#### Abstract

We consider the particle creation scenario in the dynamical Chern-Simons modified gravity in the presence of perfect fluid equation of state . 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.

#### 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 , [14], ) [15, 16], [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 (where , , and 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 .

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. Sections 3 and 4 contain the illustration of GSLT. In Section 5, we analyze the stability of thermodynamical equilibrium for constant as well as variable . 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 as for any horizon [37], where , , and 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 entropy 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 CDM 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 (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 where is the plank’s length and , 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 area . 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] where , is dimensionless constant, and 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 is thought to be a nonextensive Tsallis entropy . One can obtain Renyi entropy [91]

The action of Chern-Simons theory is given by [19–21]where , , , , , and 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]where is constant, is a Hubble parameter, and is the scale factor. The equation of continuity for this model can be described asThe particle creation pressure representing the gravitationally induced process for particle creation and is the fluid expansion. The total number of particles in an open thermodynamics iswhere is the creation rate of number of particles in comoving volume, i.e., having two phases negative and positive. The negative relates with the particle annihilation and the positive relates to production of particle. Equations (7) and (8) with Gibbs relations can be written asThe equation related to creation pressure and can be determined asUnder 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 . Differentiation of (6) will giveInserting (10) and in the above equation, we getFor flat FRW universe, Hubble parameter relates to the apparent horizon as . Differentiating the apparent horizon with respect to time, we haveThe deceleration parameter is of the form

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 asFor the sake of convenience, we consider . The differential is the amount of energy crossing the apparent horizon which can be evaluated as [28]

*Bekenstein Entropy*. From (1), the differential of surface entropy at apparent horizon leads toThe above equation with horizon temperature leads toHence becomesWith the help of (19) we observe that the first law of thermodynamics holds (i.e., ) when .

*Logarithmic Corrected Entropy*. The differential form of (2) is given aswhich leads toCombining (16) and (21), we getFrom the above equation, we observe the validity of first law of thermodynamics for .

*Power Law Correction.* Differentiating (3), we getwhich yieldsHence, takes the formFrom (25), it can be analyzed that the first law of thermodynamics remains valid for the following particle creation rate:

*Renyi Entropy.* Equation (4) gives the differential of entropy aswhich gives rise toUsing (16) and (28), we getFrom (29), it can be seen that the first law of thermodynamics is showing the validity when .

#### 3. Generalized Second Law of Thermodynamics

We discuss the GSLT of an isolated macroscopic physical system where the total entropy must satisfy the following conditions ; i.e., entropy function cannot be decreased. In this relation, and appear as the entropy at apparent horizon and the entropy of cosmic fluid enclosed within the horizon, respectively. The Gibbs equation is of the formwhere is the temperature of the cosmic fluid and is the energy of the fluid . The evolution equation for fluid temperature having constant entropy can be described as [92]Using (12), we get ; hence, (31) leads to the integralIntegration of the above equation leads towhere is the constant of integration. Equation (30) yields the differential of fluid entropy asNext, 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) aswhere . The plot of versus is shown in Figure 1 for three values of . We observe the validity of GSLT at the present epoch by setting the constant values , , , and . It can be analyzed from the figure that for all values of leads to the validity of GSLT.

*Logarithmic Corrected Entropy.* Using (20) and (34), the differential of total entropy is given byThe plot of versus with respect to the three values of by fixing the constant values , , and is shown in Figure 2. The trajectories in the plot remain which lead to the validity of GSLT in the presence of logarithmic entropy.

*Power Law Correction*. From (24) and (34), we get the differential of total entropy asThe plot of versus for three values of by setting the constant values as , , and is shown in Figure 3. We can see remains positive for all values of which leads to the validity of GSLT.

*Renyi Entropy.* We observe the validity of GSLT with for which (27) and (34) take the formThe plot of versus for three values is shown in Figure 4 for . It can be seen that GSLT is satisfying for all values of which leads to the validity of GSLT.

#### 4. Thermodynamical Equilibrium

We discuss the two scenarios for thermodynamical equilibrium by taking particle creation rate and for which entropy function attains a maximum entropy state, i.e., .

##### 4.1. = Constant

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

*Bekenstein Entropy*. We get second-order differential equation for constant by using (35):The graphical behavior of versus is shown in Figure 5 for different values of parameter . We observe that for all values of 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 (36) asThe graphical behavior between and is shown in Figure 6 for three values of . We observe that all the trajectories are showing the negative increasing behavior, which exhibits the validity of thermodynamical equilibrium.

*Power Law Correction*. To find the validity of thermodynamical equilibrium, we take for which second-order differential equation can be expressed with the help of (37) asThe plot of versus is shown in Figure 7 for three values of . We analyze that thermodynamical equilibrium is satisfying the condition for all values of , which leads to validity of thermodynamical equilibrium.

*Renyi Entropy.* In present scenario, we observe the validity of thermodynamical equilibrium by keeping as constant for which second-order differential equation is given by using (38) asThe graphical behavior of versus for three values of is shown in Figure 8. It can be seen that is negative for all values of satisfying the condition which exhibits the thermodynamical equilibrium.

##### 4.2.

Secondly, we consider particle creation rate as a variable, i.e., .

*Bekenstein Entropy.* The differentiation of (35) leads toThe plot of versus for three values of is shown in Figure 9 by keeping the constant values same as for constant . It is observed that the thermodynamical is obeying the condition 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 as variable for which (36) reduces to second-order differential equation asThe plot of versus is shown in Figure 10 for three values of by keeping the same values as in above case; we observe that thermal equilibrium condition is fulfilled which leads to the validity of thermodynamical equilibrium.

*Power Law Correction*. For variable , the differentiation of (37) turns out to beThe plot of versus for three values of by keeping the same values is above mentioned (Figure 11); one can observe easily the validity of thermodynamical equilibrium for all values of with .

*Renyi Entropy.* For variable , (38) givesThe graphical behavior of versus for three values of is shown in Figure 12 by keeping the constant values same as in above. One can see that is negative for all values of satisfying the condition which leads to the validity of thermodynamical equilibrium.

#### 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 EoS 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 for . However, GSLT remains valid for all values of with . Further, we have analyzed the validity of thermodynamical equilibrium for constant and variable . From Figures 5 and 9, we observe that thermodynamical equilibrium is satisfying the condition for all values of with .(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 when . We have also investigated the validity of GSLT on apparent horizon satisfying the condition (Figure 2). The graphical behavior of versus is shown in Figures 6 and 10. We observe the validity of thermodynamical equilibrium for all values of , for all values of , for constant as well as variable .(iii)Power law corrected entropy: for power law corrected entropy, we have investigated that first law of thermodynamics holds at apparent horizon for From Figure 3 we can analyze that the GSLT is valid for all values of with . Further, we have investigated the validity of thermodynamical equilibrium obeying the condition as shown in Figures 7 and 11 for all values of with for both constant and variable .(iv)For Renyi entropy: in this entropy, we have observed that first law of thermodynamics holds when . The graphical behavior of Figure 12 shows that all trajectories remain positive for all values of with which leads to the validity of GSLT. Moreover, thermodynamical equilibrium condition is satisfied for all values of with all values of for constant and variable .

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.