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Jmp 10 07 Content 201906051 9 772153119007 0706 Journal of Modern Physics, 2019, 10, 687-880 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Table of Contents Volume 10 Number 7 June 2019 Statistical Induced Dynamic of Self-Gravitating System B. I. Lev…………………………………………………………………………………………………………………687 Comparison between Theoretical and Experimental Radial Electron Temperature Profiles in a Low Density Weakly Ionized Plasma A. R. Galaly, G. Van Oost……………………………………………………………………………...………………699 Zero Strength of the Lorentz Force and Lack of a Classical Energy Radiation in the Bohr’s Hydrogen Atom S. Olszewski…………………………………………………………………………………………………….………717 On the Final State of Black Holes B. Lehnert………………………………………………………………………………………………………………725 The Solution Cosmological Constant Problem J. Foukzon, E. Men’kova, A. Potapov…………………………………………………………………………………729 Space-Time Properties as Quantum Effects. Restrictions Imposed by Grothendieck’s Scheme Theory L. Lutsev………………………………………………………………………………………………………………795 Templates for Quantifying Clay Type and Clay Content from Magnetic Susceptibility and Standard Borehole Geophysical Measurements D. K. Potter, A. Ali, S. Abdalah……………………….………………………………………………………………824 A Physical Explanation for Particle Spin D. J. Pons, A. D. Pons, A. J. Pons……………………………………………………………………………………835 Negative Mass: Part One E. Tannous…………………………………………………………..…………………………………………………861 The figure on the front cover is from the article published in Journal of Modern Physics, 2019, Vol. 10, No. 7, pp. 699-716 by Ahmed Rida Galaly, Guido Van Oost. Journal of Modern Physics (JMP) Journal Information SUBSCRIPTIONS The Journal of Modern Physics (Online at Scientific Research Publishing, www.SciRP.org) is published monthly by Scientific Research Publishing, Inc., USA. Subscription rates: Print: $89 per issue. To subscribe, please contact Journals Subscriptions Department, E-mail: [email protected] SERVICES Advertisements Advertisement Sales Department, E-mail: [email protected] Reprints (minimum quantity 100 copies) Reprints Co-ordinator, Scientific Research Publishing, Inc., USA. E-mail: [email protected] COPYRIGHT Copyright and reuse rights for the front matter of the journal: Copyright © 2019 by Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Copyright for individual papers of the journal: Copyright © 2019 by author(s) and Scientific Research Publishing Inc. Reuse rights for individual papers: Note: At SCIRP authors can choose between CC BY and CC BY-NC. Please consult each paper for its reuse rights. Disclaimer of liability Statements and opinions expressed in the articles and communications are those of the individual contributors and not the statements and opinion of Scientific Research Publishing, Inc. We assume no responsibility or liability for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained herein. We expressly disclaim any implied warranties of merchantability or fitness for a particular purpose. If expert assistance is required, the services of a competent professional person should be sought. PRODUCTION INFORMATION For manuscripts that have been accepted for publication, please contact: E-mail: [email protected] Journal of Modern Physics, 2019, 10, 687-698 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196 Statistical Induced Dynamic of Self-Gravitating System B. I. Lev Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Metrolohichna, Ukraine How to cite this paper: Lev, B.I. (2019) Abstract Statistical Induced Dynamic of Self-Gra- vitating System. Journal of Modern Phys- The self-gravitating systems a priory are non-equilibrium. The new approach, ics, 10, 687-698. based on non-equilibrium statistical operator, which allows taking into ac- https://doi.org/10.4236/jmp.2019.107049 count inhomogeneous distribution of particles and temperature, has been Received: April 29, 2019 proposed. Such method employs a saddle point procedure to find dominant Accepted: June 3, 2019 contributions to the partition function and permits to obtain all thermody- Published: June 6, 2019 namic parameters of the system. Statistical induced dynamic and behavioral peculiarities of self-gravitating systems for different conditions have been Copyright © 2019 by author(s) and Scientific Research Publishing Inc. predicted. This work is licensed under the Creative Commons Attribution International Keywords License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Nonequilibrium Statistical Operator, Self-Gravitating System, Open Access Inhomogeneous 1. Introduction The statistical description of a system of interacting particles has a fundamental physical interest. In that context, the interacting system provides a model of fundamental interest for which ideas of statistical mechanics and thermody- namics can be tested and developed. For self-gravitating system the thermody- namical ensemble can be nonequivalent [1] [2]. In particular, since energy is not-additive, and cannot use the canonical ensemble to study the system with long-range interaction. Since equilibrium states are only local entropy maxi- mum. Two type approaches (statistical and thermodynamic) have been developed to determine the equilibrium states of the interacting system and describe possible phase transition. It is generally believed that mean field theory is exact for sys- tems. Since in mean field theory any thermodynamical function depends on the thermodynamical variables only through the dimensionless combinations and DOI: 10.4236/jmp.2019.107049 Jun. 6, 2019 687 Journal of Modern Physics B. I. Lev the system is thermodynamically stable but the thermodynamical limit does exist [3]. Formation of the spatially inhomogeneous distribution of interaction particles is a typical problem in condensed matter physics and requires non-conventional methods of the statistical description of the system tailored to gravitational inte- racting particles with regard for an arbitrary spatially inhomogeneous particles distribution. Statistical description must employ the procedure to find dominant contributions to the partition function and to avoid entropy divergences for infinite system volume. Formation of the inhomogeneous distribution of inte- raction particles requires a non-conventional method, which was proposal in [4] [5] [6] [7] [8]. This method is based on Hubbard-Stratonovich representation of statistical sum [9] which is now extended and applied to the self-gravitating sys- tem to find solution for particles distribution without using spatial box restric- tions. It is important that this solution has no divergences in thermodynamic limits. For this goal using the saddle point approximation took into account the conservation number of particles in limiting space and provided to nonlinear equation. The partition function in the case of homogeneous distribution of the particles, and in case of inhomogeneous distribution was obtained in [5] [6] [10], but in this approach described only condition in the formation of the poss- ible inhomogeneous distribution of the interacting particles. Systems with long-range interactions, such as self-gravitating system do not relax to the usual Boltzmann-Gibbs thermodynamic equilibrium, but become trapped in quasi-stationary states, the lifetime of which diverges with the number of particles. The instability threshold was predicted for spontaneous symmetry breaking for a class of d-dimensional systems [11]. Non-equilibrium stationary states of systems were described in article [12] where concluded that three-dimensional systems do not evolve to thermodynamic equilibrium but trap in non-equilibrium quasi-stationary states. Mostly due to the fact that self-gravitating system exist in a states that far from equilibrium and the relaxation time to equilibrium state is very long. The ho- mogeneous particle distribution in a self-gravitating system is not stable. The particles distribution in such a system is inhomogeneous in space, from the be- ginning. There is standardized behavior of the interacting system, it is described differently for different equilibrium ensembles. That approach does not seem to be very consistent, due to the fact that, the state equation has to come from the definition of partition function, but the definition of it is unknown for the space inhomogeneous systems [13] [14]. Therefore, there is a dilemma, either to take the postulates of equilibrium statistical mechanics to obtain only instability criteria or not to try to account for the space in homogeneity using different ap- proach. Such inhomogeneous distribution of the particles, temperature and chemical potential can be accounted for in non-equilibrium statistical operator approach [15], where possibility of local change of thermodynamic parameters is included. This system is non-equilibrium and inhomogeneous distribution of particle can motivate the inhomogeneous distribution of temperature and DOI: 10.4236/jmp.2019.107049 688 Journal of Modern Physics B. I. Lev chemical potential and other thermodynamic parameter. In this article are proposed the new approach, based on non-equilibrium statistic operator [15], which are more suitable for the description of a self- gravitating systems. The equation of state and all thermodynamic characteris- tics needed are defined by the equations which contribute the most to the par- tition function. Therefore, there
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