Gibbs ensembles in relativistic classical and quantum mechanics

Horwitz, L. P.; Schieve, William C.; Piron, C.

doi:10.1016/0003-4916(81)90199-8

Cited by 99 publications

(127 citation statements)

References 9 publications

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“…As shown in Fig.1, the t-averaged (+) and τ -averaged (•) PDFs agrees well, respectively, with Jüttner function with β = 7.62 (dotted red line) and Jüttner function with the same β, divided by 'E' (dashed blue line). This result confirms the relation described in equation (9). The more important question, however, is the correct interpretation of the right hand side of this equation.…”

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confidence: 87%

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Time parametrization and stationary distributions in a relativistic gas

Ghodrat

Montakhab

2010

Phys. Rev. E

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In this paper we consider the effect of different time parameterizations on the stationary velocity distribution function for a relativistic gas. We clarify the distinction between two such distributions, namely the Jüttner and the modified Jüttner distributions. Using a recently proposed model of a relativistic gas, we show that the obtained results for the proper-time averaging does not lead to modified Jüttner distribution (as recently conjectured), but introduces only a Lorentz factor γ to the well-known Jüttner function which results from observer-time averaging. We obtain results for rest frame as well as moving frame in order to support our claim.

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supporting

confidence: 87%

“…Although Jüttner distribution has been generally accepted and used in high energy and astrophysics [2,3,4,5,6,7,8], some authors suggested several alternatives [9,10,11,12] that can be summarized in terms of the following η-parameterized probability distribution functions (PDFs),…”

mentioning

confidence: 99%

Time parametrization and stationary distributions in a relativistic gas

Ghodrat

Montakhab

2010

Phys. Rev. E

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“…The above conventions define the simplest interacting model system that (i) complies with all principles of SR, (ii) does not require the introduction of interaction fields, (iii) can be simulated without further approximation, and (iv) exhibits a universal stationary equilibrium state. Hence, this model system provides an optimal test case for probing the predictions of different relativistic kinetic theories by means of numerical experiments [5][6][7][8]14,16]. Moreover, as we shall see below, it helps to clarify long-standing controversial questions regarding the definition and meaning of temperature and thermal equilibrium in SR.…”

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confidence: 99%

Thermal Equilibrium and Statistical Thermometers in Special Relativity

et al. 2007

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There is an intense debate in the recent literature about the correct generalization of Maxwell's velocity distribution in special relativity. The most frequently discussed candidate distributions include the Jüttner function as well as modifications thereof. Here we report results from fully relativistic one-dimensional molecular dynamics simulations that resolve the ambiguity. The numerical evidence unequivocally favors the Jüttner distribution. Moreover, our simulations illustrate that the concept of ''thermal equilibrium'' extends naturally to special relativity only if a many-particle system is spatially confined. They make evident that ''temperature'' can be statistically defined and measured in an observer frame independent way. DOI: 10.1103/PhysRevLett.99.170601 PACS numbers: 05.70.ÿa, 02.70.Ns, 03.30.+p At the beginning of the last century, it was commonly accepted that the one-particle velocity distribution of a dilute gas in equilibrium is described by the Maxwellian probability density function (PDF)[m: rest mass of a gas particle; v: velocity; T k B ÿ1 : temperature; k B : Boltzmann constant; d: space dimension; throughout, we adopt natural units such that the speed of light c 1]. When Einstein [1,2] had formulated the theory of special relativity (SR) in 1905, Planck and others noted immediately that f M is in conflict with the fundamental relativistic postulate that velocities cannot exceed the light speed c. A first solution to this problem was put forward by Jüttner [3]. Starting from a maximum entropy principle, he proposed the following relativistic generalization of Maxwell's PDF:[Z J Z J m; J ; d : normalization constant; E m v m 2 p 2 1=2 : relativistic particle energy; p mv v : momentum with Lorentz factor v 1 ÿ v 2 ÿ1=2 , jvj < 1]. Jüttner's distribution (2) became widely accepted among theorists during the first threequarters of the 20th century [4 -8]-although a rigorous microscopic derivation is lacking due to the difficulty of formulating a relativistically consistent Hamilton mechanics of interacting particles [9][10][11][12][13]. Doubts about the Jüttner function f J began to arise in the 1980s, when Horwitz et al. [14,15] proposed a ''manifestly covariant'' relativistic Boltzmann equation, whose stationary solution differs from Eq. (2) and, in particular, predicts a different mean energy-temperature relation in the ultrarelativistic limit T ! 1 [16]. Since then, partially conflicting results and proposals from other authors [17][18][19][20][21] have led to an increasing confusion as to which distribution actually represents the correct generalization of the Maxwellian (1). For example, a recently discussed alternative to Eq. (2) is the ''modified'' Jüttner function [18,19] The distribution (3) can be obtained, e.g., by combining a maximum relative entropy principle and Lorentz symmetry [20]. Compared with f J at the same parameter values J MJ & 1=m, the modified PDF f MJ exhibits a significantly lower particle population in the high energy tail because of the additional 1=E pre...

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“…This set of 2N constraints given by (14), (21) and (22) determines covariant world lines parameterized by the initial data at equal τ .…”

Section: Micro-dynamics Of the Real Gasmentioning

confidence: 99%

“…Another interesting approach following possibility 1) is the one of Horwitz et al [14] which is based on the many-body theory of the relativistic mechanics proposed by Stückelberg [15]. The main characteristics of this method are the separation of the center of mass motion in the same way as in the non-relativistic theory and the allowance of the particles being off-shell resulting finally in a new potential corresponding to the mass degree of freedom of the relativistic system.…”

Section: Introductionmentioning

confidence: 99%

Covariant equilibrium statistical mechanics

Lehmann¹

2006

Journal of Mathematical Physics

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Abstract:A manifest covariant equilibrium statistical mechanics is constructed starting with a 8N dimensional extended phase space which is reduced to the 6N physical degrees of freedom using the Poincaré-invariant constrained Hamiltonian dynamics describing the micro-dynamics of the system. The reduction of the extended phase space is initiated forcing the particles on energy shell and fixing their individual time coordinates with help of invariant time constraints.The Liouville equation and the equilibrium condition are formulated in respect to the scalar global evolution parameter which is introduced by the time fixation conditions. The applicability of the developed approach is shown for both, the perfect gas as well as the real gas.As a simple application the canonical partition integral of the monatomic perfect gas is calculated and compared with other approaches. Furthermore, thermodynamical quantities are derived.All considerations are shrinked on the classical Boltzmann gas composed of massive particles and hence quantum effects are discarded.

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Gibbs ensembles in relativistic classical and quantum mechanics

Cited by 99 publications

References 9 publications

Time parametrization and stationary distributions in a relativistic gas

Time parametrization and stationary distributions in a relativistic gas

Thermal Equilibrium and Statistical Thermometers in Special Relativity

Covariant equilibrium statistical mechanics

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