Download Thermodynamical Model of the Universe

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]
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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)
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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
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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.
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