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EJTP 9, No. 26 (2012) 269–276 Electronic Journal of Theoretical Physics Thermodynamical Model of the Universe Naseer Iqbal1,2∗ , M. S. Khan1 , Tabasum Masood1 and Ibrahim Selim3 1 Department of Physics,University of Kashmir, Srinagar, 190 006, India Inter-University Center for Astronomy and Astrophysics, Pune 411 007, India 3 National Research Institute of Astronomy and Geophysics, Helwn Cairo-Egypt 2 Received 28 August 2010, Accepted 10 December 2010, Published 17 January 2012 Abstract: We inquire in to the physics of a self gravitating medium in quasi-static equilibrium, using the phenomenological approach of thermodynamics. Gravitational galaxy clustering is statistical and its origin is dynamical one. Hence the aspects of clustering must be understood in order to arrive at a proper appreciation of the subject of the formation and evolution of the large scale structure of the universe. Long range gravitational forces modify the thermodynamic functions and equations of state. The thermodynamical model is discussed at many levels. First we discuss the importance of thermodynamics as applicable to the gravitational clustering problem and extend our remarks to study various thermodynamic functions like free energy, entropy, pressure, internal energy and others. The various results that we discuss have interested implications for the study of large scale structure in the universe.They support the view that an easy and simple approach can be made an alternative study while discussing the central issues of cosmology ”large scale structure of universe”. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Large Scale Structure; Cosmology-Galaxies; Thermodynamics; Clustering PACS (2010): 98.80.-k; 98.65.-r; 98.65.Hb; 05.70.-a; 95.30.Tg 1. Introduction Large scale structure in the universe are central to several issues in cosmology. We assume that our universe can be considered as an infinite gas consisting of large number of molecules. Each molecule is treated to be the constituent particle(galaxy) of universe. Galaxies are treated to be the building blocks of universe. The description of gravitational galaxy clustering becomes therefore an important problem to study the large scale structure of universe. Various theories of cosmological many body problem have been developed mainly from a thermodynamic point of view. The physical validity of the ap∗ [email protected] 270 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 plication of thermodynamics in the clustering of galaxies and galaxy clusters has been discussed on the basis of N-body computer simulation results [1]. The observed universe contains structures at different scales. At the same time it also exhibits a remarkable level of uniformity as regards the large scale properties. It has been a challenge to explain both these features in a consistent theory. The conventional wisdom attempts to do so by considering the applicability of thermodynamics to a universe consisting of galaxies which are clustering together under the influence of gravitational force. The present review discusses several aspects of galaxy clusters .The emphasis, throughout,is on theoretical aspects and physical basis for the thermodynamical model rather than on the detailed observational features. Many of the concepts are developed in a self contained manner. This has made the review somewhat more pedagogical than a usual one, but has the advantage that even a non-expert astronomer or physicist will be able to understand and appreciate the contents. We begin in this paper with a discussion of an expanding universe in which galaxies cluster under the influence of there mutual gravitational force.The basic laws which govern the dynamics of galaxy clusters are Newton’s laws of motion.The review of [4] and [5] is formulated in detail in this work. Thermodynamics is based on some basic postulates called laws of thermodynamics from which all the thermodynamic properties can be calculated. We also discusses here the connectivity of thermodynamics with statistical mechanics as it bridges the gap between mechanics and thermodynamics. The applicability of thermodynamics to the cosmological many body problem suggest that statistical mechanics should also apply.The general conditions under which statistical mechanics may describe the cosmological many body problems are closely related to those for the applicability of thermodynamics described in detail by [7]. The modification of thermodynamics by gravity is itself an interesting physical problem. This paper initiates a deeper understanding of the physics of self gravitating medium. The first part of the paper discusses the basic thermodynamics and its applicability to the clustering problem, the second part focusses on the statistical mechanical aspects of gravitating systems. The third part illustrates the description of gravitational quasi-equilibrium thermodynamics. We deeply examine various findings and predict some future remarks. 2. Thermodynamics and Gravitational Clustering The basic laws which govern the dynamics of galaxy clusters are Newton’s laws of motion and Newton’s law of gravitation given by: d2 r i dt2 (1) mi mj rij2 (2) F i = mi Fij = G Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 271 2 Where mi is the mass of the ith particle(galaxy). ddtr2i is the acceleration of the ith galaxy with respect to the origin of an inertial system. Fi is the total force acting on the ith galaxy and Fij is the force on the ith galaxy due to its interaction by the j th galaxy. rij denotes the radius vectors of the ith and j th galaxies respectively. For the sake of simplicity in describing our model we start with the assumption that all the galaxies look like point mass particles, therefore we omit the inclusion of the parameters like mi and mj . However, galaxies have extended structures and the point mass concept is only an approximation one. In this review article we also deliberate on the extended nature of galaxies. For a gaseous medium like our universe the thermodynamical quantities are related by the equation of state as P V = N kT (3) It has been suggested by [8] that for large mass M of gas with volume V and temperature T containing N molecules under boundary pressure P, the above equation is modified to 1 P V = N kT − αGM 2 V − 3 (4) Where k is Boltzmann’s constant, G is gravitational constant, α is a constant depending on the shape of mass. The proposed correction of equation(3) is because of the fact that for a very large mass M, we have to take account of the gravitational interaction between various molecules (galaxies) of the system. Standard statistical thermodynamic results [2] showed that, for a system of identical particles interacting via a pair wise attractive coulomb potential, the internal energy U and pressure P are given by: 3 (5) U = N T (1 − 2b) 2 NT (1 − b) (6) V Where T is the temperature of the system, N = n̄ V is the number of particles each is the of mass m in a grand canonical ensemble of a system of volume V, and b = −W 2K measure of the ratio of correlation energy to twice the kinetic energy due to the peculiar velocities, and is also called as measuring correlation parameter which ranges between 0 to 1. When b =0, i.e no gravitational interaction of galaxies exists, the behaviour of the ensemble of galaxies in a universe is like that of an ideal gas. As b goes on changing from 0 to 1, the clustering of galaxies becomes more and more dominant and the whole system of particles (galaxies) attain viral limit. With out additional physical information, the functional form of b is undetermined, except by a dimensional argument [6] which shows that for the inverse square law force, b must be of some function of the combination n̄T −3 . The thermodynamic description of the system of galaxies in the universe is completely specified only after the choice of n̄T −3 has been made by [3]. The measuring correlation P = Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 272 parameter b is defined by: −W 2πGm2 n̄ b = = 2K 3T ξ(n̄, T, r)rdr (7) V W is the potential energy and K the kinetic energy of the particles in a system.T is the temperature, V the volume, G is the universal gravitational constant. ξ(n̄, T, r) is the two particle correlation function and r is the inter-particle distance. An overall study of ξ(n̄, T, r) has already been discussed by [3]. The one dimensional partial differential equation relating ξ2 (r) with n̄, T and r is in the form of [3] ∂ξ ∂ξ ∂ξ +T −r =0 (8) 3n̄ ∂ n̄ ∂T ∂r In a universe of finite age, ξ2 (r) is appreciable only over a limited region which may grow slowly with time. The value of b depends on the form of ξ2 (r), and previously, it has been shown that[3] (9) ξ2 (r) ∝ r−2 However, the form of b chosen by [4] is b= β n̄T −3 1 + β n̄T −3 (10) Where β is independent of n̄ and T. This seems to be the simplest analytical form for b that has the appropriate ideal gas and virilized limits. All the equations(3-10) define a thermodynamic description for a system of particles(galaxies) of point mass approximation. Actually galaxies have extended structures as a whole and therefore the above equations can be modified by making use of softening parameter . The galaxies are initially assigned co-ordinates Vi = Hri ; appropriate to the starting value of Hubble’s constant. To represent the galaxies as extended mass distribution, the interaction potensial is given by: Gmi mj φij = − (11) 1 (r2 + 2 ) 2 Where the softening parameter may be associated with the characteristic of a galaxy. The measuring correlation parameter b for extended mass structure is : b = β n̄T −3 α( R ) 1 + β n̄T −3 α( R ) (12) and is related to b for point mass system by: b = bα( R ) 1 + b α( R ) − 1 (13) Equation(13) is the relation between point mass and extended mass galaxies clustering under the influence of /R. The effect shows that b has s strong dependence on the Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 273 combination of n̄T −3 and the parameter α(/R). The value of b decreases if /R is large and increases for smaller values of mutual gravitation. Equations (5) and (6) can be extended to [5]: 3N T (14) (1 − 2b ) U= 2 NT P = (15) (1 − Jb ) V where the expression for the constant J comes out to be equal to J= 2 5 − 3 3α 1 + (16) 2 R2 Also the expression for the value of the constant α is given by: 2 2 R α = 1 + 2 + 2 ln 2 R R 1+ 1+ (17) R2 3. Statistical Mechanical Aspects of Gravitating Systems Statistical Mechanics has been found extremely useful tool for solving cosmological many body problems. The basic tool discussed here is the theory of ensembles. We discuss here an extension of the review of basic thermodynamics to various statistical aspects. As already assumed that universe is an infinite gas consisting of large number of particles (galaxies), we consider an ensemble of various cells and study them on the basis of grand canonical partition function. From the statistical treatment of [5], the gravitational partition function for gravitational galaxy clustering is: 3N N −1 1 2πmkT 2 N ZN (T, V ) = V 1 + β n̄T −3 2 N! Λ (18) N ! is due to the distinguishingly of particles. Λ represents the volume of a phase space cell. N is the number of particles (galaxies) in a system of volume V.This is valid for a point mass model approximation. The extension of this equation to a system of particles clustering gravitationally having extended mass structures is 3N 1 2πmkT 2 N N −1 −3 ZN (T, V ) = V 1 + β n̄T α( ) N! Λ2 R The entropy of the system can be calculated by using the relation ∂A S= − ∂T N,V 3 S = N ln(n̄−1 T 2 ) + (N − 1)ln(1 + β n̄T −3 ) − 3N (19) (20) β n̄T −3 5N 3N 2πmk + (21) + ln 2 1 + β n̄T −3 2 2 Λ Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 274 we substitute the expression for b defined by equation (10 )in the above equation, the entropy result for the extended mass structure of galaxies leads to: 3 S = N ln(n̄−1 T 2 +(N −1)ln(1+β n̄T −3 )−3N β n̄T −3 α(/R) 5N 3N 2πmk + + ln 2 (22) −3 1 + β n̄T α(/R) 2 2 Λ The overall clustering of galaxies is characterized by the full set of distribution function F(N). A simple objective description of the distribution function is to count their number in cells of a given size which is distributed uniformly over the sky. On the basis of statistical mechanics and the work carried out by [5] the distribution function is given by: N −1 −N̄ (1−b)−N b N̄ (1 − b) e (23) N̄ (1 − b) + N b F (N ) = N! The above one is valid for the distribution of point mass particles. However, for extended mass structures the distribution is: F (N, ) = N −1 −N̄ (1−b J)−N b J N̄ (1 − b ) e N̄ (1 − b ) + N b N! (24) If = 0, then both the distributions are same. 4. Gravitational Quasi-equilibrium Thermodynamics A thermodynamic approach seems likely to meet the requirements for the description of an equilibrium theory. The thermodynamics however, is being changed by gravity. Obviously for a system with time dependent parameters we use a quasi-equilibrium type of approach. The rigorous meaning of equilibrium is a fundamental problem for all thermodynamics, but especially for a system with long range forces. The gravitational quasi-equilibrium distribution functions (equations 23 and 24) gives the probability that a randomly placed cells of size V have exactly N particles in an expanding universe, the rate of clustering of galaxies is more and more dominant. It does not appear to be in an equilibrium state; in fact the various gravitating systems do not have any non-singular equilibrium state, because of their long range, unshielded, attractive interaction. The use of thermodynamics to study the gravitational galaxy clustering in an expanding universe is applicable as long as the evolution is quasi-equilibrium. The rate of clustering of galaxies in a cosmological many body problem using the aspect of thermodynamics enables us to make a description about the expansion of the system of galaxies. With the expansion of universe, clustering generates entropy, which is reflected in the increase in b and b . The expansion scale for galaxy clustering in an expanding universe on the basis of thermodynamic equation(equation of state) is written as: 1 a (t) = a∗ b8 7 (1 − b) 8 (25) Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 For extended mass structures, the above equation is modified in the form of: 1 J 2 +4J−6 2−J 1 (t) 8 a = a∗ b (1 − b ) 8(1−J) (1 − Jb ) 8J(1−J) e 4J(1−Jb ) 275 (26) quation(26) is the new derived expression for the expansion scale of a system consisting of extended mass galaxies clustering gravitationally. This shows that the time evolution of b as a function of the expansion scale is having an overall dependence on the time evolution of b, the cell size R and the softening parameter . Future Remarks and Conclusions We have extensively discussed the applicability of thermodynamics to the gravitational galaxy clustering and have come to know that both dynamical and statistical aspects are crucial to a deeper understanding of large scale structure of universe. The non-linear gravitational clustering in an expanding universe can be studied with the help of two point correlation function ξ2 by solving equation (8). The characteristic solution of equation (8) along the path of integration clearly gives scale invariant solution as ξ(n̄, T, r) = ξ(n̄T −3 , T r). This confirms the earlier result of b which has a specific combination dependence on n̄T −3 . The functional form of ξ(n̄, T, r) can be obtained from various physical and boundary conditions of gravitational clustering. It can be witnessed easily that ξ2 also satisfies earlier results of pebbles power law i.e ξ2 ∝ r−2 , which is a well established result of gravitational galaxy clustering. It is interesting to note that the functional form of b(n̄T −3 ) emerges directly from the equation of state itself with out making any kind of approximation. Till date all the theories on gravitational galaxy clustering are confined to point mass systems only. However, the results shown in equations( 12, 14, 15, 22, 24 and 26) indicate that both the thermodynamical and statistical aspects can be used to study the extended nature of galaxies in the clustering problem as galaxies have actually extended structures, while as point mass concept of macroscopic bodies is only an approximation one. The various results are clearly affected by the induction of softening parameter , which clearly indicates the scale dependence of two point correlation function in gravitational galaxy clustering. We describe the partition function and the free energy for both the models on the basis of statistical mechanics(viral coefficient technique) and then the evaluation of all the thermodynamic quantities follow easily. From the grand canonical partition function, the distribution function of galaxies follows directly in equations (23) and (24). The introducing of softening parameter enables us to have a significance of dark matter present in the galaxies which is an important challenge in the modern cosmology. The clustering of galaxies in an expanding universe generates entropy which is reflected in the increase in b and b . For homogenous universe, the expansion observed locally in co-moving coordinates would be adiabatic and satisfy the first law of thermodynamics.Equations (25) and (26) describe the adiabatic evolution of large scale of b and b as a function of expansion scale a(t) . The future remarks of our work indicate that we need to extend the results of two point correlation function to three point, four point and so on higher orders of correlation 276 Electronic Journal of Theoretical Physics 9, No. 26 (2012) 269–276 functions. In addition to this it is important to develop b2 , b3 , b4 , —- for higher orders of clustering. Acknowledgements The authors are grateful to Interuniversity Center for Astronomy and Astrophysics (IUCAA) Pune India for providing warm hospitality and excellent facilities during the preparation of this review article. References [1] Itoh, M; Inagaki, S; and Saslaw,W. C, 1993 Ap. J; Vol. 403, 476. [2] Hill, T. L. 1956, Statistical Mechanics:Principles and statistical applications (New York: MC Graw Hill). [3] Iqbal, N; Ahmad, F and Khan, M. S, 2006 Journal of Astrophysics and Astronomy; Vol. 27, 373-379. [4] Saslaw, W. C; and A. J. S Hamilton, 1984 Ap. J; Vol. 276, 13. [5] Ahmad, F; Saslaw, W. C; and Bhat, N. I,2002 Ap. J; Vol. 571, 576. [6] Landau, L. D and Lifshitz,E. M, 1959 course of theoretical physics, Vol. 5. Pergamon press, Oxford. [7] Saslaw, W. C; 2000 The distribution of galaxies gravitational clustering in cosmology, Cambridge university press, Cambridge. [8] Terletsky, Y. P. Teor. Fiz,1952; Vol .22, 506.