Vol.3, No.1, 65-68 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.31009
Entropy changes in the clustering of galaxies in an
expanding universe
Naseer Iqbal1,2*, Mohammad Shafi Khan1, Tabasum Masood1
1Department of Physics, University of Kashmir, Srinagar, India; *Corresponding Author:
2Interuniversity Centre for Astronomy and Astrophysics, Pune, India.
Received 19 October 2010; revised 23 November 2010; accepted 26 November 2010.
ABSTRACT
In the present work the approach-thermody-
namics and statistical mechanics of gravitating
systems is applied to study the entropy change
in gravitational clustering of galaxies in an ex-
panding universe. We derive analytically the
expressions for gravitational entropy in terms of
temperature T and average density n of the par-
ticles (galaxies) in the given phase space cell. It
is found that during the initial stage of cluster-
ing of galaxies, the entropy decreases and fi-
nally seems to be increasing when the system
attains virial equilibrium. The entropy changes
are studied for different range of measuring
correlation parameter b. We attempt to provide a
clearer account of this phenomena. The entropy
results for a system consisting of extended
mass (non-point mass) particles show a similar
behaviour with that of point mass particles
clustering gravitationally in an expanding uni-
verse.
Keywords: Gravitational Clustering;
Thermodynamics; Entropy; Cosmology
1. INTRODUCTION
Galaxy groups and clusters are the largest known
gravitationally bound objects to have arisen thus far in
the process of cosmic structure formation [1]. They form
the densest part of the large scale structure of the uni-
verse. In models for the gravitational formation of struc-
ture with cold dark matter, the smallest structures col-
lapse first and eventually build the largest structures;
clusters of galaxies are then formed relatively. The clus-
ters themselves are often associated with larger groups
called super-clusters. Clusters of galaxies are the most
recent and most massive objects to have arisen in the
hiearchical structure formation of the universe and the
study of clusters tells one about the way galaxies form
and evolve. The average density n and the temperature T
of a gravitating system discuss some thermal history of
cluster formation. For a better larger understanding of
this thermal history it is important to study the entropy
change resulting during the clustering phenomena be-
cause the entropy is the quantity most directly changed
by increasing or decreasing thermal energy of intraclus-
ter gas. The purpose of the present paper is to show how
entropy of the universe changes with time in a system of
galaxies clustering under the influence of gravitational
interaction.
Entropy is a measure of how disorganised a system is.
It forms an important part of second law of thermody-
namics [2,3]. The concept of entropy is generally not
well understood. For erupting stars, colloiding galaxies,
collapsing black holes - the cosmos is a surprisingly or-
derly place. Supermassive black holes, dark matter and
stars are some of the contributors to the overall entropy
of the universe. The microscopic explanation of entropy
has been challenged both from the experimental and
theoretical point of view [11,12]. Entropy is a mathe-
matical formula. Standard calculations have shown that
the entropy of our universe is dominated by black holes,
whose entropy is of the order of their area in planck
units [13]. An analysis by Chas Egan of the Australian
National University in Canberra indicates that the col-
lective entropy of all the supermassive black holes at the
centers of galaxies is about 100 times higher than previ-
ously calculated. Statistical entropy is logrithmic of the
number of microstates consistent with the observed
macroscopic properties of a system hence a measure of
uncertainty about its precise state. Statistical mechanics
explains entropy as the amount of uncertainty which
remains about a system after its observable macroscopic
properties have been taken into account. For a given set
of macroscopic quantities like temperature and volume,
the entropy is a function of the probability that the sys-
tem is in various quantumn states. The more states
available to the system with higher probability, the

N. Iqbal et al. / Natural Science 3 (2011) 65-68 66
greater the disorder and thus greater the entropy [2]. In
real experiments, it is quite difficult to measure the en-
tropy of a system. The technique for doing so is based on
the thermodynamic definition of entropy. We discuss the
applicability of statistical mechanics and thermodynam-
ics for gravitating systems and explain in what sense the
entropy change S – S0 shows a changing behaviour with
respect to the measuring correlation parameter b = 0 – 1.
2. THERMODYNAMIC DESCRIPTION OF
GALAXY CLUSTERS
A system of many point particles which interacts by
Newtonian gravity is always unstable. The basic insta-
bilities which may occur involve the overall contraction
(or expansion) of the system, and the formation of clus-
ters within the system. The rates and forms of these in-
stabilities are governed by the distribution of kinetic and
potential energy and the momentum among the particles.
For example, a finite spherical system which approxi-
mately satisfies the viral theorem, contracts slowly
compared to the crossing time ~

12
G 
 due to the
evaporation of high energy particles [3] and the lack of
equipartition among particles of different masses [4]. We
consider here a thermodynamic description for the sys-
tem (universe). The universe is considered to be an infi-
nite gas in which each gas molecule is treated to be a
galaxy. The gravitational force is a binary interaction
and as a result a number of particles cluster together. We
use the same approximation of binary interaction for our
universe (system) consisting of large number of galaxies
clustering together under the influence of gravitational
force. It is important to mention here that the characteri-
zation of this clustering is a problem of current interest.
The physical validity of the application of thermody-
namics in the clustering of galaxies and galaxy clusters
has been discussed on the basis of N-body computer
simulation results [5]. Equations of state for internal
energy U and pressure P are of the form [6]:

312
2
NT
Ub (1)

1
NT
Pb
V
 (2)
b defines the measuring correlation parameter and is
dimensionless, given by [8]

2
0
2,,
23
Wn
bGmnTrrdr
KT


  (3)
W is the potential energy and K the kinetic energy of
the particles in a system. nNV is the average
number density of the system of particles each of mass
m, T is the temperature, V the volume, G is the universal
gravitational constant.


,,nTr  is the two particle
correlation function and r is the inter-particle distance.
An overall study of


,,nTr  has already been dis-
cussed by [7]. For an ideal gas behaviour b = 0 and for
non-ideal gas system b varies between 0 and 1. Previ-
ously some workers [7,8] have derived b in the form of:
3
3
1
nT
bnT




 (4)
Eq.4 indicates that b has a specific dependence on the
combination 3
nT

.
3. ENTROPY CALCULATIONS
Thermodynamics and statistical mechanics have been
found to be equal tools in describing entropy of a system.
Thermodynamic entropy is a non-conserved state func-
tion that is of great importance in science. Historically
the concept of entropy evolved in order to explain why
some processes are spontaneous and others are not; sys-
tems tend to progress in the direction of increasing en-
tropy [9]. Following statistical mechanics and the work
carried out by [10], the grand canonical partition func-
tion is given by

3
21
3
2
12
,1
!
N
N
N
N
mkT
ZTVV nT
N









 (5)
where N! is due to the distinguishability of particles.

represents the volume of a phase space cell. N is the
number of paricles (galaxies) with point mass approxi-
mation. The Helmholtz free energy is given by:
ln N
A
TZ (6)
Thermodynamic description of entropy can be calcu-
lated as:
,
N
V
A
ST






(7)
The use of Eq.5 and Eq.6 in Eq.7 gives

3
12
0lnln 13SSnTb b






 (8)
where S0 is an arbitary constant. From Eq.4 we write

3
1
b
nbT


 (9)
Using Eq.9, Eq.8 becomes as
3
2
03lnSSb bT

 


(10)
Again from Eq.4

N. Iqbal et al. / Natural Science 3 (2011) 65-68
6767

1
32
21nb
Tb
 


(11)
with the help of Eq.11, Eq.10 becomes as

0
11
lnln 13
22
SSnb bb

 
 

(12)
This is the expression for entropy of a system consist-
ing of point mass particles, but actually galaxies have
extended structures, therefore the point mass concept is
only an approximation. For extended mass structures we
make use of softening parameter

whose value is
taken between 0.01 and 0.05 (in the units of total radius).
Following the same procedure, Eq.8 becomes as

3
2
0lnln 13
N
SS NTNbNb
V



 

 (13)
For extended structures of galaxies, Eq.4 gets modi-
fied to


3
3
1
nT R
bnT R





 (14)
where

is a constant, R is the radius of a cell in a
phase space in which number of particles (galaxies) is N
and volume is V. The relation between b and b

is
given by:

11
b
bb



 (15)
b

represents the correlation energy for extended mass
particles clustering gravitationally in an expanding uni-
verse. The above Eq.10 and Eq.12 take the form respec-
tively as;
 
3
2
0
3
ln 1111
bT b
SS bb




 

 


(16)

 
1
2
0
1
13
ln ln
21111
bb b
SSn bb



 


 

 


(17)
where
22
2
1ln
11
R
RRR
R




 
 
 
 



(18)
If

= 0,

= 1 the entropy equations for extended
mass galaxies are exactly same with that of a system of
point mass galaxies approximation. Eq.10, Eq.12 , Eq.16
and Eq.17 are used here to study the entropy changes in
the cosmological many body problem. Various entropy
change results S – S0 for both the point mass approxima-
tion and of extended mass approximation of particles
(galaxies) are shown in (Figures 1 and 2). The results
have been calculated analytically for different values of
Figure 1. (Color online) Comparison of isothermal entropy
changes for non-point and point mass particles (galaxies) for
an infinite gravitating system as a function of average relative
temperature T and the parameter b. For non-point mass

=
0.03 and R = 0.06 (left panel),

= 0.04 and R = 0.04 (right
panel).

N. Iqbal et al. / Natural Science 3 (2011) 65-68 68
Figure 2. (Color online) Comparison of equi-density entropy
changes for non-point and point mass particles (galaxies) for
an infinite gravitating system as a function of average relative
density n and the parameter b. For non-point mass

= 0.03
and R = 0.04.
R (cell size) corresponding to different values of soften-
ing parameter

. We study the variations of entropy
changes S – S0 with the changing parameter b for differ-
ent values of n and T. Some graphical variations for S –
S0 with b for different values of n = 0, 1, 100 and aver-
age temperature T = 1, 10 and 100 and by fixing value of
cell size R = 0.04 and 0.06 are shown. The graphical
analysis can be repeated for different values of R and by
fixing values of

for different sets like 0.04 and 0.05.
From both the figures shown in 1 and 2, the dashed line
represents variation for point mass particles and the solid
line represents variation for extended (non-point mass)
particles (galaxies) clustering together. It has been ob-
served that the nature of the variation remains more or
less same except with some minor difference.
4. RESULTS
The formula for entropy calculated in this paper has
provided a convenient way to study the entropy changes
in gravitational galaxy clusters in an expanding universe.
Gravity changes things that we have witnessed in this
research. Clustering of galaxies in an expanding universe,
which is like that of a self gravitating gas increases the
gases volume which increases the entropy, but it also
increases the potential energy and thus decreases the
kinetic energy as particles must work against the attrac-
tive gravitational field. So we expect expanding gases to
cool down, and therefore there is a probability that the
entropy has to decrease which gets confirmed from our
theoretical calculations as shown in Figures 1 and 2.
Entropy has remained an important contributor to our
understanding in cosmology. Everything from gravita-
tional clustering to supernova are contributors to entropy
budget of the universe. A new calculation and study of
entropy results given by Eqs.10, 12 , 16 and 17 shows
that the entropy of the universe decreases first with the
clustering rate of the particles and then gradually in-
creases as the system attains viral equilibrium. The
gravitational entropy in this paper furthermore suggests
that the universe is different than scientists had thought.
5. ACKNOWLEDGEMENTS
We are thankful to Interuniversity centre for Astronomy and Astro-
physics Pune India for providing a warm hospitality and facilities
during the course of this work.
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