dS + u j (cid:10) d P j (cid:11) , (3.12) (cid:10) P j (cid:11) := ∞ X n =1 w n P jn , (cid:10) d P j (cid:11) := ∞ X n =1 w n s X l =1 ∂ P jn ∂λ l dλ l ! . Keeping in mind that u j = (cid:16) − γ ( v ) ~vc , γ ( v ) (cid:17) and P jm = (cid:16) ~ P m , E m c (cid:17) one can rewrite (3.12) in the followingform d (cid:10) E (cid:11) = T γ ( v ) dS + ~v · (cid:0) d (cid:10) ~ P (cid:11) − (cid:10) d ~ P (cid:11)(cid:1) + (cid:10) d E (cid:11) . (3.13)Assuming that, analogously as in the proper frame K the statistical definition of thermodynamic work δL in any inertial frame reads δL = (cid:10) d E (cid:11) (3.14)(see [47]) and comparing (2.15) with (3.13) we obtain δQ = T γ ( v ) dS + ~v · (cid:0) d (cid:10) ~ P (cid:11) − (cid:10) d ~ P (cid:11)(cid:1) . (3.15)8rom (3.15) with (2.12) and (3.8) it follows that statistical considerations give the transformation rule(2.18) for heat if either ~v · (cid:0) d (cid:10) ~ P (cid:11) − (cid:10) d ~ P (cid:11)(cid:1) = 0 or one decides to change Def. (3.14) of δL into δL → δL ′ = (cid:10) d E (cid:11) + ~v · (cid:0) d (cid:10) ~ P (cid:11) − (cid:10) d ~ P (cid:11)(cid:1) . (3.16)The latter possibility explains from the statistical physics point of view the reason why A. Einsteinin 1907, M. Planck in 1908 and others found transformation formula (2.18). Employing the detailedphenomenological analysis of C. Møller [10] one can expect that (cid:10) d ~ P (cid:11) = (cid:0) γ ( v ) (cid:1) ~vc V dp. (3.17)(As yet we have not been able to derive (3.17) using laws of statistical physics only).Then from (2.20b) with (3.9) and (3.17) we have ~v · (cid:0) d (cid:10) ~ P (cid:11) − (cid:10) d ~ P (cid:11)(cid:1) = γ ( v ) v c δQ . (3.18)Finally, inserting (3.18) into (3.15) one arrives at the transformation rule (2.17) of H. Ott, H. Arzeli`esand C. Møller.Moreover, (2.20a) and (2.20b) with (3.9), (2.12), (2.16), (3.13)–(3.18) lead to the Lorentz invariantformulation of the first law of relativistic thermodynamics d (cid:10) P j (cid:11) − (cid:10) d P j (cid:11) = T j dS , T j := T c u j . (3.19) T j is the 4–vector of temperature [10], [11]. One quickly finds that by (2.27) T = T = γ ( v ) T c = Tc . (3.20)(Analogous approach to the relation between relativistic statistical physics and relativistic phenomeno-logical thermodynamics had been presented by P. G. Bergmann in 1951 [48] (see also [49]) many yearsbefore computer simulations were done by D. Cubero it et al. [36], [37] and C. Rasinariu [38]).It seems that the statistical considerations of this section, which follow directly from the relativisticGibbs distribution (3.1) and consequently, from the J¨uttner distribution, give evidence that the correcttransformation rules for heat and temperature are given by (2.17) and (2.27) respectively. T First, using (2.12), (2.27) and (2.31) we rewrite (2.20a) and (2.20b) in the form dE = T dS − pdV + (cid:0) γ ( v ) (cid:1) v c V dp, (4.1a) d ~P = ~vc T dS + (cid:0) γ ( v ) (cid:1) ~vc V dp. (4.1b)From (4.1a) one quickly finds d (cid:16) E − (cid:0) γ ( v ) (cid:1) v c V p (cid:17) = T dS − (cid:0) γ ( v ) (cid:1) pdV. (4.2)Hence T = (cid:18) ∂E∂S (cid:19) V − (cid:0) γ ( v ) (cid:1) v c V (cid:18) ∂p∂S (cid:19) V , (4.3a)9 − (cid:0) γ ( v ) (cid:1) v c V (cid:18) ∂p∂V (cid:19) S = − (cid:18) ∂E∂V (cid:19) S (4.3b)and (cid:18) ∂T∂V (cid:19) S = − (cid:0) γ ( v ) (cid:1) (cid:18) ∂p∂S (cid:19) V . (4.4)Moreover, from (4.2) and (4.1b) we obtain T = (cid:0) γ ( v ) (cid:1) (cid:18) ∂E∂S (cid:19) V, ~P (4.5)(compare with formula (9) from the paper [27]).As we showed in Sec. 2, the sum E + pV transforms according to the rule (2.10). We identify it withenthalpy H. Hence we see that H = γ ( v ) H . (4.6)Applying (4.2) and definition of enthalpy we get T = (cid:18) ∂H∂S (cid:19) p . (4.7)Define the free energy F F := E − T S − (cid:0) γ ( v ) (cid:1) v c pV = γ ( v ) F (4.8)(the last equality follows from (2.27), (2.31) and (2.9a)). F = E − T S is the free energy of the systemin its proper frame K . Inserting (4.8) into (4.2) we get dF = − SdT − (cid:0) γ ( v ) (cid:1) pdV. (4.9)From (4.9) one finds S = − (cid:18) ∂F∂T (cid:19) V , (cid:0) γ ( v ) (cid:1) p = − (cid:18) ∂F∂V (cid:19) T , E = F − T (cid:18) ∂F∂T (cid:19) V − v c V (cid:18) ∂F∂V (cid:19) T (4.10)and one of the relativistic Maxwell identities (cid:18) ∂S∂V (cid:19) T = (cid:0) γ ( v ) (cid:1) (cid:18) ∂p∂T (cid:19) V . (4.11)We define the Gibbs function G in a standard way G := E − T S + pV = γ ( v ) G (4.12)(use (2.27), (2.31) and (2.9a)), with G = E − T S + p V . Differentiating (4.12) and substituting (4.1a) we obtain the formula dG = − SdT + (cid:0) γ ( v ) (cid:1) V dp (4.13)from which we get the relations S = − (cid:18) ∂G∂T (cid:19) p , (cid:0) γ ( v ) (cid:1) V = (cid:18) ∂G∂p (cid:19) T , E = G − T (cid:18) ∂G∂T (cid:19) p − (cid:0) γ ( v ) (cid:1) − p (cid:18) ∂G∂p (cid:19) T (4.14)and the Maxwell identity (cid:18) ∂S∂p (cid:19) T = − (cid:0) γ ( v ) (cid:1) (cid:18) ∂V∂T (cid:19) p . (4.15)10e conclude that the thermodynamic potentials H, F, G are defined like in a rest frame and satisfy thesame transformation rule.In the celebrated monograph of L. D. Landau and E. M. Lifshitz [13] the formula (4.15) with γ ( v ) = 1is employed to find the absolute temperature as a function of empirical temperature in nonrelativisticthermodynamics. Modifying slightly those considerations we can obtain an analogous result in relativisticthermodynamics.Namely, writing (cid:0) ∂V∂T (cid:1) p = (cid:16) ∂V∂T (cid:17) p dT dT and remembering that p = p , S = S , from (4.15) one gets dT dT = − (cid:0) γ ( v ) (cid:1) T (cid:16) ∂V∂T (cid:17) p T (cid:16) ∂S ∂p (cid:17) T . (4.16)This formula gives the derivative dT dT in terms of measurable quantities. Indeed, (cid:16) ∂V∂T (cid:17) p is determined bythe change of the volume V under the change of T for the constant pressure p. Moreover, T (cid:16) ∂S ∂p (cid:17) T isdetermined by the heat δQ supplied to the thermodynamic system so that under the change of pressure p the temperature T remains unchanged T (cid:18) ∂S ∂p (cid:19) T = lim ∆ p → (cid:18) Q ∆ p (cid:19) T . Assuming also that lim T → T = 0 (4.17)what is consistent with the third law of thermodynamics (note (2.31)) one finds that formula (4.16)determines T = T ( T , v ) . Therefore, formula (4.16) indicates an operational definition of the absolute temperature T. Another operational definition of the absolute temperature T is provided by the Clausius equation(2.24) when applied to an appropriate thermodynamic engine. This idea was presented by C. Møller [11]and then considered in Refs. [1], [24]. We remind briefly main points of Møller’s construction. Let R and R be two reservoirs. R is at rest with respect to the inertial frame K and R is at rest with respect tothe system K. The absolute temperature of R with respect to K is T and it is equal to the absolutetemperature of the reservoir R with respect to the system K. Let us consider a thermodynamic engine operating between the reservoirs R and R according to someCarnot cycle. The whole process is analyzed from the frame K . First, the engine absorbs isothermallythe amount of heat Q from the reservoir R at the absolute temperature T , being at rest with respectto the reference system K . Then the engine is accelerated adiabatically to the velocity of the frame K equal to − ~v. The amount of heat Q (with respect to the system K i.e. Q with respect to the frame K ) is released isothermally from the engine to the reservoir R at the temperature T (with respect to thereference frame K i.e. the temperature T with respect to the system K ). The final step is adiabaticdeceleration of the engine so that it returns to its initial state.The efficiency η of this cycle reads η = 1 − QQ = 1 − TT . (4.18)Consequently, by measuring η one finds transformation rules of heat and of temperature. Thus both(2.17) and (2.27) can be verified experimentally . (About the relation between the absolute tem-perature and the efficiency of the Carnot cycle see also [50]).Taking the scalar product of both sides of Eq. (4.1b) with ~v and subtracting the result from (4.1a)we get dE = (cid:0) γ ( v ) (cid:1) − T dS − pdV + ~v · d ~P . (4.19)11n the case of an ideal gas one can employ (3.18) and then (4.19) is brought to the form dE = γ ( v ) T dS − pdV + ~v · (cid:10) d ~P (cid:11) . (4.20)The quantity (cid:10) d ~P (cid:11) is interpreted as the mechanical momentum supplied to the system [10], [11].With this interpretation the formula (4.20) is true also for an ideal liquid. ( (cid:10) d ~P (cid:11) corresponds to ∆ ~J inRefs. [10], [11]). Observe that in general (cid:10) d ~P (cid:11) is not the differential of any thermodynamic function.Returning to identities in relativistic thermodynamics we should note that we have considered themunder the assumption that the number of particles N and the velocity of the thermodynamic system ~v are constant. Consequently, many of our formulas differ from the respective relations of relativisticthermodynamics presented in Ref. [51] (compare, for example, our (4.3a) with Eq.(15) of [51] ).It is a straightforward matter to generalize all results to the case when N and ~v are not constant.Differentiating (2.9a) and (2.9b) and employing the first law of thermodynamics in the rest frame dE = T dS − p dV + µ dN (4.21)where µ is the chemical potential with respect to the rest frame and N = N, after some simplemanipulations one gets dE = T dS − pdV + (cid:0) γ ( v ) (cid:1) v c V dp + (cid:0) γ ( v ) (cid:1) E + p V c ~v · d~v + µdN, (4.22a) d ~P = ~vc T dS + (cid:0) γ ( v ) (cid:1) ~vc V dp + (cid:0) γ ( v ) (cid:1) E + p V c (cid:20) d~v + 1 c ~v × ( ~v × d~v ) (cid:21) + µ ~vc dN, (4.22b)where T is given by (2.27) and µ = γ ( v ) µ . (4.23)From (4.22a) and (4.22b) one infers that analogously to the 4–vector of temperature T j (see (3.19) and(3.20)) we can introduce the 4–vector of chemical potential µ j = µ c u j ⇒ u = γ ( v ) µ c = µc . (4.24)Then, from (4.22a) and (4.22b) we quickly find that the thermodynamic work δL reads δL = − pdV + (cid:0) γ ( v ) (cid:1) v c V dp + (cid:0) γ ( v ) (cid:1) E + p V c ~v · d~v (4.25)and δ~L (compare with (2.21)) is δ~L = (cid:0) γ ( v ) (cid:1) ~vc V dp + (cid:0) γ ( v ) (cid:1) E + p V c (cid:20) d~v + 1 c ~v × ( ~v × d~v ) (cid:21) . (4.26)It is an easy matter to generalize all results obtained before. For example, in the present case the identity(4.13) reads dG = − SdT + (cid:0) γ ( v ) (cid:1) V dp + (cid:0) γ ( v ) (cid:1) E + p V c ~v · d~v + µdN. (4.27)Hence µ = (cid:18) ∂G∂N (cid:19) T,p,~v . (4.28)Writing G in the form G = N g ( T, p, ~v ) ⇒ g ( T, p, ~v ) = (cid:18) ∂G∂N (cid:19)
T,p,~v (4.29)and comparing (4.28) with (4.29) we get the formula well known in nonrelativistic thermodynamics G = N µ. (4.30)12hen the potential Ω := F − µN takes the form (use (4.8) and (4.12))Ω = F − µN = F − G = E − T S − (cid:0) γ ( v ) (cid:1) v c pV − (cid:0) E − T S + pV (cid:1) = − (cid:0) γ ( v ) (cid:1) pV. (4.31)Of course Ω = − γ ( v ) p V = γ ( v )Ω (4.32)and N = − (cid:18) ∂ Ω ∂µ (cid:19) T,V,~v = − (cid:18) ∂ Ω ∂µ (cid:19) T ,V = N . (4.33)Now we can easily find an identity which has been used by many authors as a starting point for relativisticthermodynamics [10], [27], [51]. Multiplying (4.22b) by ~v and subtracting from (4.22a) one gets dE = (cid:0) γ ( v ) (cid:1) − T dS − pdV + ~v · d ~P + (cid:0) γ ( v ) (cid:1) − µdN. (4.34)In this approach the energy E is considered as a function of ( S, V, ~P , N ) . A motion of the system withvelocity ~v increases its number of degrees of freedom. New conjugated parameters ~v and ~P appear.Variables S, V, ~P and N are natural variables for energy E. Unfortunately, neither (cid:0) ∂E∂S (cid:1)
V, ~P ,N is thetemperature nor (cid:0) ∂E∂N (cid:1)
S,V, ~P is the chemical potential.Then F = F ( T, V, ~P , N ) and Ω = Ω(
T, V, ~P , µ ). Note that from (2.9b) and (2.10) we have γ ( v ) = q ( E + p V ) + c ~P E + p V , (4.35a) E + pV = q ( E + p V ) + c ~P , (4.35b) ~v = c ~P q ( E + p V ) + c ~P . (4.35c)Keeping in mind that S = S , p = p ( S , V , N ) , V = (cid:0) γ ( v ) (cid:1) − V , N = N and E = E ( S , V , N ) , from (4.35a) and (4.35c) one finds that ~v as a function of ( S, V, ~P , N ) . This enables us to pass from(4.22a) where E = E ( S, V, ~v, N ) to (4.34) where E = E ( S, V, ~P , N ) . In distinguished papers on relativistic thermodynamics by R.K. Pathria [27] and H. Callen and G.Horwitz [46] it has been shown that it is more convenient and natural to use the independent variables(
S, p, N ) rather than (
S, V, N ) and the enthalpy H = E + pV instead of energy E. This is because
S, p and N are Lorentz invariants (but V is not) and ( ~P , Hc ) constitute a 4–vector (see Sec. 2).Adding d ( pV ) to both sides of (4.22a) one obtains dH = T dS + (cid:0) γ ( v ) (cid:1) V dp + (cid:0) γ ( v ) (cid:1) H c ~v · d~v + µdN. (4.36)Here H = H ( S, p, ~v, N ) . Analogously, adding d ( pV ) to both sides of (4.34) we have dH = (cid:0) γ ( v ) (cid:1) − T dS + V dp + ~v · d ~P + (cid:0) γ ( v ) (cid:1) − µdN, (4.37)where H = H ( S, p, ~P , N ) is given by (4.35b) H = q ( E + p V ) + c ~P = q H + c ~P . (4.38)13n particular, from (4.37) with (4.38) one finds V = (cid:18) ∂H∂p (cid:19) S, ~P,N = V H H = V E + p V q ( E + p V ) + c ~P , (4.39a) ~v = (cid:18) ∂H∂ ~P (cid:19) S,p,N = c ~PH = c ~P q ( E + p V ) + c ~P . (4.39b)As it can be seen, (4.39b) is exactly (4.35c).We end the present section with an important remark. Thermodynamic identity (4.22a) with (4.25)lead to the Ott–Arzeli`es–C. Møller transformation of absolute temperature (2.27) and to the trans-formation of chemical potential as given by (4.20). However, the identity (4.34) suggests the Plancktransformation of temperature (2.28) T (Planck) = (cid:0) γ ( v ) (cid:1) − T = (cid:0) γ ( v ) (cid:1) − T and the transformation ofthe chemical potential given by µ (Planck) = (cid:0) γ ( v ) (cid:1) − µ = (cid:0) γ ( v ) (cid:1) − µ under the assumption that thethermodynamic work reads δL ′ = − pdV + ~v · d ~P . (4.40)Substituting ~v · d ~P calculated from (4.22b) we obtain δL ′ = − pdV + (cid:0) γ ( v ) (cid:1) v c V dp + (cid:0) γ ( v ) (cid:1) E + p V c ~v · d~v + v c T dS + v c µdN. (4.41)Comparing (4.41) with (4.25) one concludes that the difference between the transformation rules of theabsolute temperature and the chemical potential derived from identity (4.22a) and the rules derivedfrom (4.34) is caused by the fact that δL ′ given by (4.40) cannot be interpreted as the relativisticthermodynamic work. In fact δL ′ consists of three groups of terms. The term − pdV + (cid:0) γ ( v ) (cid:1) v c V dp + (cid:0) γ ( v ) (cid:1) E + p V c ~v · d~v defines the thermodynamic work δL, the component equal to v c T dS corresponds toa part of relativistic heat supplied to the system and, finally, the part v c µdN is an element of dE whichcorresponds to the increment dN of the number of particles.If dN = 0 then a statistical interpretation of δL ′ is given by (3.16) with (3.14). In the preceding sections we have argued that in relativistic phenomenological thermodynamics onemeets two operationally well defined notions of temperature: the empirical temperature which can beidentified with the proper absolute temperature T and which is a Lorentz invariant, and the absolutetemperature T which follows from the second law of thermodynamics for reversible processes. Accordingto the Clausius equation (2.24) this absolute temperature can be experimentally determined as indicatedby measurement of the efficiency η (4.18) of an appropriate thermodynamic engine. We have also arguedthat the transformation rule for T is given by formula (2.27) and, consequently, T can be experimentallydetermined as it is indicated by Eq. (4.16).Now we are going to consider the question, how one can define temperature in relativistic statisticalthermodynamics of an ideal gas. To this end we employ the results of Sections 2, 3 and 4 to find somethermodynamic functions for the ideal gas.First, performing integration in (3.4) one gets Z = Z = 1 N ! (cid:18) m cV π ~ β K ( βmc ) (cid:19) N , (5.1)where K ( x ) denotes the modified Bessel function of the second kind of order 2. Its integral representationis of the form K ν ( x ) = Z ∞ e − x cosh t cosh νt dt, x > . (5.2)14nserting (5.1) into the thermodynamic equation (cid:10) E (cid:11) = − (cid:18) ∂ ln Z ∂β (cid:19) V (5.3)and employing the recurrence relation dK ν ( x ) dx + νx = − K ν − ( x ) (5.4)we easily obtain E ≡ (cid:10) E (cid:11) = 3 Nβ + N mc K ( βmc ) K ( βmc ) . (5.5)Substituting (5.1) and (5.5) into (3.8) and for large N applying the well known Stirling formula ln N ! ≈ N ln Ne one gets S = S = kN (cid:20) ln (cid:18) m c π ~ β V N K ( βmc ) (cid:19) + βmc K ( βmc ) K ( βmc ) + 4 (cid:21) . (5.6)From (3.6a), (3.8), (3.9) and (4.8) we have F = − kT ln Z = γ ( v ) (cid:0) − kT ln Z (cid:1) = γ ( v ) F . (5.7)Then from (4.10) with (5.1) one obtains p = p = kT (cid:18) ∂ ln Z ∂V (cid:19) T = N kT V = ⇒ p V = N kT . (5.8)Note that the formula (5.7) justifies the name ‘free energy’ for F (see Section 4, Eq. (4.8)).We are ready to propose a model of a ‘natural’ statistical thermometer. We start from the observationthat the average energy of the ideal gas in its proper frame K in nonrelativistic thermodynamics is givenby E nonrel . = lim c →∞ (cid:0)(cid:10) E (cid:11) − N mc (cid:1) (5.9)where (cid:10) E (cid:11) is defined by (5.5). Applying the principle of equipartition of energy (cid:28) p µ ∂H nonrel . ∂p µ (cid:29) = kT , ( do not sum over µ !) (5.10)where H nonrel . denotes the Hamilton function of a system, we immediately obtain T = E nonrel . N k . (5.11)By analogy to (5.11) one can introduce the statistical temperature in relativistic thermodynamics.The generalized principle of equipartition of energy in a moving frame for a fixed index j is of theform (see also [15], [39]) D p µ ∂H∂p µ E = kT γ + v µ (cid:10) p µ (cid:11) ( do not sum over µ !) . (5.12)Then in a similar way to (5.11) we propose T (st) := 13 N k (cid:18)(cid:10) E (cid:11) − N mc (cid:28) mc H particle (cid:29) − ~v · (cid:10) ~ P (cid:11)(cid:19) . (5.13)Thus T (st) = T (Planck) = T γ ( v ) . The statistical temperature T (st) measures the average kinetic energymodulo the contribution of translatory motion of the vessel. The term D mc H particle E = D γ particle ( v ) E with15 particle being the Hamilton function of an individual particle says how relativistic is the motion of thesystem. In the rest frame K from (5.13) we can reconstruct both: nonrelativistic T = E nonrel . Nk and theultrarelativistic T = P Ni =1 √ ~p i c Nk limits. Notice that there is some freedom of the choice of the factor inDef. (5.13). Instead of Nk we can put ( γ ( v )) r Nk . Thus for r = 1 the statistical temperature is an invariantand for r = 2 we obtain the transformation rule (2.27).Another possibility is suggested by the general formula (5.8). First, we recall that the pressure p isdefined as the average momentum transported per second per unit area through the surface element inthe direction given by the unit positive normal ~n to this element. It can be easily shown that using theabove definition one gets [27] p = p = NV D ( ~p · ~n )[( ~w − ~v ) · ~n ] E = NV D mγ ( w )( ~w · ~n )[( ~w − ~v ) · ~n ] E , (5.14)where ~w is the velocity of the particle with respect to the frame K , ~p as before is the particle momentumin the system K and ~v denotes the velocity of the vessel containing gas relative to K. From (2.27), (5.8) and (5.14), remembering that V = ( γ ( v )) − V we find T = ( γ ( v )) k D mγ ( w )( ~w · ~n )[( ~w − ~v ) · ~n ] E . (5.15)This formula shows that the absolute temperature T can be measured by the ‘statistical thermometer’defined by the right hand side of (5.15). However, the term ( γ ( v )) appearing in (5.15) is somewhatartificial and is taken into account only in order to reproduce the absolute temperature T. Other terms( γ ( v )) r can also be used.Thus one can find the temperature as a Lorentz invariant [15], [36] T (L) = T = γ ( v ) 1 k D mγ ( w )( ~w · ~n )[( ~w − ~v ) · ~n ] E (5.16)or the temperature transforming according to (2.28) [38] (moving system appears to be cooler) T (Planck) = 1 k D mγ ( w )( ~w · ~n )[( ~w − ~v ) · ~n ] E . (5.17)Analogous arbitrariness appears when one looks for the temperature of a moving black body by employingthe Planck distribution [18], [19], [20]. Acknowledgment
Authors wish to thank Dr. A. Montakhab and Dr. G. L. Sewell for indicatingsome interesting papers related to the present article.
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