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1 Entropy, 007, 9, Letter to the Editor entropy ISSN by MDPI Entropy of Relativistic Mono-Atomic Gas and Temperatre Relativistic Transformation in Thermodynamics Edward Bormashenko The College of Jdea and Samaria, The Research Institte, 44837, Ariel, Israel. Received: 19 April 007 / Accepted: 4 September 007 / Pblished: 11 September 007 Abstract: It is demonstrated that the entropy of the ideal mono-atomic gas comprising identical spherical atoms is not conserved nder the Planck-Einstein like relativistic temperatre transformation, as a reslt of the change in the nmber of atomic degrees of freedom. This fact spports the idea that there is no niversal relativistic temperatre transformation. Keywords: relativistic temperatre transformation, ideal gas, entropy, degrees of freedom. PACS nmbers: a, Tg. Introdction The problems of relativistic thermodynamics and statistics attracted significant attention recently [1 4]. The interest to the field was boosted by modern astrophysics investigations. One of the most debated problems is the relativistic temperatre transformation. The Planck-Einstein (PE) approach to the temperatre transformation reslted in the well-known relation [5]: = = c 1 T 1 0 = T = 1 T0 (1) c γ 1 dq = 1 dq0 = dq0 () c γ where γ γ ( ) 1 T and dq are the temperatre and the heat respectively. Ott in his early work conclded that the relativistic temperatre transformation is at least ambigos, becase the

2 Entropy, 007, energy transfer from one relativistic system cold be carried ot in two ways: nder the constant velocity and nder the constant momentm which are not proportional in the relativity theory [6]. Since that the relativistic temperatre transformation was debated extensively [7 13]. F. Anglo- Brown et al in his recent extended stdy of the problem have smmarized the attempts to obtain the relativistic temperatre transformation in sch a way: a T ( ) = γ ( ) T, (3) dq [ ] 0 a [ γ ( ) ] dq0 =, (4) where a is the Balesc parameter which is not known with certainty and a = 1, if the PE transformation is sed, and a = 1 nder the Ott considerations [1, 6, 8, 11]. The idea that the temperatre might be invariant was also sggested, ths the sitation a = 0, cold be considered as well [1]. On the other hand Landsberg and Matsas in their recent stdy asserted that a niversal continos Lorentz transformation of temperatre cannot exist [9, 10]. This was conclded on theoretical gronds that an observer moving in a heat reservoir cold not detect a blackbody spectrm. In or present letter we spply additional argments holding the viewpoint that the Lorentz transformation of the temperatre does not exist. Entropy of Ideal Gas and Temperatre Relativistic Transformation One of the basic argments spporting PE temperatre-heat transformations (1) () resided in the fact that they remain the entropy of the thermodynamic system constant: d Q d Q0 Δ S = Δ S0 =. (5) T T0 Indeed, the entropy defined according to S = lnw (we se the system of nits in which the Boltzmann constant k B = 1 [14 15]), is a dimensionless vale and has to be invariant relatively to the transformations of frames of reference [5]. We will demonstrate that actally sitation is more complicated and even the entropy of mono-atomic ideal gas is not constant nder PE transformations. Let s consider the ideal gas, comprising ideal identical spherical atoms (for example Helim or Argon) confined within a vessel from the viewpoint of the observer in frame XYZ which rests relatively to the vessel (Figre 1 A). An interaction between atoms is neglected, and in this meaning the gas is ideal. The entropy of the gas for this observer is given by: i S ( T, c, i) = N lnt N ln c + const, (6) where N and c are the nmber and particles concentration respectively, i nmber of degrees of freedom of the particle (i = 6 for spheres) [5]. Actally it is established experimentally that the molar heat capacity of Helim, Neon and Argon C v eqals (3/)R in a broad range of at temperatres, with a high accracy, ths, actally for these gases i = 3 [16]. Hence it is clear that rotational degrees of freedom are not excited for monoatomic gases. The reason for this cold not be explained in the realm of classical physics, bt instead reqires qantm mechanics argments discssed below. I will restrict the treatment with a classical (non-qantm) monoatomic relativistic gas comprising ideal spherical particles with i = 6. The entropy of the same gas established by the observer in frame

3 Entropy, 007, X'Y'Z' moving with velocity, may be changed for several reasons. First of all, the entropy depends on the temperatre and concentration of the particles. However, PE transformation provides: S'(c',T') = S(c,T) when i = const [5]. For or discssion it is mch more important that S = S(i), and the entropy will be changed as a reslt of the change in the qantity of degrees of freedom of the particle. The change of i is stiplated by the fact, that the moving observer will see the gas of ellipsoids (as depicted in Figre 1B) and not on ideal spheres, (de to the relativistic length contraction), ths for this observer i = 5 (three translational and two rotational degrees of freedom, becase the orientation of ellipsoids central axis is fixed), and i = i(γ()). Hence the entropy trns ot to be sensitive to the frames of reference transformations: S = S(γ()). It has to be emphasized that involving qantm mechanics argments comes to help and remedies the paradox. Indeed entropy defined according to the well-known Sackr-Tetrode formla still depends on the nmber of degrees of freedom of the particle: S = S(i) [14 15]. However the rotational degrees of freedom will be excited only for temperatres greater than T h r = 4π where h is the I Planck's constant, I is the moment of inertia of the particle (the Boltzmann constant k B = 1). These temperatre are nrealistically high for real gaseos systems, ths for monoatomic gases i =3 for both spherical and ellipsoidal particles, hence S trns ot to be insensitive to the frames of reference transformations There exist several ways to srmont the difficlty nder discssion within frameworks of special relativity: 1) to assme that the PE transformation and transformations (3) (4) are applicable for point particles only, ths no realistic thermodynamic system cold be described with them; ) to sggest that the qantity of degrees of freedom is somewhat like magic nmber and it is insensitive to the transformations of frames of references; 3) to introdce artificial relativistic temperatre transformation T = f ( i) T0 1, taking into accont change in the degrees of freedom of the c particles; 4) to assme that there is no niversal relativistic temperatre transformation. We incline to sppose that the last sggestion coinciding with the approach reported recently by Landsberg [9 10] is tre. Indeed relativistic temperatre transformation is faced with mltiple principle difficlties, already described, which seem to be insperable. On the other hand invoking qantm mechanics argments brings to the conclsion that nmber of atomic degrees of freedom is insensitive to the frames of reference transformations for monoatomic systems. Ths it cold be conclded that consistent classical relativistic thinking becomes impossible in this case.

4 Entropy, 007, z z O y O y x x A B Figre 1. Scheme illstrating entropy jmp stiplated by the change in the nmber of degrees of freedom. Acknowledgements The athor is thankfl to Professor A. Voronel, Dr. G. Whyman, Mr. A. Sheshnev for fritfl discssions, and to Mrs. Albina Msin for her help in the preparing of the manscript. The athor is thankfl to anonymos referees for thoghtfl refereeing of the manscript. References 1. Ares de Parga, G.; López-Carrera, B.; Anglo-Brown, F. A proposal of relativistic transformations in thermodynamics. J. Phys. A: Math. Gen. 005, 38, Lavagno, A. Relativistic nonextensive thermodynamics. Physics Letters A 00, 301, Banach, Z. An extended Gibbs relation for the relativistic Boltzmann entropy: classical and qantm gases. Physica A 000, 75, Banach, Z.; Larecki, W. Decomposition of entropy and the choice of non-eqilibrim variables for relativistic classical and qantm gases. Physica A 001, 93,

5 Entropy, 007, Tolman, R.C. Relativity, Thermodynamics and Cosmology, Oxford University Press: Oxford, Ott, H. Z. f. Phys. 1965, 175, Öttinger, H. C. Beyond Eqilibrim Thermodynamics, Wiley & Sons: Hoboken, New York, Sienitycz, S. Thermodynamic and relativistic aspects of thermal inertia in flids. Physical Review E 1998, 58(6), Landsberg, P.T.; Matsas, G.E.A. The impossibility of a niversal relativistic temperatre transformation. Physica A 004, 340, Landsberg, P.T.; Matsas, G.E.A. Laying the ghost of the relativistic temperatre transformation. Physics Letters A 1996, 3, Balesc, R. Relativistic statistical thermodynamics. Physica 1968, 40 (3), Landsberg, P. T. Does a moving body appear cool? Natre 1967, 14, Arias-Hernández, L.A.; Anglo-Brown, F. A general property of endreversible thermal engines. J. Appl. Phys. 1997, 81, Landa, L.D.; Lifshitz, E.M. Statistical Physics, Btterworth-Heinemann: Oxford, Kittel, C. Elementary Statistical Physics, John Wiley & Sons: New York, Pohl, R. W. Mechanic, Akstik and Wärmelehre, Springer: Berlin, Göttingen, by MDPI ( Reprodction is permitted for noncommercial prposes.