Universe consisting of diffusive dark fluids: thermodynamics and   stability analysis

Subhayan Maity; Pritikana Bhandari; Subenoy Chakraborty

arXiv:1902.08037·gr-qc·February 22, 2019

Universe consisting of diffusive dark fluids: thermodynamics and stability analysis

Subhayan Maity, Pritikana Bhandari, Subenoy Chakraborty

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TL;DR

This paper investigates the thermodynamics and stability of a universe modeled with non-interacting diffusive cosmic fluids, analyzing how diffusion affects stability conditions and the universe's evolution.

Contribution

It introduces a thermodynamic stability analysis for diffusive cosmic fluids within the FLRW universe, deriving new stability conditions based on the equation of state.

Findings

01

Stability conditions depend on the equation of state parameter.

02

Diffusive nature influences thermodynamic stability.

03

Constraints on cosmic fluid parameters are established.

Abstract

The present work deals with homogeneous and isotropic FLRW model of the Universe having a system of non-interacting diffusive cosmic fluids with barotropic equation of state (constant or variable equation of state parameter). Due to diffusive nature of the cosmic fluids, the divergence of the energy momentum tensor is chosen to be proportional to the diffusive current. The thermodynamic stability analysis of individual fluids is done and the stability conditions are expressed as restrictions on the equation of state parameter.

Figures1

Click any figure to enlarge with its caption.

Tables3

Table 1. Table 1: Conditions for Stability

range of $ω_{i}$	Stability Condition
$ω_{i} \geq 0$	${[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq \frac{1 + ω_{i}}{ω_{i}}$
$- 1 \leq ω_{i} \leq 0$	$- (1 + ω_{i}) \leq {[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}}$ and ${[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \leq \frac{1 + ω_{i}}{ω_{i}} simultaneously,$ which is not possible hence unstable.
$- 2 \leq ω_{i} \leq - 1$	Unstable .
$ω_{i} \leq - 2$	${[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq - (1 + ω_{i})$

Table 2. Table 2: Conditions for Stability Criteria

range of $ω_{i}$	Stability Condition
$ω_{i} \geq 0$	$\frac{d \ln ω_{i}}{d \ln T} \leq \frac{2 + ω_{i}}{ω_{i}} and {[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq m a x [- (1 + ω_{i}), (\frac{1 + ω_{i}}{ω_{i}} - \frac{d \ln ω_{i}}{d \ln T})]$
$- 1 \leq ω_{i} \leq 0$	$\frac{d \ln ω_{i}}{d \ln T} \geq \frac{2 + ω_{i}}{ω_{i}} and$ ${[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq m a x [- (1 + ω_{i}), (\frac{1 + ω_{i}}{ω_{i}} - \frac{d \ln ω_{i}}{d \ln T})] if \frac{d \ln ω_{i}}{d \ln T} \leq \frac{1 + 3 ω_{i} + 2 ω_{i}^{2}}{ω_{i}^{2}} .$ $Again - (1 + ω_{i}) \leq {[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \leq (\frac{1 + ω_{i}}{ω_{i}} - \frac{d \ln ω_{i}}{d \ln T}) if \frac{d \ln ω_{i}}{d \ln T} \geq \frac{1 + 3 ω_{i} + 2 ω_{i}^{2}}{ω_{i}^{2}}$
$ω_{i} \leq - 1$	$\frac{d \ln ω_{i}}{d \ln T} \leq \frac{2 + ω_{i}}{ω_{i}} and {[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq m a x [- (1 + ω_{i}), (\frac{1 + ω_{i}}{ω_{i}} - \frac{d \ln ω_{i}}{d \ln T})]$

Table 3. Table 3: Conclusion on the result in Dark energy range ( − 0.33 < ω i < 0 0.33 subscript 𝜔 𝑖 0 -0.33<\omega_{i}<0 )

Nature of Dark Energy	Constraint on diffusion parameter $γ_{i}$	Restriction on $\frac{d \ln ω_{i}}{d \ln T}$ from graphs
Constant equation of state parameter	Always unstable	Always unstable
Variable equation of state parameter	${[\frac{\partial {\ln (1 + \frac{γ_{i}}{ρ_{i 0}} A_{i})}}{\partial \ln T}]}_{p_{i}} \geq m a x [- (1 + ω_{i}), (\frac{1 + ω_{i}}{ω_{i}} - \frac{d \ln ω_{i}}{d \ln T})]$	$\frac{2 + ω_{i}}{ω_{i}} \leq \frac{d \ln ω_{i}}{d \ln T} \leq \frac{1 + 3 ω_{i} + 2 ω_{i}^{2}}{ω_{i}^{2}}$

Equations118

p_{i} = w_{i} ρ_{i},

p_{i} = w_{i} ρ_{i},

3 H^{2}

3 H^{2}

2 \dot{H} + 3 H^{2}

T_{i}^{μν}; μ = 3 k^{2} N_{i}^{ν},

T_{i}^{μν}; μ = 3 k^{2} N_{i}^{ν},

i \sum N_{i}^{ν} = 0.

i \sum N_{i}^{ν} = 0.

\frac{\partial ρ}{\partial t} + 3 H (1 + ω_{i}) ρ_{i} = γ_{i} a^{- 3}

\frac{\partial ρ}{\partial t} + 3 H (1 + ω_{i}) ρ_{i} = γ_{i} a^{- 3}

E_{i} = ρ_{i} v

E_{i} = ρ_{i} v

T d S_{i} = d E_{i} + p_{i} d v .

T d S_{i} = d E_{i} + p_{i} d v .

ρ_{i} = a^{- 3 (1 + ω_{i})} [ρ_{i 0} + γ_{i} \int_{t_{0}}^{t} a^{3 ω_{i}} d t]

ρ_{i} = a^{- 3 (1 + ω_{i})} [ρ_{i 0} + γ_{i} \int_{t_{0}}^{t} a^{3 ω_{i}} d t]

d (ln ρ_{i}) = - (1 + ω_{i}) d (ln v) + d [ln (ρ_{i 0} + γ_{i} A_{i})]

d (ln ρ_{i}) = - (1 + ω_{i}) d (ln v) + d [ln (ρ_{i 0} + γ_{i} A_{i})]

d ln T = - ω_{i} d ln v + \frac{w _{i}}{1 + ω _{i}} d [ln (ρ_{i 0} + γ_{i} A_{i})] .

d ln T = - ω_{i} d ln v + \frac{w _{i}}{1 + ω _{i}} d [ln (ρ_{i 0} + γ_{i} A_{i})] .

\frac{T}{ρ _{i} v} = β [ρ_{i 0} + γ_{i} A_{i}]^{- \frac{1}{1 + ω _{i}}},

\frac{T}{ρ _{i} v} = β [ρ_{i 0} + γ_{i} A_{i}]^{- \frac{1}{1 + ω _{i}}},

\frac{T _{0}}{ρ _{i 0} v _{0}} = β ρ_{i 0}^{- \frac{1}{1 + ω _{i}}} .

\frac{T _{0}}{ρ _{i 0} v _{0}} = β ρ_{i 0}^{- \frac{1}{1 + ω _{i}}} .

E_{i} = E_{i 0} (\frac{T}{T _{0}}) [1 + \frac{γ _{i}}{ρ _{i 0}} A_{i}]^{\frac{1}{1 + ω _{i}}} .

E_{i} = E_{i 0} (\frac{T}{T _{0}}) [1 + \frac{γ _{i}}{ρ _{i 0}} A_{i}]^{\frac{1}{1 + ω _{i}}} .

C_{v} = [\frac{\partial Q}{\partial T}]_{v} = [\frac{\partial E}{\partial T}]_{v}

C_{v} = [\frac{\partial Q}{\partial T}]_{v} = [\frac{\partial E}{\partial T}]_{v}

C_{p} = [\frac{\partial Q}{\partial T}]_{p} = [\frac{\partial H}{\partial T}]_{p},

T = T_{a} (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})

T = T_{a} (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})

T_{a} = T_{0} a^{- 3 ω_{i}} .

T_{a} = T_{0} a^{- 3 ω_{i}} .

E_{i} = E_{i 0} (\frac{1}{T _{0}}) T^{\frac{2 + ω _{i}}{1 + ω _{i}}} T_{a}^{- \frac{1}{1 + ω _{i}}} .

E_{i} = E_{i 0} (\frac{1}{T _{0}}) T^{\frac{2 + ω _{i}}{1 + ω _{i}}} T_{a}^{- \frac{1}{1 + ω _{i}}} .

C_{i v} = \frac{2 + ω _{i}}{1 + ω _{i}} (\frac{E _{i 0}}{T _{0}}) (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}},

C_{i v} = \frac{2 + ω _{i}}{1 + ω _{i}} (\frac{E _{i 0}}{T _{0}}) (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}},

C_{i p} = (2 + ω_{i}) (\frac{E _{i 0}}{T _{0}}) (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}} [1 - \frac{1}{2 + ω _{i}} (\frac{T}{T _{a}}) (\frac{\partial T _{a}}{\partial T})_{p_{i}}] .

C_{i p} = (2 + ω_{i}) (\frac{E _{i 0}}{T _{0}}) (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}} [1 - \frac{1}{2 + ω _{i}} (\frac{T}{T _{a}}) (\frac{\partial T _{a}}{\partial T})_{p_{i}}] .

C_{i p} = \frac{E _{0}}{T _{0}} (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}} [(1 + ω_{i}) + T {\frac{\partial}{\partial T} ln (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})}_{p_{i}}] .

C_{i p} = \frac{E _{0}}{T _{0}} (\frac{T}{T _{a}})^{\frac{1}{1 + ω _{i}}} [(1 + ω_{i}) + T {\frac{\partial}{\partial T} ln (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})}_{p_{i}}] .

d v = v (α d T - K_{T} d p)

d v = v (α d T - K_{T} d p)

α = \frac{1}{v} (\frac{\partial v}{\partial T})_{p}

α = \frac{1}{v} (\frac{\partial v}{\partial T})_{p}

K_{T} = - \frac{1}{v} (\frac{\partial v}{\partial p})_{T} .

K_{T} = - \frac{1}{v} (\frac{\partial v}{\partial p})_{T} .

K_{S} = - \frac{1}{v} (\frac{\partial v}{\partial p})_{S} .

K_{S} = - \frac{1}{v} (\frac{\partial v}{\partial p})_{S} .

\frac{α}{K _{T}} = (\frac{\partial p}{\partial T})_{v} .

\frac{α}{K _{T}} = (\frac{\partial p}{\partial T})_{v} .

\frac{C _{p}}{C _{v}} = \frac{K _{S}}{K _{T}} .

\frac{C _{p}}{C _{v}} = \frac{K _{S}}{K _{T}} .

α_{i} = - \frac{1}{ω _{i} T} + \frac{1}{1 + ω _{i}} [\frac{\partial}{\partial T} {ln (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})}]_{p_{i}} .

α_{i} = - \frac{1}{ω _{i} T} + \frac{1}{1 + ω _{i}} [\frac{\partial}{\partial T} {ln (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})}]_{p_{i}} .

K_{i T} = \frac{α _{i} v}{ω _{i} C _{i v}}

K_{i T} = \frac{α _{i} v}{ω _{i} C _{i v}}

K_{i S} = \frac{α _{i} v}{ω _{i} C _{i p}} .

K_{i T} = \frac{v}{ω _{i} C _{i v}} [- \frac{1}{ω _{i} T} + \frac{1}{1 + ω _{i}} {\frac{\partial}{\partial T} ln (1 + \frac{γ _{i}}{ρ _{i 0}} A_{i})}_{p_{i}}]

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Full text

Universe consisting of diffusive Dark fluids: Thermodynamics and stability analysis

Subhayan [email protected]

Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India.

Pritikana [email protected]

Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India.

Subenoy [email protected]

Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India.

I Abstract

The present work deals with homogeneous and isotropic FLRW model of the Universe having a system of non-interacting diffusive cosmic fluids with barotropic equation of state (constant or variable equation of state parameter). Due to diffusive nature of the cosmic fluids, the divergence of the energy momentum tensor is chosen to be proportional to the diffusive current. The thermodynamic stability analysis of individual fluids is done and the stability conditions are expressed as restrictions on the equation of state parameter.

Keywords : Thermodynamical equilibrium, Diffusive cosmic fluid, Equation of State, Thermodynamic Properties.

PACS Numbers : 95.30.Sf, 98.80.Cq, 66.10.Cb, 74.25.Bt

II Introduction

The greatest challenge of standard cosmology today is to accommodate the recent observational predictions Perlmutter:1998np ; Riess:1998cb ; deBernardis:2000sbo ; Percival:2001hw ; Spergel:2003cb ; Jain:2003tba ; Tegmark:2003ud ; Eisenstein:2005su ; Komatsu:2010fb . The modern cosmology is facing the challenging issue of explaining the present accelerated expansion of the universe. In the framework of Einstein gravity, cosmologists are speculating some hypothetical exotic matter (known as dark energy (DE) having large negative pressure) to explain this accelerating phase. It is estimated Ade:2015xua that about 70 percent of the cosmic fluid consists of this unknown DE component. The simplest as well as the common candidate for this dark fluid is the cosmological constant (the zero point energy of the quantum fields). Although a large number of available observational data are in support of this cosmological parameter as a DE candidate but there are severe drawback of it at the interface of Cosmology and Particle Physics : the cosmological constant problem Weinberg:2000yb ; Padmanabhan:2002ji and the coincidence problem Steinhardt:2003st . As a result there are several alternative proposed dynamical DE models into the picture. These DE models have been studied for the last several years Amendola1 , yet the cosmological constant is still the best observationally supported DE candidate. However, these different dynamic DE models cannot be compared from observational view point as these models try to adjust the data seamlessly. As a result, cosmologists have been trying with interacting DE model to have a better understanding of the mechanism of this cosmic acceleration.

From recent past, interacting dark fluid models have been receiving much attention as they can provide small value of the cosmological constant and due to their ability of explaining the cosmic coincidence problem Bolotin:2013jpa ; Wang:2016lxa ; Sharov:2017iue . Moreover, recent observed data Salvatelli:2014zta ; Sola:2015wwa ; Sola:2016ecz ; Nunes:2016dlj ; Kumar:2016zpg ; vandeBruck:2016hpz favor interacting dark fluid models and it is possible to have an estimate of the coupling parameter in the interaction term by various observationsNunes:2016dlj ; Kumar:2016zpg ; vandeBruck:2016hpz ; Cai:2015zoa ; Yang:2014gza ; Yang:2014hea ; Yang:2016evp ; Xia:2016vnp ; Caprini:2016qxs ; Murgia:2016ccp . Further,the cosmologists are of the opinion that these interacting dark fluid models may have the solution to the current tensions on $\sigma_{8}$ and the local value of the Hubble constant $(H_{0})$ Kumar:2016zpg ; vandeBruck:2016hpz ; Cai:2015zoa ; Yang:2014gza ; Yang:2014hea ; Yang:2016evp ; Xia:2016vnp ; Caprini:2016qxs ; Murgia:2016ccp ; Pourtsidou:2016ico . Moreover, it is speculated that cosmological perturbation analysis may be affected by the interaction terms and consequently, the lowest multipoles of the CMB spectrum Zimdahl:2005bk ; Wang:2006qw should have an imprint of it.

On the other hand, the unknown nature of DE may have some clue from the thermodynamic laws which are applicable to all types of macroscopic systems and are based on experimental evidence. However, unlike classical mechanics or electromagnetism, thermodynamical analysis can not predict any definite value for observables, it may only give limit on physical processes. So it is reasonable to believe that thermodynamical study of dark cosmic fluid may indicate some unknown character of it. Investigation in this direction has been initiated recently by Barboza et al Barboza:2015rsa . But their stability conditions demand that the DE should have constant (-ve) equation of state and it is not supported by observation.

Subsequently, Chakraborty and collaborators Bhandari:2017cow ; Bhandari:2018fon ; Mamon:2018flx ; Haldar:2017tqd in a series of works have shown the stability criteria for different type of dark fluids and have presented the stability conditions (in tabular form) for different ranges of the equation of state parameter. The present work is an extension of the interacting dark fluid model. Here a particular realization of the non-conservation of the energy momentum tensor is used with a diffusion of dark matter in a fluid of dark energy. The change of the energy-density is proportional to the particle density. The resulting non-equilibrium thermodynamics is studied and stability conditions are explicitly determined. The paper is organized as follows: Section III deals with thermodynamical analysis of non-interacting cosmic fluids with constant equation of state . The stability criteria for non-interacting cosmic fluids with variable equation of state has been organized in Section IV . Finally, the paper ends with a brief discussion and the conclusion from the result in dark energy range in Section V .

III Thermodynamic analysis of non-interacting diffusive cosmic fluids having constant equation of state parameter

The universe is assumed to have different types of non-interacting cosmic fluids (including dark energy and dark matter). Here it will be examined whether all kinds of diffusive fluids are thermodynamically stable or not in an adiabatic universe . Let the equation of state of these cosmic fluids are barotropic in nature having explicit form (for i-th fluid )

[TABLE]

where $\omega_{i}$ ,a constant ,is the barotropic index of the fluid and $p_{i}$ and $\rho_{i}$ are the pressure and energy density of the fluid respectively. So the Friedmann equations for the whole system take the form

[TABLE]

where $H=a^{-1}\dot{a}$ is the Hubble parameter , $a(t)$ is called the scale factor of the universe as all physical distance is scaled with same factor “ $a$ ” due to homogeneity and isotropy of space-time. Now due to the diffusive nature of the fluids, they do not obey the matter conservation equation ( ${T_{i}^{\mu\nu}}_{;\mu}=0$ ), rather they follow,

[TABLE]

where $N_{i}^{\nu}$ is the current of diffusion corresponding to that fluid and $k$ is a constant. But for the whole universe,the total stress-energy tensor is conserved i.e.( $\sum_{i}{{T_{i}^{\mu\nu}}_{;\mu}=0}$ )

[TABLE]

In simplest form, the conservation equation(non conservation) takes the form for FLRW universe Amendola:2006dg ; Bohmer:2009zz ; Lima:2014hda as,

[TABLE]

where, $\gamma_{i}$ is a constant for a particular fluid but it is different for different fluids with $\sum_{i}\gamma_{i}=0$ . Now, in the context of classical thermodynamics,the contribution to the energy of the universe by i-th fluid is given by

[TABLE]

where, $v=v_{0}a^{3}(t)$ is the physical volume of universe at given time $t$ and at present time $t_{0}$ ,the physical volume is $v_{0}$ with $a(t_{0})=1$ . According to the 1st law of thermodynamics(which is an energy conservation equation),

[TABLE]

Integrating equation (6) one can easily find the expression for energy density of the i-th fluid as,

[TABLE]

where $\rho_{i0}$ is the value of $\rho_{i}$ at $t=t_{0}$ i.e. at the present time .The energy density of i-th fluid, from equation (9) can be arranged as

[TABLE]

where $A_{i}(t)=\frac{1}{v_{0}^{\omega_{i}}}\int_{t_{0}}^{t}v^{\omega_{i}}dt$ . Now, taking entropy $S_{i}$ as a function of two independent thermodynamic variables namely volume( $v$ ) and temperature ( $T$ ) and also considering $dS_{i}$ to be an exact differential, one obtains from equation (8),

[TABLE]

Now combining equations (10) and (11), a new relation among energy density ,temperature and time can be written as

[TABLE]

where $\beta$ is an integration constant. Effectively at present time $t=t_{0}$ ,

[TABLE]

Now equations (12) and (13) imply the evolution of energy of the i-th fluid with time and temperature as,

[TABLE]

This can be termed as the modified equation of state of this fluid where as before $E_{i}=\rho_{i}v$ and $E_{i0}=\rho_{i0}v_{0}$ .

III.1 Determination of thermodynamic derivatives : $C_{p},C_{v},K_{S},K_{T}$

Now, the motivation of the present work is to find the criteria of thermodynamic stability of the universe . So the conditions which may be satisfied by the parameters , indicate the thermodynamic features of the evolution of the expanding universe. For any system the thermodynamic derivatives i.e. the heat capacities,compressibility and isobaric expansibility determine whether the system is stable or not. So it will be very important to analyze these parameters for each fluid in the present context.

From the 1st law of thermodynamics in equation (8), one can determine the two heat capacities : heat capacity at constant volume $C_{v}$ and heat capacity at constant pressure $C_{p}$ by the relations ,

[TABLE]

where $H=E+pv$ , is called the enthalpy of the system.

Now the relation between temperature of a system at a constant volume $(T_{a})$ and the physical equilibrium temperature is given by (according to Haba:2016bpt )

[TABLE]

The temperature at a constant volume $T_{a}$ also evolves as

[TABLE]

So from equations (17) and (18),

[TABLE]

Now from equations (15)and (19) one obtains,

[TABLE]

and equations (1) ,(16) and (19) yield ( using the definition of enthalpy)

[TABLE]

Also equation (21) can be rewritten (using 17) as

[TABLE]

Further assuming temperature and pressure as independent thermodynamic variables , the variation of volume can be expressed as (See ref. Barboza:2015rsa and Bhandari:2017cow )

[TABLE]

where

[TABLE]

is familiar as thermal expansibility. The isothermal compressibility $(K_{T})$ is given by

[TABLE]

Similarly the adiabatic compressibility is

[TABLE]

In isothermal process,

[TABLE]

Also one has the well established relation amongst heat capacities and compressibilities as Barboza:2015rsa

[TABLE]

So one can find the expression for isothermal expansibility $\alpha$ from equation (11) as,

[TABLE]

Now equations (27) and (28) yield

[TABLE]

Hence one obtains the expressions for compressibilities from equations (29) , (30) and (31) containing the dependence of diffusion parameter $\gamma_{i}$ as,

[TABLE]

III.2 Stability Conditions for cosmic fluids :

For thermodynamic stability of any fluid , it must follow the conditions Barboza:2015rsa ; Bhandari:2017cow namely $C_{p},C_{v},K_{T},K_{S}\geq 0$ . For the present work, using the expressions of these thermodynamical parameters , the stability conditions are presented in the following table 1 :

IV Stability criteria with variable equation of state parameter

So far we have examined the conditions for stability of cosmic fluids with constant equation of state parameters. In the present section the stability conditions will be discussed for cosmic fluids having variable equation of state parameters . The fluids are diffusive as before with no interaction amongst them. In such context, the solution of the conservation equation (6) yields,

[TABLE]

where,

[TABLE]

Taking

[TABLE]

one can write equation (34) as

[TABLE]

IV.1 Derivation of the relation between physical equilibrium temperature (T) and temperature at constant volume $(T_{a})$ :

From 1st law of thermodynamics (i.e. equation (8)), by choosing entropy (S) as a function of two independent variables volume (v) and temperature (T), the condition for dS to be an exact differential gives

[TABLE]

Now, from equation (39) one obtains the energy density as a function of temperature as

[TABLE]

Again, combining equations (38) and (40) the expression for entropy can be written as,

[TABLE]

Now, as for an adiabatic system $\Delta S=0$ , so one can determine from equation (41), the relation between $T_{0}$ and $T_{a}$ (i.e. evolution of $T_{a}$ ) as

[TABLE]

Now combining equations (35) , (36) and (42) one obtains ,

[TABLE]

or equivalently

[TABLE]

$\mbox{~{}~{}where~{}~{}}z_{i}=\frac{\rho_{i0}v_{0}}{3T_{0}}\mbox{~{}~{}and~{}~{}}B(A_{i})=T_{0}+\frac{\gamma_{i}T_{0}}{\rho_{i0}}A_{i}(t)$ .

Now as the adiabatic contribution of entropy for the i-th fluid (excluding the diffusive entropy $(S_{id}=\int\frac{v_{o}\gamma_{i}}{T}dT))$ namely,

[TABLE]

does not vary (i.e. $d\ln(S_{i(ad)})=0$ ) , so temperature can be written as,

[TABLE]

where , $L$ is a constant . Now combining (45) and (46) yields

[TABLE]

with $Q=\frac{\omega_{i0}L}{1+\omega_{i0}}$ , a constant .

IV.2 Thermodynamic derivatives

Similar to the above section , one can determine the heat capacities and compressibilities of the present thermodynamical system as follows: now the solution of equation (6),(or equivalently equation ( 34)) can be expressed as

[TABLE]

Again from equation (39), one finds

[TABLE]

Combining the above equations (48) and (49) one can write,

[TABLE]

Now imposing the initial conditions , one can formulate the modified equation of state in terms of evolution of energy with time and temperature as

[TABLE]

The above equation (51) can be written according to equation (47) as

[TABLE]

So using equations (15) , (16), (47) and (51) ,one can easily express the heat capacities as

[TABLE]

and hence one obtains the relation

[TABLE]

Again equations (48) and (49) yield

[TABLE]

which implies along with equation (47) that

[TABLE]

So from equations (27) , (53) , (55) and (57) one writes

[TABLE]

and from (28),one obtains

[TABLE]

Now, using equation (56) the expression for $\alpha_{i}$ takes the form

[TABLE]

and the expressions for $K_{T}$ and $K_{S}$ are given by (using equations (58) , (59) and (60))

[TABLE]

Now according to the conditions of thermodynamic stability , we analyze the equations (53) , (54) , (61) and (62) to find the restrictions of thermodynamic stability in different ranges of $\omega_{i}$ in the following Table 2 .

V brief discussions

A detailed thermodynamic study of non-interacting cosmic fluids having diffusive nature has been organized in the present work. As a result, individual fluid does not obey the energy conservation relation ,rather the non-conservation is proportional to the corresponding diffusive current. Each component of the cosmic fluids is assumed to have barotropic equation of state with constant or variable equation of state parameter. The stability conditions are expressed in the form of inequalities for different ranges of equation of state parameter. Also, the stability conditions for both constant (Table 1 )and variable (Table 2) equation of state parameter in the range $-1<\omega_{i}<0$ (i.e. for dark energy region) have been presented graphically in figure 1 and 2 respectively. From figure 2, different restrictions on diffusion parameter $\gamma_{i}$ are found depending on whether the values of $\frac{d\ln\omega_{i}}{d\ln T}$ (i.e. relative variation of $\omega_{i}$ with respect to the temperature T) belong to REGION-I or REGION-II . However due to expanding nature of the universe, the isobaric expansibility ( $\alpha_{i}$ ) should always be positive. This condition restricts the function $\frac{d\ln\omega_{i}}{d\ln T}$ only in REGION-II,which is acceptable for expanding universe (Table 3).

Further, it is interesting to note that, for constant equation of state parameter, (i.e. Table 1 ), the system cannot be thermodynamically stable for $-1<\omega_{i}\leq 0$ . This result is very much similar to that of Barboza:2015rsa . However, the present work is thermodynamically stable in phantom region in contrary with non-diffusive cosmic fluids Barboza:2015rsa . Finally, it is found that diffusive fluid with variable equation of state parameter is thermodynamically stable for all possible values of $\omega_{i}$ under some restrictions on the diffusion parameter $\gamma_{i}$ and nature of variation of $\omega_{i}$ with respect to temperature.

VI Acknowledgement

The authors are thankful to IUCAA, Pune, India for research facilities at Library. The author SM acknowledges UGC-JRF and PB acknowledges DST-INSPIRE (File no: IF160086) for awarding Research fellowship. Also SC acknowledges the UGC-DRS Program in the Department of Mathematics, Jadavpur University.

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Universe consisting of diffusive Dark fluids: Thermodynamics and stability analysis

I Abstract

II Introduction

III Thermodynamic analysis of non-interacting diffusive cosmic fluids having constant equation of state parameter

III.1 Determination of thermodynamic derivatives : Cp,Cv,KS,KTC_{p},C_{v},K_{S},K_{T}Cp​,Cv​,KS​,KT​

III.2 Stability Conditions for cosmic fluids :

IV Stability criteria with variable equation of state parameter

IV.1 Derivation of the relation between physical equilibrium temperature (T) and temperature at constant volume (Ta)(T_{a})(Ta​):

IV.2 Thermodynamic derivatives

V brief discussions

VI Acknowledgement

III.1 Determination of thermodynamic derivatives : $C_{p},C_{v},K_{S},K_{T}$

IV.1 Derivation of the relation between physical equilibrium temperature (T) and temperature at constant volume $(T_{a})$ :