Thermodynamical behaviour of the Variable Chaplygin gas

The thermodynamical behaviour of the Variable Chaplygin gas (VCG) model is studied, using an equation of state like $P = - \frac{B}{\rho}$, where $B = B_{0}V^{-\frac{n}{3}}$. Here $B_{0}$ is a positive universal constant, $n$ is also a constant and $V$ is the volume of the fluid. From the consideration of thermodynamic stability, it is seen that only if the values of $n$ are allowed to be negative, then $(\frac{\partial P}{\partial V})_{S} <0$ throughout the evolution. Again thermal capacity at constant volume $c_{V}$ shows positive expression. Using the best fit value of $n = -3.4$ as previously found by Guo \emph{et al} \cite{guo1} gives that the fluid is thermodynamically stable through out the evolution. The effective equation of state for the special case of, $n = 0$ goes to $\Lambda$CDM model. Again for $n <0$ it favours phantom-like cosmology which is in agreement with the current SNe Ia constraints like VCG model. The deceleration parameter is also studied in the context of thermodynamics and the analysis shows that the \emph{flip} occurs for the value of $n < 4$. Finally the thermal equation of state is discussed which is an explicit function of temperature only. It is also observed that the third law of thermodynamics is satisfied in this model. As expected the volume increases as temperature falls during adiabatic expansions. In this case, for $T \rightarrow 0$, the thermal equation of state reduces to $(- 1 + \frac{n}{6})$ which is identical with the equation of state for the case of large volume.