Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions


Ratio Mathematica                                        Volume 39, 2020, pp. 261-268 

 

 
 

261 

 

 

 

Thermodynamic behavior of the 

polytropic gas in cosmology 
 

 

 

Prasanta Das* 

Kangujam Priyokumar Singh† 
 

 

 

Abstract 

In this paper, we investigate on the thermodynamic behavior of Polytropic 

gas as a candidate for dark energy by considering the relation  𝑃 =

𝐾𝜌
1+

1

𝑛  , where 𝐾 and 𝑛 are the Polytropic constant and Polytropic index 
respectively. Furthermore, 𝑃 indicates the pressure and 𝜌 is the energy 

density of the fluid such that 𝜌 =
𝑈

𝑉
 where 𝑈and 𝑉 represent the internal 

energy and volume, respectively. At first, we find an exact expression for 

the energy density of the Polytropic gas using thermodynamics and later 

on, discuss different physical parameters. Finally our study shows that the 

Polytropic gas may be used to describe the expansion history of the 

universe from the dust dominated era to the current accelerated era and it 

is thermodynamically stable. 

Keywords: Cosmology; Dark energy; Polytropic gas; Thermodynamics. 

2010 AMS subject classification: 83F05, 37D35, 82B30.‡ 
 

 
 

 
*Department of Mathematical Sciences, Bodoland University, Kokrajhar, BTR, Assam-

783370, India; prasantadasp4@gmail.com. 
†Department of Mathematical Sciences, Bodoland University, Kokrajhar, BTR, Assam-

783370, India; pkangujam18@gmail.com. 
‡Received on October 4th, 2020. Accepted on December 17th, 2020. Published on December 

31st, 2020. doi: 10.23755/rm.v39i0.564. ISSN: 1592-7415. eISSN: 2282-8214. © Prasanta Das 

et al. This paper is published under the CC-BY licence agreement. 

mailto:prasantadasp4@gmail.com
mailto:pkangujam18@gmail.com


Prasanta Das and Kangujam Priyokumar Singh 

 

262 

 

1. Introduction 
 

Cosmologists suggest that our universe expands under an accelerated 

expansion [1]-[7]. In the standard Friedman Lemaitre Robertson Walker 

(FLRW) cosmology, a new energy with negative pressure, called dark energy 

(DE) is responsible for this expansion [8]. The nature of the DE is still 

unknown and various problems have been proposed by the researchers in this 

field. About 70% of the present energy of the universe is contained in the DE. 

The cosmological constant with the time independent equation of state is the 

earliest, simplest and most traditional candidate for the dark energy which can 

be taken into account as a perfect fluid satisfying the relation 𝜌 + 𝑃 = 0. But it 
has some problems like fine-tuning and cosmic coincidence puzzles [9], [10].  

Besides the cosmological constant, the other dark energy models are 

quintessence [11], phantom [12], tachyon [13], holographic dark energy [14] 

[15], K-essence [16] and Chaplygin gas models with various equation of state.  

Polytropic gas is one of the dynamical dark energy models [17].  

In the present study, we want to investigate the thermodynamic   

behavior of the Polytropic gas. K. Karami et al. investigated the interaction 

between the Polytropic gas and cold dark matter and found that the Polytropic 

gas behaves as the phantom dark energy [18]. K.  Karami and S. Ghaffari 

showed that the generalized second law of thermodynamics is always satisfied 

by a universe filled with a Polytropic gas and a cold dark matter [19]. K. 

Kleidis and N.K. Spyron used the first law of thermodynamics in the 

Polytropic gas model and they show that the Polytropic gas behaves as dark 

energy and this model leads to a suitable fitting with the observational data 

about the current expanding era [20]. H. Moradpour, A. Abri and H. Ebadi, 

investigated the thermo dynamical behavior and stability of the Polytropic gas 

[21]. M. Salti et al. discussed validity of the first and generalized second law 

of thermodynamics in locally rotationally symmetric Bianchi-type II space 

time which is dominated by a combination of Polytropic gas and baryonic 

matter[22]. Moreover, Muzaffer Askin et al. studied the cosmological 

scenarios of the Polytropic gas dark matter-energy proposal in a Friedmann- 

Robertson- Walker universe and they found an exact expression for the energy 

density of the Polytropic gas model according to the thermo dynamical point 

of views and a relationship between a homogeneous minimally coupled scalar 

field and the Polytropic gas [23].This paper is organized as follows: in section 

2 we construct the basic thermodynamic formalism of the Polytropic gas 

model and discuss the thermodynamic behavior of this model. Finally in 

section 3 we provide a brief discussion.     

 



Thermodynamic behavior of the Polytropic Gas in Cosmology 

 

263 

 

2. Basic Formalism  
 

In this work, we consider the following equation of state which is well known 

as Polytropic gas equation of state  

𝑃 = 𝐾𝜌
1+

1

𝑛                (1) 
Here 𝐾(> 0) and 𝑛(< 0) are Polytropic constant and Polytropic index 
respectively. Moreover, 𝑃 is the pressure and 𝜌 is the energy density of the 
fluid such that 

𝜌 =
𝑈

𝑉
             (2) 

Where 𝑈and 𝑉 are the internal energy and volume filled by the fluid 
respectively. 

First of all, we try to find the internal energy 𝑈 and energy density 𝜌 of the 
polytropic gas as a function of its volume 𝑉 and entropy 𝑆.  
From the general thermodynamics, we have 

(
𝜕𝑈

𝜕𝑉
)

𝑆
= −𝑃                     (3) 

From the equations (1), (2) and (3), we get 

(
𝜕𝑈

𝜕𝑉
)

𝑆
= −𝐾 (

𝑈

𝑉
  )

1+
1

𝑛
                      (4) 

Integrating the equation (4), we get 

 

𝑈 = (−1)−𝑛 (𝐾𝑉
−

1

𝑛 + ξ)
−𝑛

            (5) 

Where the parameter ξ is the constant of integration which may be a universal 

constant or a function of entropy 𝑆 only 
The equation (5) also can rewrite in the following form 

𝑈 = (−1)−𝑛𝐾−𝑛𝑉 (1 + (
V

ε
)

1

n
)

−𝑛

           (6) 

Where        𝜀 = (
𝐾

ξ
)

𝑛

         (7) 

And it has a dimension of volume. 

Therefore, the energy density 𝜌 of the Polytropic gas is 

𝜌 =
𝑈

𝑉
=  (−1)−𝑛𝐾−𝑛 (1 + (

V

ϵ
)

1

n
)

−𝑛

                    (8) 

When  𝑛 < 0 then equation (8) gives 

                                   𝜌 ∼  (−1)−𝑛𝐾−𝑛
𝜀

𝑉
                            (9) 

Now we will use these equations to discuss different physical parameters. 

 

 

 



Prasanta Das and Kangujam Priyokumar Singh 

 

264 

 

a) Pressure: 
 

Using the equation (8) in the equation (1) we get the pressure of the Polytropic 

gas as a function of entropy 𝑆 and volume 𝑉 in the following form 

𝑃 = (−1)𝑛+1𝐾−𝑛 (1 + (
V

ϵ
)

1

n
)

−(𝑛+1)

                             (10) 

We can rewrite the equation (10) in the following form 

𝑃 = −
𝜌

1+(
V

ϵ
)

1
n

                                                 (11) 

 

When 𝑛 < 0 and   𝜀 does not diverge then for small volume i.e. at early stage 

of universe,  𝑉 ≪ 𝜀 ie 
𝑉

𝜀
≪ 1 , we get 

P ≃ 0 , which represents a dust dominated universe. When 𝑛 < 0 and  𝜀 does 
not diverge then for large volume i.e. at late stage of universe, 𝑉 ≫ 𝜀 ie  
𝑉

𝜀
≫ 1, we get P ≃ −𝜌, which indicates an accelerated expansion of the 

universe.   

 

b) Caloric equation of state: 
 

Now from the equations (8) and (10) we get the caloric equation of state 

parameter as 

𝜔 =
𝑃

𝜌
= −

1

1+(
V

ϵ
)

1
n

                                                          (12) 

When 𝑛 < 0 and   𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie  
𝑉

𝜀
≪ 1 , 

we get 𝜔 ≃ 0  (Dust dominated)  

When 𝑛 < 0 and  𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀 ie  
𝑉

𝜀
≫ 1 , 

we get 𝜔 ≃ −1 (Cosmological constant) 
Thus the equation of state parameter (  𝜔) of the Polytropic gas with 𝑛 < 0 is 
decreased from 𝜔 ≃ 0  (for small volume) to 𝜔 ≃ −1 (for large volume). It 
indicates that the universe expands from the dust dominated era to the current 

accelerating era. 

 

c) Deceleration parameter: 

We get the deceleration parameter of the Polytropic gas with the help of 

equation (12) 



Thermodynamic behavior of the Polytropic Gas in Cosmology 

 

265 

 

   𝑞 =
1

2
+

3

2

𝑃

𝜌
=

1

2
−

3

2

1

1+(
V

ϵ
)

1
n

            (13) 

When 𝑛 < 0 and   𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie  
𝑉

𝜀
≪ 1 , 

we get 𝑞 > 0,  which correspond to the deceleration universe.    

When 𝑛 < 0 and  𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀 ie  
𝑉

𝜀
≫ 1 , 

we get 𝑞 < 0,  which correspond to the accelerated universe.  
 

d) Square velocity of sound: 
 

From the equation (11) we get the velocity of sound (𝑉𝑠 )  as 

     𝑉𝑠
2

= (
𝜕𝑃

𝜕𝜌
)

𝑆
= −

1

1+(
V

ϵ
)

1
n

                (14) 

When 𝑛 < 0 and   𝜀 does not diverge then for small volume 𝑉 ≪ 𝜀 ie 
𝑉

𝜀
≪ 1, 

we get 𝑉𝑠
2

≃ 0  .Since velocity of sound is zero in vacuum. Therefore the 
Polytropic gas behaves like a pressure less fluid at the early stage of the 

universe. When 𝑛 < 0 and  𝜀 does not diverge then for large volume 𝑉 ≫ 𝜀  ie  
𝑉

𝜀
≫ 1, we get 𝑉𝑠

2 ≃ −1, which gives an imaginary speed of sound leading to a 

perturbation cosmology. 

 

e) Thermodynamic stability: 
 

The conditions of the thermodynamic stability of a fluid are  

(
𝜕𝑃

𝜕𝑉
)

𝑆
< 0                     (15) 

And                                    𝐶𝑉 > 0                (16) 
Here 𝐶𝑉  is the thermal capacity at constant volume. From the equation (10) 
we have 

 

(
𝜕𝑃

𝜕𝑉
)

𝑆
= − (1 +

1

𝑛
)

𝑃

𝑉

1

1+(
V

ϵ
)

−
1
n

              (17) 

If −1 < 𝑛 < 0 and 𝜀 < 0 then from (17), we have 

(
𝜕𝑃

𝜕𝑉
)

𝑆
< 0 

Thus the stability condition (15) of thermodynamics is satisfied. 

Now we have to verify the positivity of the thermal capacity at constant 

volume  𝐶𝑉   where   𝐶𝑉 = 𝑇 (
𝜕𝑆

𝜕𝑇
)

𝑉
   (18) 



Prasanta Das and Kangujam Priyokumar Singh 

 

266 

 

Now we determine the temperature 𝑇 of the Polytropic gas as a function of its 
entropy 𝑆 and its volume 𝑉. The temperature 𝑇 of the Polytropic gas is 
determined from the relation 

𝑇 = (
𝜕𝑈

𝜕𝑆
)

𝑉
      (19) 

Using (6) in (19) we get 

𝑇 = (−1)𝑛+1𝑉
1+

1

𝑛 (𝐾 + 𝜉𝑉
1

𝑛)
−(𝑛+1)

𝑑𝜉

𝑑𝑆
                         (20) 

This gives the temperature of the Polytropic gas.  

We can rewrite the equation (20) in the following form 

𝑇 = −𝑛
𝜌𝑉

1+
1
𝑛

1+(
V

ϵ
)

1
n

𝑑𝜉

𝑑𝑆
              (21) 

From (5) we have 
[𝜉]−𝑛 = [𝑈]             (22) 

Since           [𝑈] = [𝑇𝑆]                          (23) 
Therefore from the equations (22) & (23) we get 

𝜉 = [𝑈]
−

1

𝑛 = [𝑇∗𝑆]
−

1

𝑛                       (24) 

Where 𝑇∗ (> 0) is a universal constant with temperature dimension. 
Differentiating (24) with respect to ‘S’ we get 

𝑑𝜉

𝑑𝑆
= −

1

𝑛
𝑇∗

−
1

𝑛𝑆
−

1

𝑛
−1

                        (25) 

Using (8) & (24) in (25) we get 

𝑇 = (−1)𝑛𝑉
1+

1

𝑛 (𝑇∗
−

1

𝑛𝑆
−

1

𝑛
−1

) [𝐾 + 𝑇∗
−

1

𝑛𝑆
−

1

𝑛𝑉
1

𝑛]

−(𝑛+1)

 (26) 

This leads to the entropy of the Polytropic gas as 

𝑆 = [(−1)
𝑛

𝑛+1 (
𝑇∗

𝑇
)

1

𝑛+1
− 1]

𝑛

     
𝑉

𝐾𝑛𝑇∗
                      (27) 

We know that entropy (𝑆) of a thermo dynamical system should be positive 
ie 𝑆 > 0  [24] 
Here 𝑆 > 0 if 𝐾𝑛𝑇∗ > 0 
Now the thermal capacity at constant volume is   

𝐶𝑉 = 𝑇 (
𝜕𝑆

𝜕𝑇
)

𝑉
 

= (−1)
2𝑛+1

𝑛+1 (
𝑛

𝑛+1
)

𝑆

[(−1)
𝑛

𝑛+1(
𝑇∗
𝑇

)

𝑛
𝑛+1

−1]

(
𝑇∗

𝑇
)

1

𝑛+1
     (28) 

Therefore, the condition  𝐶𝑉 > 0  is satisfied if 𝐾
𝑛 𝑇∗ > 0. Thus both the 

conditions of thermo dynamic stability are satisfied. So the Polytropic gas is 

thermo dynamically stable. 

 



Thermodynamic behavior of the Polytropic Gas in Cosmology 

 

267 

 

3. Discussion  
 

We have studied the thermo dynamical behavior of the Polytropic gas. Here, 

we have considered the value of   𝑛 < 0  to study the whole work done in this 
article. Some important results are given below:  

(i) As we have considered 𝑛 < 0 , the pressure goes more and 
more negative as volume increases. 

(ii) The equation of state parameter (𝜔) of the Polytropic gas is 
𝜔 ≃ 0 at early stage of the universe and 𝜔 ≃ −1 at late stage of 
the universe. This indicates that the universe expands from the 

dust dominated era to the present accelerated era. 

(iii) The deceleration parameter (𝑞) is investigated in the context of 
thermodynamics as well as Polytropic gas and our analysis 

shows that universe is decelerated (𝑞 > 0) at early stage of the 
universe and accelerated (𝑞 < 0) at late stage of the universe. 

Both the conditions of the thermo dynamical stability of the Polytropic gas are 

studied for 𝐾𝑛 𝑇∗ > 0 and our analysis shows that the Polytropic gas is 
thermodynamically stable.   

  

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