In this paper, by invoking the laws of irreversible thermody\ud
namics the accelerated expansion of the universe is\ud
explained. It is shown that the entropy of the universe, at an\ud
y particular instant of time, plays a significant role\ud
in the accelerated expansion of the universe. Considering t\ud
he universe to be filled with a classical mono-atomic,\ud
homogeneous and isotropic gas under classical non-equilib\ud
rium situations, two generalized forces causing the\ud
expansion of the universe are arrived at. One of the two force\ud
s, the trivial force, has affinity to volume expansion\ud
and the non-trivial force has affinity to spatial expansion.\ud
The acceleration of the expansion of the universe is due\ud
to the spatial expansion caused by the non-trivial force and\ud
which in turn might account for the presence of the dark\ud
energy. It is shown in this paper that the non-trivial genera\ud
lized force and the dark energy, providing the negative\ud
pressure for spatial expansion, can be explained with irrev\ud
ersible thermodynamics

Rajak, D

In this paper, by invoking the laws of irreversible thermody

namics the accelerated expansion of the universe is

explained. It is shown that the entropy of the universe, at an

y particular instant of time, plays a significant role

in the accelerated expansion of the universe. Considering t

he universe to be filled with a classical mono-atomic,

homogeneous and isotropic gas under classical non-equilib

rium situations, two generalized forces causing the

expansion of the universe are arrived at. One of the two force

s, the trivial force, has affinity to volume expansion

and the non-trivial force has affinity to spatial expansion.

The acceleration of the expansion of the universe is due

to the spatial expansion caused by the non-trivial force and

which in turn might account for the presence of the dark

energy. It is shown in this paper that the non-trivial genera

lized force and the dark energy, providing the negative

pressure for spatial expansion, can be explained with irrev

ersible thermodynamics

Research Archive of Indian Institute of Technology Hyderabad

PS1 :: National Conference on CICAHEP, Dibrugarh (2015), 01, 63 – 67Explanation of the accelerated expansion of the universe and dark energywith irreversible thermodynamics∗Debobrata RajakDepartment of Physics, Indian Institute of Technology Hyderabad, Yeddumailaram-502205, Telangana, Indiaph15mscst11009@iith.ac.in; rajak.vishal@yahoo.in; COS.5, COS.8, Poster, CICAHEP15.99In this paper, by invoking the laws of irreversible thermodynamics the accelerated expansion of the universe isexplained. It is shown that the entropy of the universe, at any particular instant of time, plays a significant rolein the accelerated expansion of the universe. Considering the universe to be filled with a classical mono-atomic,homogeneous and isotropic gas under classical non-equilibrium situations, two generalized forces causing theexpansion of the universe are arrived at. One of the two forces, the trivial force, has affinity to volume expansionand the non-trivial force has affinity to spatial expansion. The acceleration of the expansion of the universe is dueto the spatial expansion caused by the non-trivial force and which in turn might account for the presence of the darkenergy. It is shown in this paper that the non-trivial generalized force and the dark energy, providing the negativepressure for spatial expansion, can be explained with irreversible thermodynamics.1. IntroductionSince the experimental validation of the accelerated expansion of the universe and the associated dimming ofthe Type Ia supernovae [1], there has been a plethora of theoretical constructions to explain the dynamics of theuniverse, each one with increasing number of complexities. There are candidates that explain this by assuming theexistence of exotic matter with negative pressure, etc. [2, 3, 4, 5, 6].The homogeneous and isotropic cosmological models based on FLRW metric, having proved to be remarkablysuccessful. But with the introduction of data from WMAP and distant supernova together with the the data fromsupernovae and galaxy distributions and cosmic microwave background anisotropies lead to introduction of cos-mological constant Λ or vacuum energy ΩΛ [7, 8]. Along these there are other approaches like that of dark energymodels which attempt to provide a dynamical explanation of the cosmological constant requiring fine-tuning [6].Together with modifications on cosmological scales of the general theory of relativity, like that of f(R) gravity the-ory which is plagued with instabilities in the metric formalism [9]. There are also in-homogeneous cosmologicalmodels based on spherically symmetric but in-homogeneous LTB metric [10].The relation between the pressure P , volume V and the internal energy of an ideal gas is related as PV = 23U .The relation like this, in general, leads to the study of thermodynamics systems of the form PV = ωU , where ω isa constant and can be positive or negative. The accelerating expansion of the universe and the associated dimmingof the Type Ia Supernovae is explained by assuming the existence of substances that exert negative pressure, withω < − 13 [3, 11]. The cold dark matter candidates of cosmological models are described phenomenologically byPV = ωU , where for ω = −1 corresponds to positive cosmological constant and ω < −1 corresponds to phantomdark energy characterised by null chemical potential [5].In this article I’ll deal only for ω = 23 which implies a universe filled with monoatomic ideal gas [4]. The normalstatistical thermodynamic equations for energy density, entropy density and chemical potential of the universeremains the same as that of an monoatomic ideal gas [3, 4].2. Entropy and the accelerated expansion of the universeIn nature reversible process are fictive. Rather all the process are irreversible in nature. This comes from the secondlaw of thermodynamics. Which states that the entropy of any isolated system always increases for any process, ormathematically we have dS ≥ 0 where S is the entropy of the system [12, 15]. Now the universe being an isolatedsystem the entropy of the universes always increases. The Sauckur-Tetrode equation for entropy of a gas obtained∗Best poster presentation in the Cosmology section.6364 Debobrata Rajakby following the Boltzmann statistic is given asS =5Nk2+Nk ln[V (2pimkT )2/3Nh3], (1)where k = 1.381× 10−23J/K is the Boltzmann constant, V is the volume, T is the temperature,N is the numberof particles,m is the mass of the particles and h = 6.626× 10−34m2Kgs−1 [15]. A simple rearrangement of theabove equation will yield the volume V of the gas as a function of entropy S given asV = exp[SNk− 52− ln (2pimkT )32Nh3]. (2)Now, since at a given instant at some point of the universe some sort of irreversible process is always going on.As from the second law of thermodynamics it follows that for every single irreversible process the entropy of theuniverse increases. Due to which the entropy of the universe is a monotonic increasing function of time. The rateof change of entropy S′ as S′ = lim∆t→0∆S∆t =dSdt . Since entropy is a non-decreasing function of time henceS′ ≥ 0. On similar grounds we have the second derivative of entropy with respect to time defined as S′′ = d2Sdt2 .From thermodynamics we have the second time derivative of entropy defined as the rate of entropy productionand is negative as for any system to attain thermodynamic equilibrium has its entropy maximal. The rate of theexpansion of the universe is calculated by differentiating equation (2) with respect to time:V ′ =ddtexp[SNk− 52− ln (2pimkT )32Nh3]. (3)Differentiating again gives the acceleration of expansion of the universe and is given asV ′′ =[(SNk)′−(ln(2pimkT )32Nh3)′]2exp[SNk− 52− ln (2pimkT )32Nh3]+[(SNk)′′−(ln(2pimkT )32Nh3)′′]exp[SNk− 52− ln (2pimkT )32Nh3]. (4)On analyzing equation (4) we find that the term, exp[ SNk − 52 − ln(2pimkT )32Nh3 ] and the squared term, [(SNk )′−(ln (2pimkT )32Nh3 )′]2 are always positive as the exponential of any term is always ≥ 0 and the square of any numberis also positive. But the term [( SNk )′′ − (ln (2pimkT )32Nh3 )′′] is negative. Assuming that the term (ln (2pimkT )32Nh3 )′′ isnegligible compared to ( SNk )′′ we have the the second term to be negative as the rate of decrease of entropy isnegative.Although we must have certain contributions due to the (ln (2pimkT )32Nh3 )′′ as the universe was definitely hotter atthe beginning than now and thus T ′ must exist and must be negative and T ′′ too is negative as the universe cooledrapidly at the beginning than now. Though we know the behavior of T but we don’t know the behavior of N,h, kandm. As these might be different at different time of the universe.3. Generalized force governing the dynamics of universeIn practice, we can consider the universe to be composite thermodynamic system, with identical and non-identicalsubsystems separated by permeable diathermal wall and non-rigid boundaries. In addition to this I’ll considernon-equilibrium situations, where physical quantities like mass, temperature, pressure, etc. are a subject to changein both time and space and the flow of matter and the dynamic change of the various physical parameters of thesubsystems are governed by the gradient of the parameters and their time dependence. In this non-equilibriumuniverse we have spatially homogeneous states, dividing the system into small cells.To any irreversible thermodynamic process, one can in principle, associate two important types of parameters: oneExplanation of the accelerated expansion of the universe and dark energy with...... 65to describe the ’force’ that drives the process and the other to describe the ’response’ to the force [12, 14]. Again,from irreversible thermodynamics if the extensive parameters of two subsystems of a system are unconstrainedthen an equilibrium is reached when the affinity vanishes [12, 14, 15]The universe itself works similar to a composite irreversible thermodynamic system. As in one part of the universeor the other we have some sort of irreversible process happening due to which the entropy of the universe is alwaysincreasing.The processes in the universe being irreversible in nature we can, in principle, safely apply the laws of non-equilibrium thermodynamics. In non-equilibrium thermodynamics if an extensive parameter of two subsystemsare unconstrained, an equilibrium is obtained when the affinity vanishes [12, 13, 14].For a universe that is expanding at an accelerated rate we have the volume to be an extensive parameter that isunconstrained. Hence for an equilibrium to be reached we must have the affinity associated with volume to bezero. The affinity associated with the volume is mathematically defined as Av = (∂S0∂V )V 0 which must be zero fora system to be in an equilibrium state [12]. But if Av is non zero we have irreversible process taking place whichtakes the system, here the universe, to a state of equilibrium. A state of equilibrium is also defined as the stateat which the entropy of the system, here the universe, is maximum. Thus we can say that the a generalized forcethat drives the universe can be written as Av . Or that Av is the force that causes the accelerated expansion of theuniverse.The response to the generalized force is defined as the rate of change of the extensive parameter Jv =dVdt[12, 13, 14].3.1 Volume element of the expanding universeThe FLRW metric gives the volume element as [16]d3V = R3(t)1√1− ar2r2dr sin θdθdφ. (5)The above metric can be written asdV = dV (k(t, r, θ, φ), τ) = k(t)τ, (6)where τ = r2dr sin θdθdφ is the normal volume element and k(t, r, θ, φ) = R3(t) 1√1−ar2 . Where k is theexpansion factor and in general can be a function of both time and space. Signifying that the expansion can varyboth spatially and temporally different . Thus from equation (6) can be written as dV = (∂V∂τ )kdτ + (∂V∂k )τdk.3.2 Affinities of expansionThe rate of production of entropy is given asdSdt=∂S∂VdVdt. (7)Substituting dVdt =∂V∂τ kdτdt +∂V∂k τdkdt in the above equation, we getdSdt=∂S∂V[∂V∂τ kdτdt+∂V∂k τdkdt]. (8)Solving further and using the mathematical definition of flux, we have equation (8) to be given asdSdt=(∂S∂τ)kJτ +(∂S∂k)τJk. (9)Thus from equation (7) and equation (9) we have the affinity due to volume expansion to be given as (∂S∂τ )k = Akand the affinity due to spatial expansion to be given as (∂S∂k )τ = AkWhere the affinity due to volume expansion (∂S∂τ )k = Ak is the generalized force responsible for the increase in66 Debobrata Rajakthe mean free path of the constituents due to the actual displacement of the constituents comprising the universe.The response to the affinity Jτ =dτdt is the response to the affinity which the flux in the universe due to the actualmovement of the particles.The affinity due to spatial expansion (∂S∂k )τ = Ak is the generalized force responsible for the increase of the spacebetween the constituents of the universe by stretching the fabric of the space itself. The response to the affinityJk =dkdt is the flux due to the expansion of the fabric of space itself.4. Negative pressure and dark energyFrom the first law of thermodynamics, we have for a quasi-static process δW = δQ− dU where, δW is the workdone, δQ is the heat exchanged, dU is the change in the internal energy of the system. We have δW = −pdV . Inthe above expression if we substitute dV with dV = (∂V∂τ )kdτ + (∂V∂k )τdk we find,δW = −p[(∂V∂τ)k]dτ − p[(∂V∂k)τdk]. (10)From equation (10) we have the first component −p[(∂V∂τ )k]dτ as the normal work done while the second compo-nent −p[(∂V∂k )τ ]dk is of some interest as it represents the work done to expand the fabric of the space. Or in otherwords it is the work done to expand the space between any two particles in an ensemble.A comparison of the equation (10) with δW = −pdV we have −p[(∂V∂τ )k] is the generalized pressure due to themotion of the constituents while −p[(∂V∂k )τ ] is the generalized pressure due to the expansion of the space.The work done to expand the space is stored in the fabric of the space. A similar analogy can be derived fromthe shearing of an iron rod. When one applies a force to expand an iron rod the work done by the force is storedbetween the molecules of the solid in the form of inter-molecular forces. a similar kind of process happens for thefabric of space. To expand the space fabric one has to do some work which is stored in the fabric of space just likein the case of the of iron rod.5. ConclusionThe exponential factors present in equation (4) might refer to an inflationary process as the accelerated rate ofexpansion of the universe is proportional to an exponential factor [2]. But this doesn’t refer to an inflationaryprocess because the equation (1) is applicable only after the first elementary particles were formed i.e duringthe quark epoch. Referring to an era where the universe was filled with the quark gluon plasma as these arefermions and bosons and temperatures was high enough to safely assume them to be a gas of fermions and bosonsi.e. the quarks and gluons couldn’t have founded any bounded structures. Thus we can safely apply the Sackur-Tetrode equation of entropy for fermions and bosons respectively. Thus the total entropy will then be Stotal =Sfermions+Sbosons, where Stotal is the total entropy of the universe and Sfermions is the entropy due to fermionicparticles and Sbosons is the entropy due to bosonic particles. The form Stotal = Sfermions+Sbosons can be writtenas entropy is a extensive parameter. The Sackur-Tetrode equation for entropy thus is valid from the Quark epochto all way through to the current state of the universe.From the equation (4) one can predict that the active regions of the universe i.e. regions where the rate of changeof entropy dSdt is very high those regions will have higher red shits compared to the regions where thedSdt is lower.Though the Hubble’s law is not compromised rather along with that if the (dSdt ) is higher for a star system the fasterwill the system recede from us. Thus system like quasars and active galaxies have higher red shifts than the oneswhich have low dSdt .Considering the universe as a thermodynamic system under non-equilibrium conditions we see that there are twogeneralized forces that causes the expansion of the universe. One is a trivial force is the affinity due to volumeexpansion while the other, the non-trivial one is the affinity due to spatial expansion. This, affinity due to spatialexpansion, is also a reason for the accelerated expansion of the universe if not the only.Initial total energy released during the big bang was either converted into matter or was used to expand the space-Explanation of the accelerated expansion of the universe and dark energy with...... 67time or for other various process. And this accounts for the missing energy or the dark energy as it is also charac-terized by a negative generalized pressure −p[(∂V∂k )τ ].References[1] A. G. Riess et al., Astron. J. 116, 1009 (1998); S. Perlmutter et al., Astrophys. J. 517, 565 (1999); P. Astieret al., Astron. Astrophys. 447, 31 (2006); A. G. Riess et al., Astro. J. 659, 98 (2007).[2] A. H. Guth, Phys. Rev. D 23, 347 (1981).[3] J. A. S. Lima and S. H. Pereira Phys. Rev. D 78, 083504 (2008).[4] S. H. Pereira, arXiv:1002.4584v2.[5] S. H. Pereira and J. A. S. Lima, Phys. Lett. B 669, 266 (2008).[6] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006); N. Straumann, Mod. Phys.Lett. A 21, 1083 (2006).[7] D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003).[8] A. G. Riess et al., Astrophys. J. 607, 665 (2004); P. Astier et al., Astron. Astrophys. 447, 31 (2006).[9] A. D. Dolgov and M. Kawasaki, Phys. Lett. B 573, 1 (2003).[10] Kari Enqvist, arXiv:astro-ph/0709.2044v1.[11] T. Padmanabhan, Phys. Rept. 380, 235 (2003); P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003);J. A. S. Lima, Braz. J. Phys. 34, 194 (2004).[12] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, J. Wiley, (1985).[13] H. C. Ottinger, Beyond Equilibrium Thermodynamics, J. Wiley,(2005).[14] I. Prigogine, Introduction to Thermodynamics of Irreversible Process, J. Wiley, (1967).[15] R. K. Pathria and P. D. Beale, Statistical Mechanics, Elsevier, (2011); P. B. Pal, An Introductory Course ofStatistical Mechanics, Narosa Publishing House, (2011).[16] S. Weinberg, Cosmology, Oxford University Press,(2008); P. Sharan, Spacetime, Geometry and Gravitation,Hindustan Book Agency,(2009); S. Weinberg, Gravitation and Cosmology, J. Wiley (2014).

Explanation of the accelerated expansion of the universe and dark energy with irreversible thermodynamics∗

Name not available

PS1 :: National Conference on CICAHEP, Dibrugarh (2015), 01, 63 – 67
Explanation of the accelerated expansion of the universe and dark energy
with irreversible thermodynamics∗
Debobrata Rajak
Department of Physics, Indian Institute of Technology Hyderabad, Yeddumailaram-502205, Telangana, India
ph15mscst11009@iith.ac.in; rajak.vishal@yahoo.in; COS.5, COS.8, Poster, CICAHEP15.99
In this paper, by invoking the laws of irreversible thermodynamics the accelerated expansion of the universe is
explained. It is shown that the entropy of the universe, at any particular instant of time, plays a significant role
in the accelerated expansion of the universe. Considering the universe to be filled with a classical mono-atomic,
homogeneous and isotropic gas under classical non-equilibrium situations, two generalized forces causing the
expansion of the universe are arrived at. One of the two forces, the trivial force, has affinity to volume expansion
and the non-trivial force has affinity to spatial expansion. The acceleration of the expansion of the universe is due
to the spatial expansion caused by the non-trivial force and which in turn might account for the presence of the dark
energy. It is shown in this paper that the non-trivial generalized force and the dark energy, providing the negative
pressure for spatial expansion, can be explained with irreversible thermodynamics.
1. Introduction
Since the experimental validation of the accelerated expansion of the universe and the associated dimming of
the Type Ia supernovae [1], there has been a plethora of theoretical constructions to explain the dynamics of the
universe, each one with increasing number of complexities. There are candidates that explain this by assuming the
existence of exotic matter with negative pressure, etc. [2, 3, 4, 5, 6].
The homogeneous and isotropic cosmological models based on FLRW metric, having proved to be remarkably
successful. But with the introduction of data from WMAP and distant supernova together with the the data from
supernovae and galaxy distributions and cosmic microwave background anisotropies lead to introduction of cos-
mological constant Λ or vacuum energy ΩΛ [7, 8]. Along these there are other approaches like that of dark energy
models which attempt to provide a dynamical explanation of the cosmological constant requiring fine-tuning [6].
Together with modifications on cosmological scales of the general theory of relativity, like that of f(R) gravity the-
ory which is plagued with instabilities in the metric formalism [9]. There are also in-homogeneous cosmological
models based on spherically symmetric but in-homogeneous LTB metric [10].
The relation between the pressure P , volume V and the internal energy of an ideal gas is related as PV = 23U .
The relation like this, in general, leads to the study of thermodynamics systems of the form PV = ωU , where ω is
a constant and can be positive or negative. The accelerating expansion of the universe and the associated dimming
of the Type Ia Supernovae is explained by assuming the existence of substances that exert negative pressure, with
ω < − 13 [3, 11]. The cold dark matter candidates of cosmological models are described phenomenologically by
PV = ωU , where for ω = −1 corresponds to positive cosmological constant and ω < −1 corresponds to phantom
dark energy characterised by null chemical potential [5].
In this article I’ll deal only for ω = 23 which implies a universe filled with monoatomic ideal gas [4]. The normal
statistical thermodynamic equations for energy density, entropy density and chemical potential of the universe
remains the same as that of an monoatomic ideal gas [3, 4].
2. Entropy and the accelerated expansion of the universe
In nature reversible process are fictive. Rather all the process are irreversible in nature. This comes from the second
law of thermodynamics. Which states that the entropy of any isolated system always increases for any process, or
mathematically we have dS ≥ 0 where S is the entropy of the system [12, 15]. Now the universe being an isolated
system the entropy of the universes always increases. The Sauckur-Tetrode equation for entropy of a gas obtained
∗Best poster presentation in the Cosmology section.
63
64 Debobrata Rajak
by following the Boltzmann statistic is given as
S =
5Nk
2
+Nk ln
[
V (2pimkT )2/3
Nh3
]
, (1)
where k = 1.381× 10−23J/K is the Boltzmann constant, V is the volume, T is the temperature,N is the number
of particles,m is the mass of the particles and h = 6.626× 10−34m2Kgs−1 [15]. A simple rearrangement of the
above equation will yield the volume V of the gas as a function of entropy S given as
V = exp
[
S
Nk
− 5
2
− ln (2pimkT )
3
2
Nh3
]
. (2)
Now, since at a given instant at some point of the universe some sort of irreversible process is always going on.
As from the second law of thermodynamics it follows that for every single irreversible process the entropy of the
universe increases. Due to which the entropy of the universe is a monotonic increasing function of time. The rate
of change of entropy S′ as S′ = lim∆t→0
∆S
∆t =
dS
dt . Since entropy is a non-decreasing function of time hence
S′ ≥ 0. On similar grounds we have the second derivative of entropy with respect to time defined as S′′ = d2Sdt2 .
From thermodynamics we have the second time derivative of entropy defined as the rate of entropy production
and is negative as for any system to attain thermodynamic equilibrium has its entropy maximal. The rate of the
expansion of the universe is calculated by differentiating equation (2) with respect to time:
V ′ =
d
dt
exp
[
S
Nk
− 5
2
− ln (2pimkT )
3
2
Nh3
]
. (3)
Differentiating again gives the acceleration of expansion of the universe and is given as
V ′′ =
[(
S
Nk
)′
−
(
ln
(2pimkT )
3
2
Nh3
)′]2
exp
[
S
Nk
− 5
2
− ln (2pimkT )
3
2
Nh3
]
+
[(
S
Nk
)′′
−
(
ln
(2pimkT )
3
2
Nh3
)′′]
exp
[
S
Nk
− 5
2
− ln (2pimkT )
3
2
Nh3
]
. (4)
On analyzing equation (4) we find that the term, exp[ SNk − 52 − ln
(2pimkT )
3
2
Nh3 ] and the squared term, [(
S
Nk )
′−
(ln (2pimkT )
3
2
Nh3 )
′]2 are always positive as the exponential of any term is always ≥ 0 and the square of any number
is also positive. But the term [( SNk )
′′ − (ln (2pimkT )
3
2
Nh3 )
′′] is negative. Assuming that the term (ln (2pimkT )
3
2
Nh3 )
′′ is
negligible compared to ( SNk )
′′ we have the the second term to be negative as the rate of decrease of entropy is
negative.
Although we must have certain contributions due to the (ln (2pimkT )
3
2
Nh3 )
′′ as the universe was definitely hotter at
the beginning than now and thus T ′ must exist and must be negative and T ′′ too is negative as the universe cooled
rapidly at the beginning than now. Though we know the behavior of T but we don’t know the behavior of N,h, k
andm. As these might be different at different time of the universe.
3. Generalized force governing the dynamics of universe
In practice, we can consider the universe to be composite thermodynamic system, with identical and non-identical
subsystems separated by permeable diathermal wall and non-rigid boundaries. In addition to this I’ll consider
non-equilibrium situations, where physical quantities like mass, temperature, pressure, etc. are a subject to change
in both time and space and the flow of matter and the dynamic change of the various physical parameters of the
subsystems are governed by the gradient of the parameters and their time dependence. In this non-equilibrium
universe we have spatially homogeneous states, dividing the system into small cells.
To any irreversible thermodynamic process, one can in principle, associate two important types of parameters: one
Explanation of the accelerated expansion of the universe and dark energy with...... 65
to describe the ’force’ that drives the process and the other to describe the ’response’ to the force [12, 14]. Again,
from irreversible thermodynamics if the extensive parameters of two subsystems of a system are unconstrained
then an equilibrium is reached when the affinity vanishes [12, 14, 15]
The universe itself works similar to a composite irreversible thermodynamic system. As in one part of the universe
or the other we have some sort of irreversible process happening due to which the entropy of the universe is always
increasing.
The processes in the universe being irreversible in nature we can, in principle, safely apply the laws of non-
equilibrium thermodynamics. In non-equilibrium thermodynamics if an extensive parameter of two subsystems
are unconstrained, an equilibrium is obtained when the affinity vanishes [12, 13, 14].
For a universe that is expanding at an accelerated rate we have the volume to be an extensive parameter that is
unconstrained. Hence for an equilibrium to be reached we must have the affinity associated with volume to be
zero. The affinity associated with the volume is mathematically defined as Av = (
∂S0
∂V )V 0 which must be zero for
a system to be in an equilibrium state [12]. But if Av is non zero we have irreversible process taking place which
takes the system, here the universe, to a state of equilibrium. A state of equilibrium is also defined as the state
at which the entropy of the system, here the universe, is maximum. Thus we can say that the a generalized force
that drives the universe can be written as Av . Or that Av is the force that causes the accelerated expansion of the
universe.The response to the generalized force is defined as the rate of change of the extensive parameter Jv =
dV
dt
[12, 13, 14].
3.1 Volume element of the expanding universe
The FLRW metric gives the volume element as [16]
d3V = R3(t)
1√
1− ar2
r2dr sin θdθdφ. (5)
The above metric can be written as
dV = dV (k(t, r, θ, φ), τ) = k(t)τ, (6)
where τ = r2dr sin θdθdφ is the normal volume element and k(t, r, θ, φ) = R3(t) 1√
1−ar2 . Where k is the
expansion factor and in general can be a function of both time and space. Signifying that the expansion can vary
both spatially and temporally different . Thus from equation (6) can be written as dV = (∂V∂τ )kdτ + (
∂V
∂k )τdk.
3.2 Affinities of expansion
The rate of production of entropy is given as
dS
dt
=
∂S
∂V
dV
dt
. (7)
Substituting dVdt =
∂V
∂τ k
dτ
dt +
∂V
∂k τ
dk
dt in the above equation, we get
dS
dt
=
∂S
∂V
[
∂V
∂τ k
dτ
dt
+
∂V
∂k τ
dk
dt
]
. (8)
Solving further and using the mathematical definition of flux, we have equation (8) to be given as
dS
dt
=
(
∂S
∂τ
)
k
Jτ +
(
∂S
∂k
)
τ
Jk. (9)
Thus from equation (7) and equation (9) we have the affinity due to volume expansion to be given as (∂S∂τ )k = Ak
and the affinity due to spatial expansion to be given as (∂S∂k )τ = Ak
Where the affinity due to volume expansion (∂S∂τ )k = Ak is the generalized force responsible for the increase in
66 Debobrata Rajak
the mean free path of the constituents due to the actual displacement of the constituents comprising the universe.
The response to the affinity Jτ =
dτ
dt is the response to the affinity which the flux in the universe due to the actual
movement of the particles.
The affinity due to spatial expansion (∂S∂k )τ = Ak is the generalized force responsible for the increase of the space
between the constituents of the universe by stretching the fabric of the space itself. The response to the affinity
Jk =
dk
dt is the flux due to the expansion of the fabric of space itself.
4. Negative pressure and dark energy
From the first law of thermodynamics, we have for a quasi-static process δW = δQ− dU where, δW is the work
done, δQ is the heat exchanged, dU is the change in the internal energy of the system. We have δW = −pdV . In
the above expression if we substitute dV with dV = (∂V∂τ )kdτ + (
∂V
∂k )τdk we find,
δW = −p
[(
∂V
∂τ
)
k
]
dτ − p
[(
∂V
∂k
)
τ
dk
]
. (10)
From equation (10) we have the first component −p[(∂V∂τ )k]dτ as the normal work done while the second compo-
nent −p[(∂V∂k )τ ]dk is of some interest as it represents the work done to expand the fabric of the space. Or in other
words it is the work done to expand the space between any two particles in an ensemble.
A comparison of the equation (10) with δW = −pdV we have −p[(∂V∂τ )k] is the generalized pressure due to the
motion of the constituents while −p[(∂V∂k )τ ] is the generalized pressure due to the expansion of the space.
The work done to expand the space is stored in the fabric of the space. A similar analogy can be derived from
the shearing of an iron rod. When one applies a force to expand an iron rod the work done by the force is stored
between the molecules of the solid in the form of inter-molecular forces. a similar kind of process happens for the
fabric of space. To expand the space fabric one has to do some work which is stored in the fabric of space just like
in the case of the of iron rod.
5. Conclusion
The exponential factors present in equation (4) might refer to an inflationary process as the accelerated rate of
expansion of the universe is proportional to an exponential factor [2]. But this doesn’t refer to an inflationary
process because the equation (1) is applicable only after the first elementary particles were formed i.e during
the quark epoch. Referring to an era where the universe was filled with the quark gluon plasma as these are
fermions and bosons and temperatures was high enough to safely assume them to be a gas of fermions and bosons
i.e. the quarks and gluons couldn’t have founded any bounded structures. Thus we can safely apply the Sackur-
Tetrode equation of entropy for fermions and bosons respectively. Thus the total entropy will then be Stotal =
Sfermions+Sbosons, where Stotal is the total entropy of the universe and Sfermions is the entropy due to fermionic
particles and Sbosons is the entropy due to bosonic particles. The form Stotal = Sfermions+Sbosons can be written
as entropy is a extensive parameter. The Sackur-Tetrode equation for entropy thus is valid from the Quark epoch
to all way through to the current state of the universe.
From the equation (4) one can predict that the active regions of the universe i.e. regions where the rate of change
of entropy dSdt is very high those regions will have higher red shits compared to the regions where the
dS
dt is lower.
Though the Hubble’s law is not compromised rather along with that if the (dSdt ) is higher for a star system the faster
will the system recede from us. Thus system like quasars and active galaxies have higher red shifts than the ones
which have low dSdt .
Considering the universe as a thermodynamic system under non-equilibrium conditions we see that there are two
generalized forces that causes the expansion of the universe. One is a trivial force is the affinity due to volume
expansion while the other, the non-trivial one is the affinity due to spatial expansion. This, affinity due to spatial
expansion, is also a reason for the accelerated expansion of the universe if not the only.
Initial total energy released during the big bang was either converted into matter or was used to expand the space-
Explanation of the accelerated expansion of the universe and dark energy with...... 67
time or for other various process. And this accounts for the missing energy or the dark energy as it is also charac-
terized by a negative generalized pressure −p[(∂V∂k )τ ].
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Explanation of the accelerated expansion of the universe and dark energy with irreversible thermodynamics∗

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