# Thermodynamics of Various Entropies in Specific Modified Gravity with Particle Creation

Authors: Abdul Jawad; Shamaila Rani; Salman Rafique
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We consider the particle creation scenario in the dynamical Chern-Simons modified gravity in the presence of perfect fluid equation of statep=(γ-1)ρ. By assuming various modified entropies (Bekenstein entropy logarithmic entropy, power law correction, and Renyi entropy), 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 thermodynamics laws and equilibrium condition remain valid under certain conditions of parameters.


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## Body

## 1. Introduction

Recently, different observations such as cosmic microwave background radiations (CMBR) [1] and sloan digital sky survey (SDSS) [2] have confirmed the accelerated expansion of universe. It is predicted that the expansion of the universe is due to the curious form of force, i.e., dark energy (DE). This discovery was unexpected, because before this invention cosmologists just 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], f(T,TG) [15, 16], f(T,T) [17, 18], and dynamical Chern-Simons modified gravity [19–22].In cosmology, attempts to disclose the connection between Einstein gravity and thermodynamics were carried out in [23–32]. 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=TδS (where δQ, T, and δS represent the change in energy, temperature, and entropy change o the system, respectively) [33]. Further, various gravity theories have been investigated to study the deep connection between gravity and thermodynamics [34–41]. Cosmological investigations of thermodynamics in modified gravity theories have been executed in [11, 42–54] (for a recent review on thermodynamic properties of modified gravity theories, see, e.g., [55]). Saha and Mondal [56] 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 Renyi 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. Sections3 and 4 contain the illustration of GSLT. In Section 5, we analyze the stability of thermodynamical equilibrium for constant as well as variable Γ. In the last section, we summarize our results.


## 2. 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 asTdS=dE+PdV for any horizon [37], 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 Hubble or apparent or event horizons) of the universe can never decrease” [57]. Most of the researchers have discussed the validity of GSLT of different system including interaction of two fluid components dark energy (DE) and dark matter (DM) [58–61] and interaction of three fluid components [62] in FRW universe.To discuss the behavior of GSLT scientist assumed horizon entropy1/4 of its area [63], power law correction [64, 65], logarithmic entropy [66], and Renyi entropy. GSLT has been discussed on the basis of gravitationally induced particle scenario which was firstly introduced by Schrodinger [67] on microscopic level. Parker et al. [68–72] extend this mechanism towards quantum field theory in curved space. Prigogine et al. [73] introduced the macroscopic mechanism of gravitationally induced particle scenario. Afterward, covariant description and difference between particle creation and bulk viscosity of creation process were given [74]. 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 [75, 76]. In addition, it was proved that phenomenological particle creation helps us to discuss the behavior of accelerating universe and paved the alternative way to the concordance ΛCDM model [77, 78].In the following discussion we will analyze the validity of first law of thermodynamics with Gibbs relation, GSLT, and thermodynamical equilibrium by assuming the following entropy corrections.(i)
Bekenstein entropy: the Bekenstein entropy and Hawking temperature of the apparent horizon are given by(8π=G=1)(1)SA=A4=RA28,TA=12πRA=4RAwhereA=4πRA2.(ii)
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 [79–85]. The logarithmic entropy corrections can be defined as(2)SA=A4Lp2+αln⁡A4Lp2+β4Lp2A,whereLP is the plank’s length and α, β are constants whose values are still under consideration.(iii)
Power law correction: in thermodynamics of apparent horizon in the standard FRW cosmology, the geometric entropy is assumed to be proportional to its horizon areaSA=A/4. The quantum corrections provided to the entropy-area relationship lead to the curvature correction in the Einstein-Hilbert action and vice versa [83, 86, 87]. 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 [88](3)SA=A4Lp21-KδA1-δ/2,whereKδ=δ(4π)δ/2-1/(4-δ)rc4-δ, δ is dimensionless constant, and rc is the crossover scale.(iv)
Renyi entropy: a novel sort of Renyi entropy has been proposed and inspected [89–91], in which not exclusively is the logarithmic corrected entropy of the original Renyi entropy utilized, yet the Bekenstein Hawking entropy SBH is thought to be a nonextensive Tsallis entropy SA. One can obtain Renyi entropy SR [91](4)SR=ln⁡1+ηSAηThe action of Chern-Simons theory is given by [19–21](5)S=116πG∫d4x-gR+l4θ⋆RρσμνRρσμν-12gμν∇μθ∇νθ+Vθ+Smat,where R, ⋆RρσμνRρσμν, l, θ, Smat, 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 Friedmann equation for flat universe turns out to be [22](6)H2=13ρ+c26a6,where c is constant, H=a˙(t)/a(t) is a Hubble parameter, and a(t) is the scale factor. The equation of continuity for this model can be described as(7)ρ˙+Θρ+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 is(8)n˙+Θn=nΓ,where Γ is the creation rate of number of particles in comoving volume, i.e., (N=na3) having two phases negative and positive. The negative Γ relates with the particle annihilation and the positive Γ relates to production of particle. Equations (7) and (8) with Gibbs relations can be written as(9)Tds=dρn+pd1n.The equation related to creation pressure Π and Γ can be determined as(10)Π=-ΓΘρ+p.Under traditional assumption, 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 nonconserved particle number. The respective EoS for this model is represented by p=(γ-1)ρ. Differentiation of (6) will give(11)H˙=-12ρ+p+Π-Hc2a6.Inserting (10) and p=(γ-1)ρ in the above equation, we get(12)H˙H2=-12H2γ3H2-c22a61-Γ3H+c2a6.For flat FRW universe, Hubble parameter relates to the apparent horizon as RA=1/H. Differentiating the apparent horizon with respect to time, we have(13)R˙A=-H˙H2=12H2γ3H2-c22a61-Γ3H+c2a6.The deceleration parameter q is of the form(14)q=-H˙H2-1=12H2γ3H2-c22a61-Γ3H+c2a6-1.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(15)-dEA=TAdSA.For the sake of convenience, we consider X=TAdSA+dEA. The differential dEA is the amount of energy crossing the apparent horizon which can be evaluated as [28](16)-dEA=12R3ρ+pHdt=γ2H23H2-c22a6.Bekenstein Entropy. From (1), the differential of surface entropy at apparent horizon leads to(17)dSA=18H3γ3H2-c22a61-Γ3H+c2a6.The above equation with horizon temperature leads to(18)TAdSA=12H2γ3H2-c22a61-Γ3H+c2a6.Hence X becomes(19)X=-γ2H23H2-c22a6+12H2γ3H2-c22a61-Γ3H+c2a6.With the help of (19) we observe that the first law of thermodynamics holds (i.e., X→0) when Γ=3H(c2/γa6(3H2-c2/2a6)-1).Logarithmic Corrected Entropy. The differential form of (2) is given as(20)dSA=γ3H2-c2/2a61-Γ/3H+c2/a61/4HLp2+2αH-16βH3Lp22H2dt,which leads to(21)TAdSA=12H2γ3H2-c22a61-Γ3H+c2a61Lp2+8H2α-64βH4Lp2.Combining (16) and (21), we get(22)X=12H2γ3H2-c22a61-Γ3H+c2a61Lp2+8H2α-64βH4Lp2-γ2H23H2-c22a6.From the above equation, we observe the validity of first law of thermodynamics for Γ=3H1-1/γ3H2-c2/a6-1γ3H2-c2/2a6/1/Lp2+8H2α-64βH4Lp2-c2/a6.Power Law Correction. Differentiating (3), we get(23)dSA=12H2γ3H2-c22a61-Γ3H+c2a614HLp2-Kδ4Lp22-δ21H3-δdt,which yields(24)TAdSA=12H2γ3H2-c22a61-Γ3H+c2a61Lp2-2-δ2KδLp21H2-δ.Hence, X takes the form(25)X=12H2γ3H2-c22a61-Γ3H+c2a61Lp2-2-δ2KδLp21H2-δ-γ2H23H2-c22a6.From (25), it can be analyzed that the first law of thermodynamics remains valid for the following particle creation rate:(26)Γ=3H1-1γ3H2-c22a6-1γ3H2-c22a61Lp2-2-δ2KδLp21H2-δ-1-c2a6.Renyi Entropy. Equation (4) gives the differential of entropy as(27)dSR=1Hγ3H2-c22a61-Γ3H+c2a61η+8H2,which gives rise to(28)TAdSR=4η+8H2γ3H2-c22a61-Γ3H+c2a6.Using (16) and (28), we get(29)X=4η+8H2γ3H2-c22a61-Γ3H+c2a6-γ2H23H2-c22a6.From (29), it can be seen that the first law of thermodynamics is showing the validity when Γ=3H(1-1/γ3H2-c2/2a6)-1γη+8H2/8H23H2-c2/2a6-c2/a6.


## 3. Generalized Second Law of Thermodynamics

We discuss the GSLT of an isolated macroscopic physical system where the total entropyST must satisfy the following conditions d(SA+Sf)≥0; i.e., entropy function cannot be decreased. In this relation, SA and Sf 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(30)TfdSf=dEf+pdV,where Tf is the temperature of the cosmic fluid and Ef is the energy of the fluid (Ef=ρV). The evolution equation for fluid temperature having constant entropy can be described as [92](31)T˙fTf=Γ-Θ∂p∂ρ.Using (12), we get (Γ-Θ)=6HH˙+3Hc2/a6/γ(3H2-c2/2a6); hence, (31) leads to the integral(32)ln⁡TfT0=γ-1γ∫2HH˙+Hc2/a6H2-c2/6a6dH.Integration of the above equation leads to(33)Tf=T0H2-c26a6γ-1/γ,where T0 is the constant of integration. Equation (30) yields the differential of fluid entropy as(34)dSf=T0-1H2H2-c26a61-γ/γ2H˙+c2a6-γH˙H23H2-c22a6.Next, we observe the validity of 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 (17) and (34) as(35)ST˙=18H3γ3H2-c22a61-Γ3H+c2a6+T0-1H2H2-c26a61-γ/γ×2H˙+c2a6-γH˙H23H2-c22a6,where ST=SA+Sf. The plot of ST˙ versus γ is shown in Figure 1 for three values of T. We observe the validity of GSLT at the present epoch by setting the constant values H=H0=67,  c=-1, a=a0=1, and q=-0.53. It can be analyzed from the figure that ST˙≥0 for all values of T leads to the validity of GSLT.Figure 1
Plot ofST˙ versus γ for Bekenstein entropy.Logarithmic Corrected Entropy. Using (20) and (34), the differential of total entropy is given by(36)ST˙=12H2γ3H2-c22a61-Γ3H+c2a614HLp2+2αH-16βH3Lp2dt+T0-1H2H2-c26a61-γ/γ2H˙+c2a6-γH˙H23H2-c22a6dt.The plot of ST˙ versus γ with respect to the three values of T by fixing the constant values α=1,  β=-0.00001, and LP=1 is shown in Figure 2. The trajectories in the plot remain which lead to the validity of GSLT in the presence of logarithmic entropy.Figure 2
Plot ofST˙ versus γ for logarithmic corrected entropy.Power Law Correction. From (24) and (34), we get the differential of total entropy as(37)ST˙=12H2γ3H2-c22a61-Γ3H+c2a614HLp2-Kδ4Lp22-δ21H3-δdt+T0-1H2H2-c26a61-γ/γ2H˙+c2a6-γH˙H23H2-c22a6dt.The plot of ST˙ versus γ for three values of T by setting the constant values as δ=1,  rc=1/67, and LP=1 is shown in Figure 3. We can see ST˙ remains positive for all values of T which leads to the validity of GSLT.Figure 3
Plot ofST˙ versus γ for power law correction.Renyi Entropy. We observe the validity of GSLT with ST˙≥0 for which (27) and (34) take the form(38)ST˙=1Hη+8H2γ3H2-c22a61-Γ3H+c2a6+T0-1H2H2-c26a61-γ/γ×2H˙+c2a6-γH˙H23H2-c22a6.The plot of ST˙ versus γ for three values is shown in Figure 4 for η=1. It can be seen that GSLT is satisfying ST˙≥0 for all values of T which leads to the validity of GSLT.Figure 4
Plot ofST˙ versus γ for Renyi entropy.


## 4. Thermodynamical Equilibrium

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


### 4.1.Γ = Constant

Firstly, we considerΓ as constant and observe the stability of thermodynamical equilibrium.Bekenstein Entropy. We get second-order differential equation for constant Γ by using (35):(39)ST¨=-3H˙c2/a6+γ1-Γ/3H-c2/2a6+3H28H4-2H˙-c26a6+H21-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0+-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0×c2a˙a7+2HH˙1-γH2-c26a6-1+1-γ/γ+γΓH˙3H2-c22a6124H5-6c2a˙8H3a7+γ1-Γ/3H3c2a˙/a7+6HH˙8H3+H2-c2/6a61-γ/γH2T0-γΓH˙3H2×3H2-c22a6-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H3c2a˙a7+6HH˙+γc2/a6+γ1-Γ/3H3H2-c2/2a62H2×3c2a˙a7+6HH˙+-6c2a˙/a7+γΓH˙3H2-c2/2a6/3H22H2+3c2a˙/a7+6HH˙2H2×γ1-Γ3H3H2-c22a6γ.The graphical behavior of ST¨ versus γ is shown in Figure 5 for different values of parameter T. We observe that ST¨<0 for all values of T which leads to the validity of thermodynamical equilibrium.Figure 5
Plot ofST¨ versus γ for Bekenstein entropy when Γ is constant.Logarithmic Corrected Entropy. In the presence of logarithmic corrected entropy second-order differential equation is obtained from (36) as(40)ST¨=-H˙1/4HLp2+2αH-16Lp2βH3c2/a6+γ1-Γ/3H3H2-c2/2a6H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a62αH˙-H˙/4Lp2H2-48Lp2βH2H˙2H2+14HLp2+2αH-16Lp2βH3-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γΓH˙3H2-c2/2a66H4+1H2T0H2-c26a61-γ/γ-γΓH˙3H2-c2/2a63H2-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H×3c2a˙a7+6HH˙+γ3c2a˙/a7+6HH˙c2/2a6+γ1-Γ/3H3H2-c2/2a62H2+γ3H2-c22a6γΓH˙3H2-c2/2a6/3H2+γ1-Γ/3H3c2a˙/a7+6HH˙2H2-6c2a˙2H2a7.The graphical behavior between ST¨ 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 exhibits the validity of thermodynamical equilibrium.Figure 6
Plot ofST¨ versus γ for logarithmic corrected entropy when Γ is a constant.Power Law Correction. To find the validity of thermodynamical equilibrium, we take Γ=constant for which second-order differential equation can be expressed with the help of (37) as(41)ST¨=-c2/a6+γ1-Γ/3H3H2-c2/2a61/4HLp2-2-δ/21/H3-δKδ/4Lp2H˙H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a6-H˙/4HLp2+3-δH˙2-δ/21/H4-δKδ/4Lp22H2+14HLp2-2-δ/21/H3-δKδ4HLp2-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γΓH˙3H2-c2/2a66H4+H2-c2/6a6-1-γ/γH2T0γΓH˙3H2-c2/2a63H2-γ3c2a˙a7+6HH˙1-Γ3H-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3+γ3c2a˙/a7+6HH˙c2/a6+γ1-Γ/3H3H2-c2/2a62H2+γ3H2-c22a6×-6c2a˙/a7+γΓH˙3H2-c2/2a6/3H2+γ1-Γ/3H3c2a˙/a7+6HH˙2H2.The plot of ST¨ versus γ is shown in Figure 7 for three values of T. We analyze that thermodynamical equilibrium is satisfying the condition ST¨<0 for all values of T, which leads to validity of thermodynamical equilibrium.Figure 7
Plot ofST¨ versus γ for power law correction when Γ is constant.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 (38) as(42)ST¨=-16H˙η+8H22c2a6+γ3H2-c22a61-Γ3H-H˙H2η+8H2×c2a6+γ3H2-c22a61-Γ3H-2H˙H2-c2/6a61-γ/γH3T03H2-c22a6×1-Γ3H-γ+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×1-Γ3H1-γc2a˙/a7+2HH˙H2-c2/6a6-1+1-γ/γγH2T0+1Hη+8H2×-6c2a˙a7+γΓH˙3H2-c2/2a63H2+γ1-Γ3H3c2a˙a7+6HH˙+1H2T0×H2-c26a61-γ/γ-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γΓH˙3H2-c2/2a63H2-γ1-Γ3H3c2a˙a7+6HH˙+-6c2a˙a7+γΓH˙3H2-c2/2a63H2+1-Γ3H×γ3c2a˙a7+6HH˙γ3H2-c2/2a62H2.The graphical behavior of ST¨ versus γ for three values of T is shown in Figure 8. It can be seen that ST¨ is negative for all values of T satisfying the condition d2ST/dt2<0 which exhibits the thermodynamical equilibrium.Figure 8
Plot ofST¨ versus γ for Renyi entropy when Γ=constant.


### 4.2.Γ=Γ(t)

Secondly, we consider particle creation rateΓ as a variable, i.e., Γ=Γ(t).Bekenstein Entropy. The differentiation of (35) leads to(43)ST¨=-3H˙c2/a6+γ1-Γ/3H-c2/2a6+3H28H4-2H˙-c26a6+H21-γ/γ×-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2H3T0+-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2γH2T0×1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ-6c2a˙8H3a7+3c2a˙/a7+6HH˙8H3×γ1-Γ3H+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H8H3+H2-c2/6a61-γ/γH2T0×-γH˙3H2-c2/2a6c2/a6+3H2-c2/2a61-Γ/3HH3-γ1-Γ3H×3c2a˙a7+6HH˙+γc2/a6+γ3H2-c2/2a61-Γ/3H3c2a˙/a7+6HH˙2H2-γ3H2-c22a6ΓH˙3H2-Γ˙3H+-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H23H2-c22a6γ.The plot of ST¨ versus γ for three values of T is shown in Figure 9 by keeping the constant values same as for constant Γ. It is observed that the thermodynamical is obeying the condition ST¨<0 which leads to the validity of thermodynamical equilibrium.Figure 9
Plot ofST¨ versus γ for Bekenstein entropy when Γ=Γ(t).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 (36) reduces to second-order differential equation as(44)ST¨=-H˙1/4HLp2+2αH-16Lp2βH3c2/a6+γ1-Γ/3H3H2-c2/2a6H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a62αH˙-H˙/4Lp2H2-48Lp2βH2H˙2H2+14HLp2+2αH-16Lp2βH3-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H2+1H2T0H2-c26a61-γ/γ-3H2-c22a6×γH˙c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H3c2a˙a7+6HH˙+γc2/a6+γ1-Γ/3H3H2-c2/2a63c2a˙/a7+6HH˙2H2-γ3H2-c22a6×ΓH˙3H2-Γ˙3H+γ3H2-c2/2a6-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ23H2-c2/2a62ΓH˙/3H2-Γ˙/3H2H2.The plot of ST¨ versus γ is shown in Figure 10 for three values of T by keeping the same values as in above case; we observe that thermal equilibrium condition ST¨<0 is fulfilled which leads to the validity of thermodynamical equilibrium.Figure 10
Plot ofST¨ versus γ when logarithmic corrected entropy Γ=Γ(t).Power Law Correction. For variable Γ, the differentiation of (37) turns out to be(45)ST¨=-1/4HLp2-2-δ/21/H3-δKδ/4Lp2c2/a6+γ1-Γ/3H3H2-c2/2a6H˙H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2γH2T0+-H˙/4HLp2+H˙3-δ2-δ/21/H4-δKδ/4Lp2c2/a6+γ3H2-c2/2a61-Γ/3H2H2+14HLp2-2-δ/21/H3-δKδ4Lp2-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H2+H2-c2/6a61-γ/γH2T0-3c2a˙a7+6HH˙×γ1-Γ3H-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×ΓH˙3H2-Γ˙3H+γ3H2-c2/2a62H2+γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-6c2a˙a7.The plot of ST¨ versus γ for three values of T by keeping the same values is above mentioned (Figure 11); one can observe easily the validity of thermodynamical equilibrium for all values of T with ST¨<0.Figure 11
Plot ofST¨ versus γ for power law correction when Γ=Γ(t).Renyi Entropy. For variable Γ, (38) gives(46)ST¨=-16H˙η+8H22c2a6+γ3H2-c22a61-Γ3H-H˙H2η+8H2×c2a6+γ3H2-c22a61-Γ3H-2H˙H2-c2/6a61-γ/γH3T03H2-c22a6×1-Γ3H-γ+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×1-Γ3H1-γc2a˙/a7+2HH˙H2-c2/6a6-1+1-γ/γγH2T0+1Hη+8H2×γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-6c2a˙a7+H2-c2/6a61-γ/γH2T0-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3c2a˙a7+6HH˙×1-Γ3H+γ3H2-c2/2a62H2-6c2a˙a7+γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-γ3H2-c22a6ΓH˙3H2-Γ˙3H.The graphical behavior of ST¨ versus γ for three values of T is shown in Figure 12 by keeping the constant values same as in above. One can see that ST¨ is negative for all values of T satisfying the condition d2ST<0 which leads to the validity of thermodynamical equilibrium.Figure 12
Plot ofST¨ versus γ for Renyi entropy when Γ=Γ(t).


## 4.1.Γ = Constant

Firstly, we considerΓ as constant and observe the stability of thermodynamical equilibrium.Bekenstein Entropy. We get second-order differential equation for constant Γ by using (35):(39)ST¨=-3H˙c2/a6+γ1-Γ/3H-c2/2a6+3H28H4-2H˙-c26a6+H21-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0+-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0×c2a˙a7+2HH˙1-γH2-c26a6-1+1-γ/γ+γΓH˙3H2-c22a6124H5-6c2a˙8H3a7+γ1-Γ/3H3c2a˙/a7+6HH˙8H3+H2-c2/6a61-γ/γH2T0-γΓH˙3H2×3H2-c22a6-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H3c2a˙a7+6HH˙+γc2/a6+γ1-Γ/3H3H2-c2/2a62H2×3c2a˙a7+6HH˙+-6c2a˙/a7+γΓH˙3H2-c2/2a6/3H22H2+3c2a˙/a7+6HH˙2H2×γ1-Γ3H3H2-c22a6γ.The graphical behavior of ST¨ versus γ is shown in Figure 5 for different values of parameter T. We observe that ST¨<0 for all values of T which leads to the validity of thermodynamical equilibrium.Figure 5
Plot ofST¨ versus γ for Bekenstein entropy when Γ is constant.Logarithmic Corrected Entropy. In the presence of logarithmic corrected entropy second-order differential equation is obtained from (36) as(40)ST¨=-H˙1/4HLp2+2αH-16Lp2βH3c2/a6+γ1-Γ/3H3H2-c2/2a6H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a62αH˙-H˙/4Lp2H2-48Lp2βH2H˙2H2+14HLp2+2αH-16Lp2βH3-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γΓH˙3H2-c2/2a66H4+1H2T0H2-c26a61-γ/γ-γΓH˙3H2-c2/2a63H2-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H×3c2a˙a7+6HH˙+γ3c2a˙/a7+6HH˙c2/2a6+γ1-Γ/3H3H2-c2/2a62H2+γ3H2-c22a6γΓH˙3H2-c2/2a6/3H2+γ1-Γ/3H3c2a˙/a7+6HH˙2H2-6c2a˙2H2a7.The graphical behavior between ST¨ 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 exhibits the validity of thermodynamical equilibrium.Figure 6
Plot ofST¨ versus γ for logarithmic corrected entropy when Γ is a constant.Power Law Correction. To find the validity of thermodynamical equilibrium, we take Γ=constant for which second-order differential equation can be expressed with the help of (37) as(41)ST¨=-c2/a6+γ1-Γ/3H3H2-c2/2a61/4HLp2-2-δ/21/H3-δKδ/4Lp2H˙H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a6-H˙/4HLp2+3-δH˙2-δ/21/H4-δKδ/4Lp22H2+14HLp2-2-δ/21/H3-δKδ4HLp2-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γΓH˙3H2-c2/2a66H4+H2-c2/6a6-1-γ/γH2T0γΓH˙3H2-c2/2a63H2-γ3c2a˙a7+6HH˙1-Γ3H-γH˙3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6H3+γ3c2a˙/a7+6HH˙c2/a6+γ1-Γ/3H3H2-c2/2a62H2+γ3H2-c22a6×-6c2a˙/a7+γΓH˙3H2-c2/2a6/3H2+γ1-Γ/3H3c2a˙/a7+6HH˙2H2.The plot of ST¨ versus γ is shown in Figure 7 for three values of T. We analyze that thermodynamical equilibrium is satisfying the condition ST¨<0 for all values of T, which leads to validity of thermodynamical equilibrium.Figure 7
Plot ofST¨ versus γ for power law correction when Γ is constant.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 (38) as(42)ST¨=-16H˙η+8H22c2a6+γ3H2-c22a61-Γ3H-H˙H2η+8H2×c2a6+γ3H2-c22a61-Γ3H-2H˙H2-c2/6a61-γ/γH3T03H2-c22a6×1-Γ3H-γ+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×1-Γ3H1-γc2a˙/a7+2HH˙H2-c2/6a6-1+1-γ/γγH2T0+1Hη+8H2×-6c2a˙a7+γΓH˙3H2-c2/2a63H2+γ1-Γ3H3c2a˙a7+6HH˙+1H2T0×H2-c26a61-γ/γ-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γΓH˙3H2-c2/2a63H2-γ1-Γ3H3c2a˙a7+6HH˙+-6c2a˙a7+γΓH˙3H2-c2/2a63H2+1-Γ3H×γ3c2a˙a7+6HH˙γ3H2-c2/2a62H2.The graphical behavior of ST¨ versus γ for three values of T is shown in Figure 8. It can be seen that ST¨ is negative for all values of T satisfying the condition d2ST/dt2<0 which exhibits the thermodynamical equilibrium.Figure 8
Plot ofST¨ versus γ for Renyi entropy when Γ=constant.


## 4.2.Γ=Γ(t)

Secondly, we consider particle creation rateΓ as a variable, i.e., Γ=Γ(t).Bekenstein Entropy. The differentiation of (35) leads to(43)ST¨=-3H˙c2/a6+γ1-Γ/3H-c2/2a6+3H28H4-2H˙-c26a6+H21-γ/γ×-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2H3T0+-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2γH2T0×1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ-6c2a˙8H3a7+3c2a˙/a7+6HH˙8H3×γ1-Γ3H+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H8H3+H2-c2/6a61-γ/γH2T0×-γH˙3H2-c2/2a6c2/a6+3H2-c2/2a61-Γ/3HH3-γ1-Γ3H×3c2a˙a7+6HH˙+γc2/a6+γ3H2-c2/2a61-Γ/3H3c2a˙/a7+6HH˙2H2-γ3H2-c22a6ΓH˙3H2-Γ˙3H+-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H23H2-c22a6γ.The plot of ST¨ versus γ for three values of T is shown in Figure 9 by keeping the constant values same as for constant Γ. It is observed that the thermodynamical is obeying the condition ST¨<0 which leads to the validity of thermodynamical equilibrium.Figure 9
Plot ofST¨ versus γ for Bekenstein entropy when Γ=Γ(t).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 (36) reduces to second-order differential equation as(44)ST¨=-H˙1/4HLp2+2αH-16Lp2βH3c2/a6+γ1-Γ/3H3H2-c2/2a6H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2γH2T0+c2/a6+γ1-Γ/3H3H2-c2/2a62αH˙-H˙/4Lp2H2-48Lp2βH2H˙2H2+14HLp2+2αH-16Lp2βH3-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H2+1H2T0H2-c26a61-γ/γ-3H2-c22a6×γH˙c2/a6+γ1-Γ/3H3H2-c2/2a6H3-γ1-Γ3H3c2a˙a7+6HH˙+γc2/a6+γ1-Γ/3H3H2-c2/2a63c2a˙/a7+6HH˙2H2-γ3H2-c22a6×ΓH˙3H2-Γ˙3H+γ3H2-c2/2a6-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ23H2-c2/2a62ΓH˙/3H2-Γ˙/3H2H2.The plot of ST¨ versus γ is shown in Figure 10 for three values of T by keeping the same values as in above case; we observe that thermal equilibrium condition ST¨<0 is fulfilled which leads to the validity of thermodynamical equilibrium.Figure 10
Plot ofST¨ versus γ when logarithmic corrected entropy Γ=Γ(t).Power Law Correction. For variable Γ, the differentiation of (37) turns out to be(45)ST¨=-1/4HLp2-2-δ/21/H3-δKδ/4Lp2c2/a6+γ1-Γ/3H3H2-c2/2a6H˙H3--γ1-Γ/3H3H2-c2/2a6+γ3H2-c2/2a6c2/a6+γ1-Γ/3H3H2-c2/2a6/2H2H3T0×2H˙H2-c26a61-γ/γ+1-γc2a˙a7+2HH˙H2-c26a6-1+1-γ/γ×-γ3H2-c2/2a61-Γ/3H+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H/2H2γH2T0+-H˙/4HLp2+H˙3-δ2-δ/21/H4-δKδ/4Lp2c2/a6+γ3H2-c2/2a61-Γ/3H2H2+14HLp2-2-δ/21/H3-δKδ4Lp2-6c2a˙/a7+γ1-Γ/3H3c2a˙/a7+6HH˙2H2+γ3H2-c2/2a6ΓH˙/3H2-Γ˙/3H2H2+H2-c2/6a61-γ/γH2T0-3c2a˙a7+6HH˙×γ1-Γ3H-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×ΓH˙3H2-Γ˙3H+γ3H2-c2/2a62H2+γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-6c2a˙a7.The plot of ST¨ versus γ for three values of T by keeping the same values is above mentioned (Figure 11); one can observe easily the validity of thermodynamical equilibrium for all values of T with ST¨<0.Figure 11
Plot ofST¨ versus γ for power law correction when Γ=Γ(t).Renyi Entropy. For variable Γ, (38) gives(46)ST¨=-16H˙η+8H22c2a6+γ3H2-c22a61-Γ3H-H˙H2η+8H2×c2a6+γ3H2-c22a61-Γ3H-2H˙H2-c2/6a61-γ/γH3T03H2-c22a6×1-Γ3H-γ+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2+γ3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3H2-c22a6×1-Γ3H1-γc2a˙/a7+2HH˙H2-c2/6a6-1+1-γ/γγH2T0+1Hη+8H2×γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-6c2a˙a7+H2-c2/6a61-γ/γH2T0-γH˙3H2-c2/2a6c2/a6+γ3H2-c2/2a61-Γ/3HH3+γ3c2a˙/a7+6HH˙c2/a6+γ3H2-c2/2a61-Γ/3H2H2-γ3c2a˙a7+6HH˙×1-Γ3H+γ3H2-c2/2a62H2-6c2a˙a7+γ1-Γ3H3c2a˙a7+6HH˙+γ3H2-c22a6ΓH˙3H2-Γ˙3H-γ3H2-c22a6ΓH˙3H2-Γ˙3H.The graphical behavior of ST¨ versus γ for three values of T is shown in Figure 12 by keeping the constant values same as in above. One can see that ST¨ is negative for all values of T satisfying the condition d2ST<0 which leads to the validity of thermodynamical equilibrium.Figure 12
Plot ofST¨ versus γ for Renyi entropy when Γ=Γ(t).


## 5. Concluding Remarks

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 EoSp=(γ-1)ρ by assuming the different entropy corrections such as Bekenstein 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:(i)
For Bekenstein entropy: we have analyzed that first law of thermodynamics is showing the validity forΓ=3H(c2/γa6(3H2-c2/2a6)-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 d2ST/dt2<0 for all values of T with 2/3≤γ≤2.(ii)
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 whenΓ=3H(1-1/γ(3H2-c2/a6)-1(γ(3H2-c2/2a6)/(1/Lp2+8H2α-64βH4Lp2)-c2/a6)). We have also investigated the validity of GSLT on apparent horizon satisfying the condition dST/dt≥0 (Figure 2). The graphical behavior of ST¨ versus γ is 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.(iii)
Power law corrected entropy: for power law corrected entropy, we have investigated that first law of thermodynamics holds at apparent horizon forΓ=3H(1-1/γ3H2-c2/2a6-1(γ(3H2-c2/2a6))(1/Lp2-(2-δ/2)Kδ/Lp2(1/H)2-δ)-1-c2/a6). From Figure 3 we can analyze 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 condition ST¨<0 as shown in Figures 7 and 11 for all values of T with 2/3≤γ≤2 for both constant and variable Γ.(iv)
For Renyi entropy: in this entropy, we have observed that first law of thermodynamics holds whenΓ=3H(1-1/γ3H2-c2/2a6-1(γ(η+8H2)/8H2(3H2-c2/2a6)-c2/a6)). The graphical behavior of Figure 12 shows that all trajectories remain positive for all values of T with 2/3≤γ≤2 which leads to the validity of GSLT. Moreover, thermodynamical equilibrium condition is satisfied for all values of T with all values of γ for constant and variable Γ.



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*Source: 9471346-2018-10-18.xml*