Entropy changes in the clustering of galaxies in an expanding universe

doi:10.4236/ns.2011.31009

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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: [email protected]

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

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 ~





 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

Ub

(1)



 (2)

b defines the measuring correlation parameter and is

dimensionless, given by [8]



2,,

bGmnTrrdr





  (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:

bnT





 (4)

Eq.4 indicates that b has a specific dependence on the

combination 3



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



mkT

ZTVV nT

















 (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

TZ (6)

Thermodynamic description of entropy can be calcu-

lated as:













(7)

The use of Eq.5 and Eq.6 in Eq.7 gives



0lnln 13SSnTb b











 (8)

where S0 is an arbitary constant. From Eq.4 we write



nbT





 (9)

Using Eq.9, Eq.8 becomes as

03lnSSb bT



 





(10)

Again from Eq.4

N. Iqbal et al. / Natural Science 3 (2011) 65-68

6767



21nb









(11)

with the help of Eq.11, Eq.10 becomes as



lnln 13

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



0lnln 13

SS NTNbNb





 



 (13)

For extended structures of galaxies, Eq.4 gets modi-

fied to



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



given by:









(15)



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;

 

ln 1111

bT b

SS bb









 



 





(16)



 

ln ln

21111

bb b

SSn bb















 



 





(17)

where

1ln

RRR









 

 

 

 







(18)



= 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

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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