Nonequilibrium thermodynamics of dark energy onthe power-law entropy-corrected apparent horizon

M. Umar Farooq and Mubasher Jamil

Abstract: We investigate the Friedmann–Robertson–Walker (FRW) universe (containing dark energy) as a nonequilibrium(irreversible) thermodynamic system by considering the power-law correction to the horizon entropy. By taking the power-law entropy area law that appears in dealing with the entanglement of quantum fields both within and outside of the hori-zon, we determine the power-law entropy-corrected apparent horizon of the FRW universe.

PACS Nos: 98.80.–k

Résumé : Nous étudions l’univers de Friedmann–Robertson–Walker (FRW), qui inclurait de l’énergie sombre, comme unsystème thermodynamique (irréversible) hors d’équilibre, en considérant la correction en loi de puissance pour l’entropie del’horizon. En prenant une loi de puissance pour la surface entropique qui apparaît lorsqu’on s’occupe de l’intrication deschamps quantiques à l’intérieur et à l’extérieur de l’horizon, nous déterminons l’horizon avec sa correction en loi de puis-sance pour l’entropie dans un univers FRW.

[Traduit par la Rédaction]

1. IntroductionIt is quite well known that black holes behave like black

bodies, emitting thermal radiation, with temperatures propor-tional to their surface gravity at the black hole horizon andwith an entropy that is proportional to the horizon area [1,2]. Further, it is also known that the Hawking temperatureand horizon entropy, together with the black hole mass, obeythe first law of thermodynamics: dM = TdS [3]. In the litera-ture, the thermodynamical features of black holes have beenstudied by taking dark energy into account [4–7], which isequally well-defined in this scenario for the apparent horizonof the black hole, while there is no more discussion concern-ing the event horizon.It is generally believed that the current universe is in an

accelerated expansion phase because of the presence of darkenergy. The holographic dark energy (HDE) has been studiedextensively in the literature [8–10] and proposed as an inter-esting candidate for dark energy that is motivated by theholographic principle [11]. In the derivation of HDE density,rD = 3c2M2

pL–2, the black hole entropy plays a key role,SBH = A/4G where A ~ L2 and A represents the area of thehorizon [12]. Here M2

p = (8pG)–1 is the modified Planckmass and 3c2 is constant and introduced for convenience.However, this definition of the entropy–area relationship

can be modified using power-law corrections to entropy thatarise in studying the entanglement of quantum fields withinand outside of the horizon of the black hole [13]. Thepower-law corrected entropy relation is given by the equation[14]

S ¼ A

4G

h1� KaA

1�ða=2Þi

ð1Þ

where a is a dimensionless constant whose value is still amatter of debate, and

Ka ¼ að4pÞa=2�1

ð4� aÞr2�ac

ð2Þ

where rc is the crossover scale. In (1), the second term is de-fined to be the power-law correction to the area law andmainly appears owing to entanglement when the wave func-tion is defined in terms of ground state and excited state[13]. So the entanglement entropy of the ground state fulfilsthe Hawking area law while the excited state is related to thecorrection, and more excitations produce more deviation fromthe area law. For more detail see ref. 15, which provides astrong basis for entanglement to be used as a source of blackhole entropy.It is important to mention that the correction term falls off

rapidly with A and hence, in the semi-classical limit (for largeA), the area law is recovered, while in the case of small blackholes the correction term will play a significant role. For in-stance, at low energies in the case of large horizon area it isdifficult to excite the modes, and therefore the ground statemodes contribute to most of the entanglement entropy. Onthe other hand, for a small area, a large number of fieldmodes will be excited and play a significant role in the cor-rection causing large deviations from the area law.In the cosmological framework, the study of nonequili-

brium thermodynamics of dark energy has some interesting

Received 17 September 2011. Accepted 22 October 2011. Published at www.nrcresearchpress.com/cjp on 17 November 2011.

M.U. Farooq. Center for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad, Pakistan;Department of Basic Sciences, College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan.M. Jamil. Center for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad, Pakistan.

Corresponding author: Mubasher Jamil (e-mail: mjamil@camp.nust.edu.pk).

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Can. J. Phys. 89: 1251–1254 (2011) doi:10.1139/P11-126 Published by NRC Research Press

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insights. To explore the hidden features of the correctionterm, Das et al. [13] computed leading-order corrections tothe entropy of a thermodynamical system caused by smallstatistical fluctuations around equilibrium. They showed thatone can obtain a general logarithmic correction related to theblack hole entropy. The thermodynamical explanation of theinteraction between HDE and dark matter is given in detailin ref. 16, and taking the logarithmic correction to the equili-brium entropy, they derived an expression for the interactionterm that was consistent with the observational data. Pavonand Wang [17] proposed a system consisting of dark energyand dark matter (with equilibrium entropies). They arguedthat if there is a transfer of energy between dark energy anddark matter, the entropy of dark matter is directly linked withthe entropy of dark energy. Further, Zhou et al. [18] studiedthe natural interaction between dark matter and dark energyin the universe by resorting to the extended thermodynamicsof irreversible processes. Moreover, Karami and Ghaffari [19]further extended the work done in ref. 18 and examined thevalidity of the generalized second-law in nonequilibrium (irre-versible) thermodynamics in a nonflat Friedmann–Robertson–Walker (FRW) universe (with dynamical horizon) in whichdark energy interacts with dark matter. In ref. 20, Wangand Wen-Biao discussed the nonequilibrium thermodynam-ics of dark energy on cosmic apparent horizons. They ar-gued that if irreversible processes are considered, theproper position for constructing thermodynamics would nolonger be the apparent horizon. The new position would berelated to the dark energy state equation and the irreversibleprocess parameters.In this paper, our aim is to extend the work done by Wang

and Wen-Biao by introducing the power-law correction to thearea law that appears in dealing with the entanglement ofquantum fields within and outside of the horizon. Consider-ing irreversible processes, we find that the nonequilibriumthermodynamic laws cannot be built on the original apparenthorizon, and develop a relationship that exhibits how the ap-parent horizon can be modified to keep the nonequilibriumthermodynamic law in effect. The outline of the paper is asfollows. The next section is devoted to the nonequilibriumthermodynamics of dark energy on the power-law entropycorrected apparent horizon. Concluding remarks are given inSect. 3.

2. Nonequilibrium thermodynamics of darkenergy on the power-law entropy-correctedapparent horizon

The metric of a homogeneous and isotropic universe in theFRW model is

ds2 ¼ �dt2 þ a2ðtÞ dr2

1� kr2þ r2dU2

� �ð3Þ

where a(t) is the scale factor and k (= –1, 0, 1) is the con-stant curvature of their spatial section. By defining~r ¼ aðtÞr, the location of the dynamical apparent horizoncan be determined by setting f = 0 in the relation

f ¼ gab~r;a~r;b ¼ 1� H2 þ k

a2

� �~r2 ð4Þ

which is

~r ¼ H2 þ k

a2

� ��ð1=2Þð5Þ

The observational data suggest that our universe is spatiallyflat with k = 0. In this case, the apparent horizon is ~r = 1/H, which is equal to the Hubble horizon RA (i.e., RA = 1/H).We can define the surface gravity, k, and Hawking temper-

ature, respectively, on the apparent horizon as [7]

k ¼ � 1

2

@f

@~r¼ ~r

R2A

ð6Þ

and

T ¼ k

2p¼ ~r

2pR2A

ð7Þ

So the Hawking temperature at the apparent horizon has theform

TA ¼ T���~r¼RA

¼ 1

2pRA

ð8Þ

Cai et al. [21] showed that, for an apparent horizon of anFRW universe, there exists Hawking radiation with tempera-ture 1/2pRA For metric (3), the first Friedmann equation be-comes

_a

a

� �2

¼ 8

3pr ð9Þ

where G = 1 and r is the energy density of dark energy. Thecontinuity equation for dark energy is

_rþ 3Hð1þ uÞr ¼ 0 ð10Þwhere u is the parameter of the equation of state of dark en-ergy, and when u < –1/3 it exhibits accelerating behavior ofthe universe. After defining e = (3/2)(1 + u) for a definiteequation of state, we obtain

aðtÞ ¼ t1=e 0 < e < 1 � 1 < u < � 1

3ð11Þ

and the apparent horizon can be written as RA = 1/H = et.The amount of energy flux crossing the apparent horizon

within the time interval dt is

�dEA ¼ 4pR2Arð1þ uÞdt ¼ edt ð12Þ

Because the process of energy flux crossing the apparent hor-izon is irreversible, internal entropy production will be gener-ated by this irreversibility. Thus, in the presence ofinteraction between dark energy and the horizon within, thetime-derivative of the nonequilibrium entropy is expressedas1

_S ¼ _Se þ _Si ð13Þwhere _Si shows the rate of change in internal entropy produc-tion of the universe while _Se appears as heat flow betweenthe universe and the horizon. If _Se = 0, then this case corre-sponds to equilibrium thermodynamics (i.e., the study ofthermodynamics dealing with no transfer of energy or en-

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tropy across the physical boundary of the system). We men-tion here that irreversible (nonequilibrium) thermodynamicshas been studied in the literature for various dynamicalspace–times in different gravitational theories [22].Following the steps given in ref. 20, we can have

_Si ¼IV

sdV ð14Þ

_Se ¼ �IS

Js � dS ð15Þ

where s and Js are the internal entropy source productiondensity and entropy flow density, respectively. Again follow-ing ref. 20, by considering the heat conduction between theuniverse and the horizon, s and Js can be expressed as

s ¼ Jq � r1

T

���~r¼RA

ð16Þ

Js ¼ JqTA

ð17Þ

where Jq is the heat current. After substituting (17) into (15),we obtain

_Se ¼ � 1

TA

IS

Jq � dS ¼ 1

TAJqA ð18Þ

where we have assumed that Jq takes the same value at anypoint on the apparent horizon with surface A = 4pR2

A. Frompower-law entropy relation (1), we can write

d

dt

�A

4

�1� KaA

½1�ða=2Þ���

¼ 2pRA_RA 1� Ka

4� a

2

� ��4pR2

A

�1�ða=2Þ" #ð19Þ

Taking the relation

dðSÞ ¼ deS ð20Þwe can have

Jq ¼ e

4pR2A

1� Ka

4� a

2

� ��4pR2

A

�1�ða=2Þ" #

e ¼ _RA ð21ÞAccording to Fourier’s law

Jq ¼ �lrT ð22Þwhere l is the thermal conductivity [20]. So there will be anenergy flux Jq if there is a ∇T in a thermodynamical system.After substituting (21) and (22) into (16), we obtain

s ¼ e2

4lR2A

1� Ka

4� a

2

� ��4pR2

A

�1�ða=2Þ" #2

ð23Þ

So after substituting the value of s into (14), we obtain

_Si ¼ e2pRA

3l1� Ka

4� a

2

� ��4pR2

A

�1�ða=2Þ" #2

ð24Þ

Now from (19) and (24) the final expression of irreversibleentropy can be written as

_S ¼ _Se þ _Si

¼ 2pRAe 1� Ka

4� a

2

� ��4pR2

A

�1�ða=2Þ" #

� 1þ e

6l� e

6lKa

4� a

2

� ��4pR2

A

�1�ða=2Þ" #ð25Þ

Therefore, the first law in irreversible thermodynamics holdsif we define

dS ¼ � dE

~TA

¼ 2p ~RAe dt ð26Þ

where

~TA ¼ 1

2p ~RA

and

~RA ¼ RA 1� Ka

4� a

2

� ��4pRA

�1�ða=2Þ" #(

� 1þ e

6l� e

6lKa

4� a

2

� ��4pR2

A

�1�ða=2Þ" #�ð27Þ

which is also called the power-law entropy corrected apparenthorizon. To check consistency, if we put a = 0, we obtain

~RA ¼ RA 1þ e

6l

� �ð28Þ

which is the same as appeared in ref. 20.

3. ConclusionIn this paper, we considered the FRW universe and inves-

tigated nonequilibrium features of it by taking the power-lawcorrection to the horizon entropy. In an irreversible scenario,as energy goes outside the horizon an internal entropy pro-duction term plays a significant role, which comes out tozero in the case of equilibrium. In ref. 20, they showed that,by considering irreversible processes, the nonequilibriumthermodynamic laws cannot hold true on the original appa-rent horizon and should be modified. They derived an ex-pression for the modified apparent horizon that depends on

1In thermodynamics, the internal entropy is defined as the variation in entropy inside the thermal system, referring to irreversible processes,defined as dS = dSe + dSi, where dSe is the entropy exchange with the surroundings or across the physical boundary of the system. Dividingby dt yields dS/dt = dSe/dt + dSi/dt or (13). In reversible thermodynamics, the entropy exchange term is zero and the total entropy is theinternal entropy itself. Note that all natural processes are irreversible, and thus the physical universe as a whole is an irreversible thermalsystem.

Farooq and Jamil 1253

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the state parameter of dark energy, e, and a nonequilibriumfactor l. Here we extended their study by taking the power-law correction to entropy which appears in dealing with theentanglement of quantum fields both within and outside ofthe horizon. The appearance of power-law correction termsto the horizon entropy is a fundamental prediction of quan-tum entanglement and must be considered in performing thethermodynamic study of a horizon for the FRW universe. Sofrom the power-law correction to entropy, we determined theentropy-corrected form of the apparent horizon of the FRWuniverse.Because the power-law entropy-corrected apparent horizon

is a cosmological horizon, it can be useful in a variety ofways. For instance, in f(T) gravity, the thermodynamic prop-erties of this horizon can be investigated following ref. 23.Also, in fractional action cosmology, one can study the gen-eralized second law of thermodynamics at the the power-lawentropy-corrected apparent horizon following ref. 24. Simi-larly, one can investigate thermal properties of interactingholographic and new-age graphic dark energy models withcorrected apparent horizon [25].

AcknowledgmentWe would like to thank the referees for giving useful com-

ments on our work.

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