Covariant equilibrium statistical mechanics

Lehmann, E.

doi:10.1063/1.2165771

Cited by 48 publications

(61 citation statements)

References 31 publications

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“…The relativistic analogue of the Maxwell-Boltzman velocity DF has been proposed by Juttner (1911). However, alternatives to a Juttner DF have been discussed by Horwitz et al (1989) and, recently, by Lehmann (2006) and Dunkel & Hanggi (2007).…”

Section: Discussionmentioning

confidence: 99%

Can electron distribution functions be derived through the Sunyaev-Zel’dovich effect?

Prokhorov

Colafrancesco

Akahori

et al. 2011

A&A

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Aims. Measurements of the Sunyaev-Zel'dovich (hereafter SZ) effect distortion of the cosmic microwave background provide methods to derive the gas pressure and temperature of galaxy clusters. Here we study the ability of SZ effect observations to derive the electron distribution function (DF) in massive galaxy clusters. Methods. Our calculations of the SZ effect include relativistic corrections considered within the framework of the Wright formalism and use a decomposition technique of electron DFs into Fourier series. Using multi-frequency measurements of the SZ effect, we find the solution of a linear system of equations that is used to derive the Fourier coefficients; we further analyze different frequency samples to decrease uncertainties in Fourier coefficient estimations. Results. We propose a method to derive DFs of electrons using SZ multi-frequency observations of massive galaxy clusters. We found that the best frequency sample to derive an electron DF includes high frequencies ν = 375, 600, 700, 857 GHz. We show that it is possible to distinguish a Juttner DF from a Maxwell-Bolzman DF as well as from a Juttner DF with the second electron population by means of SZ observations for the best frequency sample if the precision of SZ intensity measurements is less than 0.1%. We demonstrate by means of 3D hydrodynamic numerical simulations of a hot merging galaxy cluster that the morphologies of SZ intensity maps are different for frequencies ν = 375, 600, 700, 857 GHz. We stress that measurements of SZ intensities at these frequencies are a promising tool for studying electron distribution functions in galaxy clusters.

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Section: Discussionmentioning

confidence: 99%

Can electron distribution functions be derived through the Sunyaev-Zel’dovich effect?

Prokhorov

Colafrancesco

Akahori

et al. 2011

A&A

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“…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.…”

mentioning

confidence: 99%

“…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 prefactor.…”

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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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“…The dotted (red) and solid (green and pink) lines are the x(y) component of velocity distribution obtained, respectively, by integrating Eqs. (12) and (13) over v y (v x ). We have also used the fact that the temperature parameters in the moving frame are the same as that of the rest frame [14,15].…”

mentioning

confidence: 99%

“…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),…”

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confidence: 99%

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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Covariant equilibrium statistical mechanics

Cited by 48 publications

References 31 publications

Can electron distribution functions be derived through the Sunyaev-Zel’dovich effect?

Can electron distribution functions be derived through the Sunyaev-Zel’dovich effect?

Thermal Equilibrium and Statistical Thermometers in Special Relativity

Time parametrization and stationary distributions in a relativistic gas

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