A manifestly covariant relativistic Boltzmann equation for the evolution of a system of events

Horwitz, L. P.; Shashoua, S.; Schieve, William C.

doi:10.1016/0378-4371(89)90471-8

Cited by 69 publications

(129 citation statements)

References 39 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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show abstract

“…Equilibrium distributions for ideal relativistic quantum gases were also derived by Jüttner [194] in 1928. A few recent papers [22,[195][196][197][198][199][200] have raised doubts about the correctness of Jüttner's results [186,194], but relativistic molecular dynamics simulations confirm Jüttner's prediction [201,202], cf. discussion in Sections 3.2.2 and 4.3 below.…”

Section: Historical Backgroundmentioning

confidence: 99%

“…Section 3.2 focuses on the thermostatistics of stationary systems, since these will play the role of a heat bath later on. In this context, particular emphasis will be placed on the relativistic generalization of Maxwell's distribution for the following reason: Nonrelativistic Brownian motion models such as the classical Ornstein-Uhlenbeck process are in obvious conflict with special relativity because they permit particles to move faster than the speed of light c. Most directly, this can be seen from the stationary velocity distribution, which is a Maxwell velocity distribution and thus non-zero for velocities |v| > c. The recent literature has seen some debate about the correct generalization of Maxwell's distribution in special relativity [22,196,198,199,232,475]. In Section 3.2 we shall discuss recent molecular dynamics simulations [202] which favor a distribution that was proposed by Jüttner [186] in 1911, i.e., six years after Einstein had formulated his theory of special relativity [2,3].…”

Section: Relativistic Equilibrium Thermostatisticsmentioning

confidence: 99%

Relativistic Brownian motion

2009

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a b s t r a c tOver the past one hundred years, Brownian motion theory has contributed substantially to our understanding of various microscopic phenomena. Originally proposed as a phenomenological paradigm for atomistic matter interactions, the theory has since evolved into a broad and vivid research area, with an ever increasing number of applications in biology, chemistry, finance, and physics. The mathematical description of stochastic processes has led to new approaches in other fields, culminating in the path integral formulation of modern quantum theory. Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuationdissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Since the velocities of relativistic particles are bounded by the speed of light, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

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“…He was motived by the problem of finding a way to improve the quantum statistical mechanics, based on the desity matrix, to treat the transport equations for superfluids [2][3][4]. Since then, the formalism proposed by Wigner has been applied in different contexts, such as quantum optics [5,6], condensed matter [7][8][9], quantum computing [10][11][12], quantum tomography [13], plasma physics [14][15][16][17][18][19]. Wigner introduced his formalism by using a kind of Fourier transform of the density matrix, ρ(q, q ′ ), giving rise to what in nowadays called the Wigner function, f W (q, p), where (q, p) are coordinates of a phase space manifold (Γ).…”

Section: Introductionmentioning

confidence: 99%

On the Symplectic Dirac Equation

2015

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Symplectic unitary representations for the Poincaré group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Klein-Gordon and Dirac equations are derived in phase space. As an application, we study the Dirac equation with electromagnetic interaction in phase space.

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A manifestly covariant relativistic Boltzmann equation for the evolution of a system of events

Cited by 69 publications

References 39 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?

Relativistic Brownian motion

On the Symplectic Dirac Equation

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