Thermodynamics of Various Entropies in Specific Modified Gravity with Particle Creation

We consider the particle creation scenario in the dynamical Chern-Simons modified gravity in the presence of perfect fluid equation of state $p=(\gamma-1)\rho$. By assuming various modified entropies (Bekenstein, logarithmic, power law correction and Reyni), we investigate the first law of thermodynamics and generalized second law of thermodynamics on the apparent horizon. In the presence of particle creation rate, we discuss the generalized second law of thermodynamics and thermal equilibrium condition. It is found that thermodynamic laws and equilibrium condition remain valid under certain conditions of parameters.


Introduction
think that the expansion of the universe would be decelerating because of the gravitational attraction of the matter in the universe. Scientists have proposed various aspects of dynamical DE models such as quintessence [3], K-essence [4], phantom [5], quintom [6], tachyon [7], holographic [8] and pilgrim DE [9]- [13]. The simplest candidate of DE is cosmological constant but its formation and mechanisms are unknown. In order to explain the cosmic acceleration, various DE models and modified theories of gravity have been predicted such as f (R), f (T ) [14]- [16], f (T, T G ) [17,18], f (T, T) [19,20], dynamical Chern-Simons modified gravity [21]- [24] etc.
In cosmology, attempts to disclose the connection between Einstein gravity and thermodynamics were carried out in [25]- [35]. The basic concept of thermodynamics comes from black hole physics. In general, there has been some profound thought on the connection among gravity and thermodynamics for a long time. The initial work was done by Jacobson who showed that the gravitational Einstein equation can be derived from the relation between the horizon area and entropy, together with the Clausius relation δQ = T δS (where δQ, T and δS represents the change in energy, temperature and entropy change o the system respectively) [36]. Further, various gravity theories has been investigated to study the deep connection between gravity and thermodynamics [37]- [44]. Cosmological investigations of thermodynamics in modified gravity theories have been executed in Refs. [45]- [51] (for a recent review on thermodynamic properties of modified gravity theories, see, e.g., [52]). Saha and Mondal [53] have studied thermodynamics on apparent horizon with the help of gravitationally induced particle scenario with constant specific entropy and particle creation rate Γ.
In the present work, we develop the particle creation scenario in the dynamical Chern-Simons modified gravity by assuming various modified entropies (bekenstein entropy logarithmic entropy, power law correction and Reyni entropy) on the apparent horizon. In the presence of particle creation rate, we discuss the generalized second law of thermodynamics (GSLT). This paper is organized as follows: In the next section, we will present the basic equations of dynamical Chern-Simons modified theory and particle creation rate. Also, we investigate the first law of thermodynamics. Section 3 contains the illustration of GSLT. In section 4, we analyze the stability of thermodynamical equilibrium for constant as well as variable Γ. Section 5 contains the comparison of results with preceding works. In the last section, we summarize our results.

Modified Entropies and Particle Creation Rate
It was shown that the differential form of the Friedmann equation in the FRW universe can be written in the form of the first law of thermodynamics on the apparent horizon. The profound connection provides a thermodynamical interpretation of gravity which makes it interesting to explore the cosmological properties through thermodynamics. It was proved that for any spherically symmetric spacetime, the field equations can be expressed as T dS = dE + P dV for any horizon [54], where E, P and V represent the internal energy, pressure and volume of the spherical system respectively. The generalized second law of thermodynamics (GSLT) has been studied extensively in the behavior of expanding universe. According to GSLT, " the sum of all entropies of the constituents (mainly DM and DE) and entropy of boundary (either it is apparent or event horizons) of the universe can never decrease." [55]. Most of the researchers have discussed the validity of GSLT of different systems including interaction of two fluid components dark energy (DE) and dark matter (DM) [56] and interaction of three fluid components [57] in FRW universe.
To discuss the behavior of GSLT, scientists assumed the horizon entropy as 1/4 of its area [58], power law correction [59], logarithmic entropy [60] and Reyni entropy. GSLT has been discussed on the basis of gravitationally induced particle scenario which was firstly introduced by Schrodinger [61] on microscopic level. Parker at al. [62]- [66] extend this mechanism towards quantum field theory in curved space. Prigogine at al. [67] introduced the macroscopic mechanism of gravitationally induced particle scenario. Afterward, covariant description and difference between particle creation and bulk viscosity of creation process was given [68]- [70]. The particle creation process can be predicted with the incorporation of backreaction term in the Einstein field equations whose negative weight may help in clarifying the cosmic acceleration. In such a way, most of the phenomenological models of particle creation have been granted [71]- [76]. In addition, it was proved that phenomenological particle creation help us to discuss the behavior of accelerating universe and paved the alternative way to the concordance ΛCDM model [76]- [80].
In the following discussion, we check the validity of first law of thermodynamics with Gibbs relation, GSLT and thermodynamical equilibrium by assuming the following entropy corrections.
• Bekenstein entropy: The Bekenstein entropy and Hawking temperature of the apparent horizon are given by (8π = G = 1) • Logarithmic corrected entropy: To study the expansion of entropy of the universe, we discuss the addition of entropy related to the horizon. Quantum gravity allows the logarithmic corrections in the presence of thermal equilibrium fluctuations and quantum fluctuations [81]- [87]. The logarithmic entropy corrections can be defined as where L P is the Planck's length and α, β are constants whose values are still under consideration.
• Power law corrected entropy: Thermodynamics of apparent horizon in the standard FRW cosmology, the geometric entropy is assumed to be proportional to its horizon area (S A = A 4 ). The quantum corrections provided to the entropy-area relationship lead to the curvature correction in the Einstein-Hilbert action and vice versa [88]. The power-law quantum correction to the horizon entropy motivated by the entanglement of quantum fields between inside and outside of the horizon is given by [89] where Here δ is a dimensionless constant and r c is the crossover scale.
• Renyi entropy: A novel sort of Renyi entropy has been proposed and inspected [90]- [92]. In which not exclusively is the logarithmic corrected entropy of the original Renyi entropy utilized yet the Bekenstein Hawking entropy S BH is thought to be a non-extensive Tsallis entropy S A . One can obtain Renyi entropy S R as [92] The action of Chern-Simons theory is given by [21]-[23] where R, ⋆ R ρσµν R ρσµν , ℓ, θ, S mat , and V (θ) are Ricci scalar, a topological invariant called the Pontryagin term, the coupling constant, the dynamical variable, the action of matter and the potential, respectively. The Freidmann equation for flat universe turns out to be [24] here c is constant, H =˙a (t) a(t) is the Hubble parameter and a(t) is the scale factor. The equation of continuity for this model can be described aṡ ρ + Θ ρ + P + Π = 0.
The particle creation pressure (Π) representing the gravitationally induced process for particle creation and Θ = 3H is the fluid expansion. The total number of n−particles in an open thermodynamics arė where Γ is the creation rate of number of particles in comoving volume i.e., (N = na 3 ) having two phases negative and positive. The negative Γ relates with the particle anhilation and the positive Γ relates with production of particle. Equations (7) and (8) with Gibbs relations can be written as The equation related to creation pressure Π and Γ can be determined as Under traditional assumption that the specific entropy of each particle is constant, i.e., the process is adiabatic or isentropic, which implies that dissipative fluid is similar to a perfect fluid with a non-conserved particle number. The respective EoS for this model represented by p = (γ−1)ρ. Differentiation of Eq.(6) givesḢ Inserting Eqs. (10) and p = (γ − 1)ρ in above equation, we geṫ For flat FRW universe, Hubble parameter relates with the apparent horizon as R A = 1 H . Differentiating the apparent horizon with respect to time, we haveṘ The deceleration parameter q is of the form

First Law of Thermodynamics
Next, we investigate the first law of thermodynamics in the presence of modified entropies. The relation between thermodynamics and Einstein field equations was found by Jacobson with the help of clausius relation at apparent horizon described as For the sake of convenance we consider X = T A dS A + dE A . The differential dE A is the amount of energy crossing the apparent horizon can be evaluated as [93] − Bekenstein entropy: From Eq.(1), the differential of surface entropy at apparent horizon leads to The above equation with horizon temperature leads to Hence X becomes With the help of Eq. (19), we observe that the first law of thermodynamics holds (i.e., X → 0) when Logarithmic corrected entropy: The differential form of Eq.(2) is given as which leads to Combining Eqs. (16) and (22), we get From above equation, we observe the validity of first law of thermodynamics if Power law corrected entropy: which yields Hence, X takes the form From Eq. (27), it can be analyzed that the first law of thermodynamics remains valid for the following particle creation rate Renyi entropy: Eq.(4) gives the differential of entropy as which gives rise to Using Eqs. (16) and (30) we get From Eq. (31), it can be seen that the first law of thermodynamics is showing the validity when

Generalized Second Law of Thermodynamics
We discuss the GSLT of an isolated macroscopic physical system where the total entropy S T must satisfies the following conditions d(S A + S f ) ≥ 0 i.e., entropy function cannot be decrease. In this relation, S A and S f appear as the entropy at apparent horizon and the entropy of cosmic fluid enclosed within the horizon, respectively. The Gibbs equation is of the form where T f is the temperature of the cosmic fluid and E f is the energy of the fluid (E f = ρV ). The evolution equation for fluid temperature having constant entropy can be described as [94] T Using Eq. (12), we get ( , hence, the above Eq. (34) leads Integration of above equation leads to where T 0 is the constant of integration. Equation (33) yields the differential of fluid entropy as Next, we observe the validity GSLT by assuming Bekenstein entropy, logarithmic corrected entropy, power law correction and Renyi entropy.

Bekenstein entropy:
In present case, we get the differential of total entropy by using Eqs. (17) and (37) aṡ where S T = S A + S f . The plot betweenṠ T versus γ is shown in Figure 1  Using Eqs. (21) and (37) the differential of total entropy is given bẏ The plot ofṠ T versus γ with respect to the three values of T by fixing the constant values α = 1, β = −0.00001 and L P = 1 as shown in Figure 2. The trajectories in the plot remains which leads to the validity of GSLT in the presence of logarithmic entropy.
Power law corrected entropy: From Eqs. (26) and (37), we get the differential of total entropy aṡ The plot ofṠ T versus γ for three values of T by setting the constant values as δ = 1, r c = 1 67 , and L P = 1 as shown in Figure 3. We can seeṠ T remains positive for all values of T which leads to the validity of GSLT.

Renyi entropy:
We observe the validity of GSLT withṠ T ≥ 0 for which Eqs. (29) and (37) takes the forṁ The plot betweenṠ T versus γ for three values of is shown in Figure 4 for η = 1. It can be seen that GSLT is satisfyingṠ T ≥ 0 for all values of T which leads to the validity of GSLT.

Thermodynamical equilibrium
We discuss the two scenarios for thermodynamical equilibrium by taking particle creation rate Γ=constant and Γ = Γ(t) for which entropy function attains a maximum entropy state i.e., d(S A + S f ) < 0.

Γ = constant
Firstly, we consider Γ as constant and observe the stability of thermodynamical equilibrium.

Bekenstein entropy:
We get second order differential equation for constant Γ by using Eq.(38) The graphical behavior betweenS T versus γ as shown in Figure 5 for different values of parameter T . We observe thatS T < 0 for all values of T which leads to the validity of thermodynamical equilibrium.

Logarithmic corrected entropy:
In the presence of logarithmic corrected entropy second order differential equation is obtained from Eq.(39) as The graphical behavior betweenS T and γ is shown in Figure 6 for three values of T . We observe that all the trajectories are showing the negative increasing behavior, which exhibit the validity of thermodynamical equilib-rium.
Power law corrected entropy: To find the validity of thermodynamical equilibrium we take Γ as constant for which second order differential equation can be expressed with the help of Eq.(40) as The plot ofS T versus γ as shown in Figure 7 for three values of T . We analyze that thermodynamical equilibrium is satisfying the conditionS T < 0 for all values of T , which leads to validity of thermodynamical equilibrium. Renyi entropy: In present scenario, we observe the validity of thermodynamical equilibrium by keeping Γ as constant for which second order differential equation is given by using Eq.(41) The graphical behavior ofS T versus γ for three values of T as shown in Figure 8. It can be seen thatS T is negative for all values of T satisfying the condition d 2 S T /dt 2 < 0 which exhibits the thermodynamical equilibrium.

Bekenstein entropy:
The differentiation of Eq.(38) leads tö The plot betweenS T versus γ for three values of T as shown in Figure 9 by keeping the constant values same as for constant Γ. It is observed that the thermodynamical is obeying the conditionS T < 0 which leads to the validity of thermodynamical equilibrium.

Logarithmic corrected entropy:
We discuss the stability analysis of thermal equilibrium in the presence of logarithmic corrected entropy by taking Γ as variable for which Eq.(39) reduces to second order differential equation as The plot ofS T versus γ as shown in Figure 10 for three values of T by keeping the same values as in above case, we observe that thermal equilibrium condi-tionS T < 0 fulfill which leads to the validity of thermodynamical equilibrium.
Power law corrected entropy: For variable Γ, the differentiation of Eq.(40) turns out to bë The plot ofS T versus γ for three values of T by keeping the same values as above mentioned (Figure 11), one can observe easily the validity of thermodynamical equilibrium for all values of T withS T < 0.

Renyi entropy:
For variable Γ, Eq. (41) gives The graphical behavior ofS T versus γ for three values of T as shown in Figure 12 by keeping the constant values same as in above. One can see thaẗ S T is negative for all values of T satisfying the condition d 2 S T < 0 which leads to the validity of thermodynamical equilibrium.

Comparison
Here we provide some literature on underlying scenario for comparison and summary of present work. Harko et al. [95] considered the possibility of a gravitationally induced particle production through the mechanism of a non-minimal curvaturematter coupling. Firstly, they have reformulated the model in terms of an equivalent scalartensor theory, with two arbitrary potentials. The particle number creation rates, the creation pressure, and the entropy production rates have explicitly obtained as functions of the scalar field and its potentials as well as of the matter Lagrangian. The temperature evolution laws of the newly created particles have also obtained. The cosmological implications of the model have briefly investigated and it is shown that the late-time cosmic acceleration may be due to particle creation process. Furthermore, it has also shown that due to the curvaturematter coupling, during the cosmological evolution a large amount of comoving entropy is also produced.
Mitra et al. [96] have studied thermodynamics laws by assuming flat FRW universe enveloped by by apparent and event horizon in the framework of RSII brane model and DGP brane scenario. Assuming extended Hawking temperature on the horizon, the unified first law is examined for perfect fluid (with constant equation of state) and Modified Chaplygin Gas model. As a result there is a modification of Bekenstein entropy on the horizons. Further the validity of the GSLT and thermodynamical equilibrium have also been investigated. By assuming the gravitationally induced particle scenario with constant specific entropy and arbitrary particle creation rate (Γ), thermodynamics on the apparent horizon for FRW universe has been discussed [97]. They have investigated the first law, GSLT and thermodynamical equilibrium by assuming the EoS for perfect fluid and put forward various constraints on Γ for which thermodynamical laws hold.
Recently, we have done the thermodynamical analysis for gravitationally induced particle creation scenario in the framework of DGP braneworld model [98] by assuming usual entropy as well as its entropy corrections (power law and logarithmic) in the flat FRW universe. We have extracted EoS parameter and obtained its various constraints with respect to quintessence, vacuum and phantom era of the universe. For variable as well as constant particle creation rate (Γ), the first law of thermodynamics, GSLT and thermal equilibrium condition is satisfied in all the cases of entropy forms within some specific ranges of γ. In the present work, we have extended this work in the dynamical Chern-Simons modified gravity taking equation of state for perfect fluid as p = (γ −1)ρ. By assuming various modified entropies (Bekenstein, logarithmic, power law correction and Reyni), we investigate the first law of thermodynamics, equilibrium condition and generalized second law of thermodynamics on the apparent horizon in the presence of particle creation rate. It is concluded that the GSLT and thermodynamical equilibrium are satisfying the conditions dS T dt ≥ 0 and d 2 S T dt 2 < 0 for all values of T throughout the range 2 3 ≤ γ ≤ 2.
In section 4, we have discussed the thermal equilibrium phenomenon for Γ as variable and constant. The cosmic history is well-established through different observational sources that the radiation phase was followed by a matter dominated era which eventually passed through to a second de Sitter phase. Accordingly, it can be expected that in the radiation dominated era the entropy increased but the thermodynamic equilibrium was not achieved [99]. If this were not true, the universe would have attained a state of maximum entropy and would have stayed in it forever unless acted upon by some external agent. However, it is a well-known fact [100] that the production of particles was suppressed during the radiation phase, so in this model there would be no external agent to remove the system from thermodynamic equilibrium. In the present work, it is very difficult to find the analytical constraints to meet the equilibrium condition as discussed in the [101] due to lengthy expressions ofS T . Therefore, we checked these conditions graphically taking specific values of model parameters.
Since the prefect fluid is the simplest model in the cosmological studies, we study the prefect fluid case. In fact, this model can provide suitable results in the Einstein theory as well as modified theories [102,103]. Moreover, the motivation of the present work in the framework of the particle creation mechanism comes from some recent related works. It has been shown in [104,105] that the entire cosmic evolution from inflationary stage can be described by particle creation mechanism with some specific choices of the particle creation rates. As these works show late-time acceleration without any concept of dark energy, so, it is very interesting to think of the particle creation mechanism as an alternative way of explaining the idea of dark energy.

Conclusion
In this work, we have investigated the validity of first law of thermodynamics, GSLT and thermodynamical equilibrium for particle creation scenario in the presence of perfect fluid EoS p = (γ − 1)ρ by assuming the different entropy corrections such as Bakenstein entropy, logarithmic corrected entropy and power law corrected entropy and Renyi entropy in a newly proposed dynamical Chern-Simons modified gravity. We have summarized our results as follows: • For Bekenstein entropy We have analyzed that first law of thermodynamics is showing the validity for Γ = 3H c 2 γa 6 3H 2 − c 2 2a 6 −1 . However, GSLT remains valid for all values of T with 2 3 ≤ γ ≤ 2. Further, we have analyzed the validity of thermodynamical equilibrium for constant and variable Γ. From Figures(5 and 9), we observe that thermodynamical equilibrium is satisfying the condition d 2 S T dt 2 < 0 for all values of T with 2 3 ≤ γ ≤ 2.

• For Logarithmic corrected Entropy
In the presence of logarithmic corrected entropy it can be seen that the first law of thermodynamics is valid on the apparent horizon when . We have also investigated the validity of GSLT on apparent horizon satisfying the condition dS T dt ≥ 0 ( Figure 2). The graphical behavior ofS T versus γ as shown in Figures (6 and 10). We observe the validity of thermodynamical equilibrium for all values of T for all values of γ for constant as well as variable Γ.

• Power law corrected entropy
For power law corrected entropy we have investigates that first law of thermodynamics is hold at apparent horizon for Γ = 3H 1 − 1 Figure  3 we can analyzed that the GSLT is valid for all values of T with 2 3 ≤ γ ≤ 2. Further, we have investigated the validity of thermodynamical equilibrium obeying the conditionS T < 0 as shown in Figures(7 and  11) for all values of T with 2 3 ≤ γ ≤ 2 for both constant and variable Γ.

• For Renyi Entropy
In this entropy we have observed that first law of thermodynamics is holds when Γ = 3H 1 − 1 γ 3H 2 − c 2 . The Graphical behavior of Figure 12 shows that all trajectories remains positive for all values of T with 2 3 ≤ γ ≤ 2 which leads to the validity of GSLT. Moreover, thermodynamical equilibrium condition satisfied for all values of T with all values of γ for constant and variable Γ.