Thermodynamic Behavior of particular Gravity Models
We investigate the thermodynamics at the apparent horizon of the FRW universe in theory under non-equilibrium description. The laws of thermodynamics have been discussed for two particular models of theory. The first law of thermodynamics is expressed in the form of Clausius relation , where is the energy flux across the horizon and is the entropy production term. Furthermore, the conditions to preserve the generalized second law of thermodynamics are established with the constraints of positive temperature and attractive gravity. We have illustrated our results for some concrete models in this theory.
Keywords: Modified Gravity; Dark Energy; Apparent Horizon;
PACS: 04.50.Kd; 04.70.Df; 95.36.+x; 97.60.Lf.
Recent astrophysical observations indicate that expansion of the universe is presently in an accelerated epoch. The most compelling evidence for this is found in measurements of supernovae type Ia (SNeIa)  which is supported by renowned observations -. The mysterious component of energy named as dark energy (DE) is often introduced to explain this behavior of the universe. However, the mechanism responsible for the accelerated expansion is still under debate.
Two approaches have been used to illustrate the issue of current cosmic acceleration. Introducing an “exotic cosmic fluid” in the framework of Einstein gravity - is one direction to deal such issue, however this approach did not fully explain current empirical data. The other way is to discuss the modified theories of gravity such as [9, 10], , where ”” is the torsion scalar in teleparallel and , where and being the Ricci sclar and the trace of energy-momentum tensor - etc. The theory modifies Einstein Lagrangian through the coupling of matter and geometry. In fact, this modified gravity generalizes the theory and necessitates an arbitrary function of and . Recently, Bamba et al.  presented a comprehensive review of the problem of DE and modified theories.
Black hole thermodynamics suggests that there is a fundamental connection between gravitation and thermodynamics . Hawking radiations  together with; proportionality relation between temperature and surface gravity, also connection between horizon entropy and area of a black hole  further support this idea. Jacobson  was the first to deduce the Einstein field equations from the Clausius relation together with entropy proportional to the horizon area. In case of a general spherically symmetric spacetime, it was shown that the field equations can be constituted as the first law of thermodynamics (FLT) .
The relation between the FRW equations and the FLT was shown in  for . The field equations for FRW background were also formulated in Gauss-Bonnet and Lovelock theories by employing the corresponding entropy relation of static spherically symmetric black holes. Eling et al.  shown that we cannot find the correct field equations simply by using the Clausius relation in nonlinear theories of gravity. They remarked that the non-equilibrium description of thermodynamics is needed, whereby the Clausius relation is modified to , where is the entropy production term. In ref.- it was shown that the FRW field equations in general relativity (GR) and modified theories can be rewritten as (unified FLT on the trapping horizon suggested by Hayward ) with work term .
Wu et al.  developed a generalized procedure to construct FLT and the generalized second law of thermodynamics (GSLT) at the apparent horizon of Friedmann universe. The validity conditions of GSLT were studied in modified theories of gravity. Gong and Wang  showed that equilibrium thermodynamics is achievable for extended theories of gravity and entropy correction terms can be confined to mass-like functions. Other alternative approaches - have also been developed to reinterpret the non-equilibrium correction. In , we have explored the GSLT in theory and found necessary conditions for its validity. It has been shown that the equilibrium description isn’t feasible by redefining the dark energy components in theory.
In present work, the thermodynamics laws are examined for two particular models of theory. We show that the FRW equations can be rewritten in a from of the FLT . We formulate the GSLT and explore the conditions to validate this law. The paper is arranged in following format: In section 2, we present a brief introduction of theory. Section 3 is devoted to discuss the FLT and GSLT corresponding to the Friedmann equations of particular models. Finally, concluding remarks are given in section 4.
2 Gravity: An Overview
The modified gravity is described by the action 
where , defines the matter substances of the universe. The matter energy-momentum tensor is defined as 
The field equations can be found by varying the action of gravity with respect to the metric tensor
where and represent derivatives of with respect to and , respectively. The field equations depend on the source term , hence every selection of generates a particular set of field equations.
We consider the perfect fluid as matter source with matter Lagrangian so that is given by
Substituting this value in Eq.(3), it follows that
The spatially homogeneous and isotropic, ()-dimensional FRW universe is defined as
where is the 2-dimensional metric, is the scale factor and is the cosmic curvature; and is the metric of a ()-dimensional sphere. For , we have ()-dimensional FRW metric in Einstein gravity, while one can have in other gravity theories.
3 Thermodynamics in gravity
Now, we discuss the laws of thermodynamics for two particular choices of gravity .
Let us consider the following model
where and are arbitrary functions of and , respectively. The corresponding field equations are
where and . The choice of implies the field equation of gravity. In FRW background, the field equations will become
where , and
3.1.1 First Law of Thermodynamics
Now, the FLT is constructed for the above model. The condition, , gives the radius of the apparent horizon
The associated temperature is , where is the surface gravity . The temperature is positive for . Applying the definition of , the condition to keep positive is expressed as
In GR, the horizon entropy is defined as -, where represents area of the apparent horizon. Wald  proposed that in modified gravitational theories, the horizon entropy is associated with a Noether charge entropy. Brustein et al.  showed that Wald entropy is equivalent to where being the effective gravitational coupling. We can define the Wald entropy in theory as 
where for the first model. Following ref., one can obtain the FLT in the following form
where the energy flux is
is the total work density . Thus, the FLT can be expressed as
is the entropy production term developed for this model. This characterizes non-equilibrium treatment of the thermodynamics. The FLT for a flat FRW universe in theory [32, 33] can be retrieved from this result. For , the term vanishes, which leads to the FLT in Einstein gravity.
3.1.2 Generalized Second Law of Thermodynamics
Now we investigate the validity of GSLT in theory for this model. The FLT determines the horizon entropy given by Eq.(17). The composition of the entire matter and energy fluids within the horizon is given by Gibb’s equation 
where and represent the temperature and entropy of all contents within the horizon, respectively. There is a relation of temperature within the horizon to  i.e., , here to assure that . Consider being the sum of matter entropy within the horizon, horizon entropy and the non-equilibrium entropy production term.
The GSLT states that the time derivative of total entropy is not decreasing with time i.e.,
We can impose the constraint so that is positive. Applying the positive temperature condition with the temperature parameter , this gives
Thus the GSLT can be satisfied, provided that . If , then the GSLT is protected only if . If the gravitational coupling constant is indeed a constant i.e., , then the GSLT always holds. The condition to preserve the GSLT in theory can be reproduced if . For and , one get the inequality already constructed by Wu et al.  in non-linear gravity. In the thermal-equilibrium limit , the constraint to protect the GSLT is
The relation (22) depends on the choice of , for instance with results in
which is the condition for validity of GSLT in Einstein gravity.
Here, we discuss the validity of GSLT for some particular forms of
In the first case, we consider the model corresponding to the power law solution 
For this model, the Hubble and deceleration parameters are and . The validity of the GSLT in ()-dimensional flat FRW universe for the model (23) requires the condition
where . Now we present some constraints for particular values of .
For this solution represents CDM model and constraint on GSLT is given by
, which is true if with .
For , the GSLT is valid if , which requires .
For , we find if with . This choice would favor the expanding universe since .
The higher powers of curvature can be made available for larger values of and one can examine the validity of GSLT. If one consider the dust case , then the possible role of and can be seen from graphical description as shown in Figure 1.
In the second case, the GSLT for the model demands the following inequality to be fulfilled
Here we consider the power law solution of the form , where and are parameters. Following  for the dust case, we set and . Thus the above inequality takes the form
This shows that the GSLT holds for the power law model and its validity is shown in Figure 2.
A more general gravity model is of the form 
where are functions of and is function of . For a dust matter source, the field equation is obtained as
An equivalent Einstein field equation can be obtained with , whereas
In the following discussion, we define .
and which includes contributions from both matter and geometry. The total energy exchange term for this model is given by
Now we analyze the validity of the FLT and GSLT for the above model.
3.2.1 First Law of Thermodynamics
The Wald entropy for function (25) becomes
In this case, FLT involves the energy flux and entropy production terms of the form 
The gravity model, involves the explicit non-minimal gravitational coupling between matter and curvature. Results obtained using this theory would be different from other models such as theory. The coupling of matter and geometry reveals that the matter energy-momentum tensor is no longer conserved and there is an energy transfer between the two components. Due to this interaction, the energy exchange term is found to be non-zero, so the entropy production term would be an additional term in this modified gravity. Hence, the FLT is established in more general gravity and entropy production is induced in a non-equilibrium treatment of thermodynamics [21, 34]. In recent papers , Bamba et al. shown that the entropy production term can be incorporated by redefinition of the field equations. However, in this theory, such treatment is not useful as shown in .
3.2.2 Generalized Second Law of Thermodynamics
To develop the GSLT for the second model, we consider the Gibbs equation (18). The horizon entropy is determined from FLT. The necessary constraint for the validity of the GSLT is shown in Eq.(19). For the model (25), we obtain
The effective gravitational coupling constant for this model is . We can impose the condition to keep . For the positive temperature condition with , Eq.(34) is reduced to
This shows that the GSLT is valid only if . If the gravitational coupling constant is indeed a constant, the GSLT is always protected. If , then the GSLT can hold only if . The GSLT in theory can be retrieved for . If , then the condition (35) becomes
In this paper, the thermodynamics properties have been discussed in more general theory. The non-equilibrium treatment of thermodynamics is addressed for two particular models of gravity. In this modified theory, accelerated expansion may produce not just from scalar curvature part to the entire energy density of universe, but may include a matter component as well. The consequences of theory may contribute to significant results when compared to other modified gravitational theories, applicable to various problems of contemporary interest such as accelerated cosmic expansion, gravitational collapse, dark matter and the detection of gravitational waves . The detection of gravitational waves could be an excellent way to test general relativity and modified theories of gravity. Corda  has investigated the detection of gravitational waves in theory and it would be appealing to explore this issue in theory.
It is shown that representation of equilibrium thermodynamics is not executable in this theory . Hence the non-equilibrium treatment of thermodynamics is employed to discuss the laws of thermodynamics. Here, we studied two particular models of theory to show the consequences of explicit coupling of matter and geometry. The gravitational coupling between matter and higher derivatives terms of curvature describes a transfer of energy and momentum beyond that normally existing in curved spaces. This interaction leads to the entropy production term in this modified gravity. The FLT is formulated by employing the Wald’s entropy relation. We remark that an entropy production term is produced in this work but no such term in present in GR, Gauss-Bonnet , Lovelock  and braneworld  theories of gravity.
The validity of GSLT has also been investigated in this work. We have found that the GSLT holds with the conditions namely, attractive nature of gravity and temperature being positive. In fact, it is natural to assume the relation and proportionality constant can be considered as unity, implying that the system is in thermal-equilibrium. Generally, the horizon temperature cannot match the temperature of all energy sources within the horizon, and the two mechanisms must experience interaction for some interval of time ahead of achieving the thermal-equilibrium. Moreover, the gravitational curvature-matter coupling in theory may produce the unscripted flow of energy between the horizon and fluid. Also, the energy fluid of dark components does not permit the effective gravitational constant to be an approximate constant. In the limiting choice of thermal-equilibrium, we assume that is very close to . We find that the GSLT is fulfilled in both phantom and quintessence regimes of the universe which seems to be consistent with refs.. Furthermore, we have also developed constraints on some concrete models corresponding to power law solution. It is significant to remark that the equilibrium treatment of thermodynamics in theory would benefit from further study.
The authors would like to thank the Higher Education Commission, Islamabad, Pakistan for its financial support through the Indigenous Ph.D. 5000 Fellowship Program Batch-VII.
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