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Advances in Relativistic Statistical Mechanics
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Advances in Relativistic Statistical Mechanics

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 July 2017) | Viewed by 28582

Special Issue Editors

Prof. Dr. Lawrence P. Horwitz

E-Mail
Guest Editor
1. Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
2. Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
3. Department of Physics, Ariel University, Ariel 40700, Israel
Interests: relativistic quantum mechanics and quantum field theory; theory of classical and quantum unstable systems and chaos; quantum theory on hypercomplex hilbert modules; complex projective spaces in quantum dynamics; relativistic statistical mechanics and thermodynamics; high energy nuclear structure
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Asher Yahalom

E-Mail Website
Guest Editor
Department of Electrical and Electronic Engineering, Ariel University, Ariel 40700, Israel
Interests: relativistic dynamics and relativistic engines; non-barotropic (entropy dependent) fluid dynamics and magnetohydrodynamics; topological conservation laws in entropy dependent flow dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Relativistic statistical mechanics, with the work of Max Planck, lies at the very foundations of quantum theory. Major theoretical steps were made by Synge, de Groot, Israel and Kandrup, Haber and Weldon, Hakim, Horwitz, Schieve and Piron, and others; recent experiments and high-precision observations have motivated growing interest and importance of this subject.

Both the classical and quantum theories of relativistic many body systems have been developed over the years, with important applications in many areas, such as plasma physics, also associated with the fusion problem, high energy particle physics (as in the work of Oppenheimer and Hagedorn, and observations and interpretations of deep inelastic scattering), high frequecy electronic devices, such as the free electron laser, relativistic electron tubes and dissipative relativistic hydrodynamics.

The significance of relativistic statistical mechanics is also of great importance in the framework of general relativity and cosmology, such as stellar structures, studies of instabilities as in supernova events, dark matter and dark energy problems and black hole physics. There have, for example, been recent attempts to define entropic processes in connection with the geometric configuration of geodesic curves on the space–time manifold.

This Special Issue of Entropy attempts to collect recent developments in order to motivate and stimulate further research in this important field.

Prof. Lawrence P. Horwitz,
Prof. Asher Yahalom
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Relativity
  • Fluid mechanics
  • Plasma physics
  • Many body physics
  • High energy scattering
  • High energy electron tubes
  • High energy nuclear structure
  • Free electron lasers
  • Stellar structure
  • Cosmology

Published Papers (7 papers)

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Research

259 KiB  
Article
Entropy Measures as Geometrical Tools in the Study of Cosmology
by Gilbert Weinstein, Yosef Strauss, Sergey Bondarenko, Asher Yahalom, Meir Lewkowicz, Lawrence Paul Horwitz and Jacob Levitan
Entropy 2018, 20(1), 6; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010006 - 25 Dec 2017
Cited by 3 | Viewed by 3527
Abstract
Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the [...] Read more.
Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ ( x ) with χ ( x ) being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi equation. We further show that the geodesic equation for a null geodesic in a double-warped spacetime leads to the same entropy equation. By applying a Robertson–Walker metric for a flat three-dimensional Euclidean space expanding as a function of time, we again reach the entropy equation stressing the connection between the chosen entropy measure and time. We finally turn to the Raychaudhuri equation for expansion, which also is a Riccati equation similar to the transformed entropy equation. Those Riccati-type equations have solutions of the same form as the Jacobi equation. The Raychaudhuri equation can be transformed to a harmonic oscillator equation, and it has been shown that the geodesic deviation equation of Jacobi is essentially equivalent to that of a harmonic oscillator. The Raychaudhuri equations are strong geometrical tools in the study of general relativity and cosmology. We suggest a refined entropy measure applicable in cosmology and defined by the average deviation of the geodesics in a congruence. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
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243 KiB  
Article
Gravitational Contribution to the Heat Flux in a Simple Dilute Fluid: An Approach Based on General Relativistic Kinetic Theory to First Order in the Gradients
by Dominique Brun-Battistini, Alfredo Sandoval-Villalbazo and Ana Laura Garcia-Perciante
Entropy 2017, 19(11), 537; https://0-doi-org.brum.beds.ac.uk/10.3390/e19110537 - 28 Oct 2017
Cited by 1 | Viewed by 3173
Abstract
Richard C. Tolman analyzed the relation between a temperature gradient and a gravitational field in an equilibrium situation. In 2012, Tolman’s law was generalized to a non-equilibrium situation for a simple dilute relativistic fluid. The result in that scenario, obtained by introducing the [...] Read more.
Richard C. Tolman analyzed the relation between a temperature gradient and a gravitational field in an equilibrium situation. In 2012, Tolman’s law was generalized to a non-equilibrium situation for a simple dilute relativistic fluid. The result in that scenario, obtained by introducing the gravitational force through the molecular acceleration, couples the heat flux with the metric coefficients and the gradients of the state variables. In the present paper it is shown, by explicitly describing the single particle orbits as geodesics in Boltzmann’s equation, that a gravitational field drives a heat flux in this type of system. The calculation is devoted solely to the gravitational field contribution to this heat flux in which a Newtonian limit to the Schwarzschild metric is assumed. The corresponding transport coefficient, which is obtained within a relaxation approximation, corresponds to the dilute fluid in a weak gravitational field. The effect is negligible in the non-relativistic regime, as evidenced by the direct evaluation of the corresponding limit. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
362 KiB  
Article
Statistics of Binary Exchange of Energy or Money
by Maria Letizia Bertotti and Giovanni Modanese
Entropy 2017, 19(9), 465; https://0-doi-org.brum.beds.ac.uk/10.3390/e19090465 - 02 Sep 2017
Cited by 4 | Viewed by 3592
Abstract
Why does the Maxwell-Boltzmann energy distribution for an ideal classical gas have an exponentially thin tail at high energies, while the Kaniadakis distribution for a relativistic gas has a power-law fat tail? We argue that a crucial role is played by the kinematics [...] Read more.
Why does the Maxwell-Boltzmann energy distribution for an ideal classical gas have an exponentially thin tail at high energies, while the Kaniadakis distribution for a relativistic gas has a power-law fat tail? We argue that a crucial role is played by the kinematics of the binary collisions. In the classical case the probability of an energy exchange far from the average (i.e., close to 0% or 100%) is quite large, while in the extreme relativistic case it is small. We compare these properties with the concept of “saving propensity”, employed in econophysics to define the fraction of their money that individuals put at stake in economic interactions. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
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397 KiB  
Article
From Relativistic Mechanics towards Relativistic Statistical Mechanics
by Luca Lusanna
Entropy 2017, 19(9), 436; https://0-doi-org.brum.beds.ac.uk/10.3390/e19090436 - 23 Aug 2017
Cited by 6 | Viewed by 4780
Abstract
Till now, kinetic theory and statistical mechanics of either free or interacting point particles were well defined only in non-relativistic inertial frames in the absence of the long-range inertial forces present in accelerated frames. As shown in the introductory review at the relativistic [...] Read more.
Till now, kinetic theory and statistical mechanics of either free or interacting point particles were well defined only in non-relativistic inertial frames in the absence of the long-range inertial forces present in accelerated frames. As shown in the introductory review at the relativistic level, only a relativistic kinetic theory of “world-lines” in inertial frames was known till recently due to the problem of the elimination of the relative times. The recent Wigner-covariant formulation of relativistic classical and quantum mechanics of point particles required by the theory of relativistic bound states, with the elimination of the problem of relative times and with a clarification of the notion of the relativistic center of mass, allows one to give a definition of the distribution function of the relativistic micro-canonical ensemble in terms of the generators of the Poincaré algebra of a system of interacting particles both in inertial and in non-inertial rest frames. The non-relativistic limit allows one to get the ensemble in non-relativistic non-inertial frames. Assuming the existence of a relativistic Gibbs ensemble, also a “Lorentz-scalar micro-canonical temperature” can be defined. If the forces between the particles are short range in inertial frames, the notion of equilibrium can be extended from them to the non-inertial rest frames, and it is possible to go to the thermodynamic limit and to define a relativistic canonical temperature and a relativistic canonical ensemble. Finally, assuming that a Lorentz-scalar one-particle distribution function can be defined with a statistical average, an indication is given of which are the difficulties in solving the open problem of deriving the relativistic Boltzmann equation with the same methodology used in the non-relativistic case instead of postulating it as is usually done. There are also some comments on how it would be possible to have a hydrodynamical description of the relativistic kinetic theory of an isolated fluid in local equilibrium by means of an effective relativistic dissipative fluid described in the Wigner-covariant framework. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
2462 KiB  
Article
Comparative Analysis of Jüttner’s Calculation of the Energy of a Relativistic Ideal Gas and Implications for Accelerator Physics and Cosmology
by John R. Fanchi
Entropy 2017, 19(7), 374; https://0-doi-org.brum.beds.ac.uk/10.3390/e19070374 - 22 Jul 2017
Cited by 3 | Viewed by 3813
Abstract
Jüttner used the conventional theory of relativistic statistical mechanics to calculate the energy of a relativistic ideal gas in 1911. An alternative derivation of the energy of a relativistic ideal gas was published by Horwitz, Schieve and Piron in 1981 within the context [...] Read more.
Jüttner used the conventional theory of relativistic statistical mechanics to calculate the energy of a relativistic ideal gas in 1911. An alternative derivation of the energy of a relativistic ideal gas was published by Horwitz, Schieve and Piron in 1981 within the context of parametrized relativistic statistical mechanics. The resulting energy in the ultrarelativistic regime differs from Jüttner’s result. We review the derivations of energy and identify physical regimes for testing the validity of the two theories in accelerator physics and cosmology. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
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304 KiB  
Article
Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity
by Claudio Cremaschini and Massimo Tessarotto
Entropy 2017, 19(7), 339; https://0-doi-org.brum.beds.ac.uk/10.3390/e19070339 - 06 Jul 2017
Cited by 18 | Viewed by 4721
Abstract
Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017) are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state [...] Read more.
Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017) are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG) wave equation, which advances the quantum state ψ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton–Jacobi quantization of the classical variational tensor field g ≡ g μ ν and its conjugate momentum, referred to as (canonical) g-quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for ψ , which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the metric tensor g ≡ g μ ν and its quantum conjugate momentum operator. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
681 KiB  
Article
The Gibbs Paradox, the Landauer Principle and the Irreversibility Associated with Tilted Observers
by Luis Herrera
Entropy 2017, 19(3), 110; https://0-doi-org.brum.beds.ac.uk/10.3390/e19030110 - 11 Mar 2017
Cited by 35 | Viewed by 4231
Abstract
It is well known that, in the context of General Relativity, some spacetimes, when described by a congruence of comoving observers, may consist of a distribution of a perfect (non–dissipative) fluid, whereas the same spacetime as seen by a “tilted” (Lorentz–boosted) congruence of [...] Read more.
It is well known that, in the context of General Relativity, some spacetimes, when described by a congruence of comoving observers, may consist of a distribution of a perfect (non–dissipative) fluid, whereas the same spacetime as seen by a “tilted” (Lorentz–boosted) congruence of observers may exhibit the presence of dissipative processes. As we shall see, the appearance of entropy-producing processes are related to the high dependence of entropy on the specific congruence of observers. This fact is well illustrated by the Gibbs paradox. The appearance of such dissipative processes, as required by the Landauer principle, are necessary in order to erase the different amount of information stored by comoving observers, with respect to tilted ones. Full article
(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics)
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