Relativistic Invariance and Hamiltonian Theories of Interacting Particles

Currie, D. G.; Jordan, Thomas F.; Sudarshan, E. C. G.

doi:10.1103/revmodphys.35.350

Cited by 464 publications

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“…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.…”

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confidence: 99%

“…By assuming strictly localized, pointlike pair interactions, one may avoid the introduction of fields which are required when considering relativistic particle interactions at a distance (the interested reader may wish to consult the original papers of Wheeler and Feynman [9], Currie, Jordan, and Sudarshan [10], and Van Dam and Wigner [11,34], who discuss in detail the difficulties associated with classical particle-particle interactions in SR). However, considering pointlike localized interactions is expedient in the 1D case only; in higher space dimensions, the collision probability would become zero, thus preventing the system from equilibration.…”

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confidence: 99%

“…Further, our simulations corroborate Landsberg's hypothesis [45,46] that the temperature of classical gaseous systems can be defined and measured in a Lorentz invariant way. The extension of the MD approach to higher space dimensions is nontrivial, due to the fundamental difficulty of treating 2D and 3D two-body collisions in a relativistically consistent manner [9][10][11][12][13]. As a first step, it should be carefully analyzed if and how specific semirelativistic interaction models affect the 2D=3D equilibrium velocity distributions.…”

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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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Thermal Equilibrium and Statistical Thermometers in Special Relativity

et al. 2007

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“…(1) (see Figure 2). This is equivalent to the no interaction theorem [36] for classical particles because it has been proven that a 4−vector total energy-momentum cannot be defined for a system of interacting particles. Indeed, this represents Pryces major idea [37] which is equivalent to the no interaction theorem.…”

Section: The Total Energy-momentum Vector For a System Of Point Partimentioning

confidence: 99%

“…In the course of the article, the no interaction theorem [36] is proven for classical particles and the case of a system of interacting point charged particles is analyzed including the reaction force. …”

Section: Introductionmentioning

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A further analysis of the energy-momentum for a system of point particles and for a perfect fluid in finite volume

Parga

González-Narváez

2017

J. Phys.: Conf. Ser.

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Abstract. A simple analysis of the total energy-momentum of a system of non-interacting point particles shows that it represents a well defined 4-vector. A generalization for interacting charged particles including the self-forces is developed. A comparison between a system of point particles in an infinite volume and the perfect fluid in a finite volume is done. The finite volume case is analyzed in order to define an energy-momentum depending on the notion of instantaneity for a perfect fluid. The relativistic transformations of thermodynamic quantities are defined in a covariant form leading to a redefined relativistic Thermodynamics.

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An Example of a Consistent Relativistic Mechanics of Point Particles

Staruszkiewicz

1970

Annalen der Physik

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Relativistic Invariance and Hamiltonian Theories of Interacting Particles

Cited by 464 publications

References 25 publications

Thermal Equilibrium and Statistical Thermometers in Special Relativity

Thermal Equilibrium and Statistical Thermometers in Special Relativity

A further analysis of the energy-momentum for a system of point particles and for a perfect fluid in finite volume

An Example of a Consistent Relativistic Mechanics of Point Particles

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