Phase Transition and Thermodynamic Stability in an Entropy-driven Universe

Abstract

Motivated by the notion that the mathematics of gravity can be reproduced from a statistical requirement of maximal entropy, we study the consequence of introducing an entropic source term in the Einstein-Hilbert action. For a spatially homogeneous cosmological system driven by this entropic source and enveloped by a time evolving apparent horizon, we formulate a modified version of the second law of thermodynamics. An explicit differential equation governing the internal entropy profile is found. Using a Hessian matrix analysis of the internal entropy we check the thermodynamic stability for a ΛCDM cosmology, a unified cosmic expanson and a non-singular ekpyrotic bounce. We find the mathematical condition for a second order phase transition during these evolutions from the divergence of specific heat at constant volume. The condition is purely kinematic and quadratic in nature, relating the deceleration parameter and the jerk parameter that chalks out an interesting curve on the parameter space. This condition is valid even without the entropic source term and may be a general property of any phase transition.

Figures (6)

• Figure 1: Evolution of entropy ($S$) as a function of cosmic time for (i) top : an ever-accelerating model ($a(t) \sim t^{\frac{4}{3}}$), (ii) middle : decelerating model ($a(t) \sim t^{\frac{2}{3}}$) and (iii) bottom : a unified expansion model ($a(t) \sim \exp [H_{0}t -\frac{H_{1}}{(n-1)t^{n-1}}]$).
• Figure 2: Plot of Eq. (\ref{['dimlesscondi']}) : the kinematic condition (jerk vs deceleration) of phase transition for the universe.
• Figure 3: Plots of $C_V$, $C_P$, $\frac{\partial^2{S}}{\partial U^2}$ and $\alpha$ as a function of cosmic time for a $\Lambda$CDM cosmology ($a(t) = \left(\frac{\Omega_m}{\Omega_{vac}}\right)^{1/3}\sinh^{2/3}(t/t_0)$).
• Figure 4: Plots of $C_V$, $C_P$, $\frac{\partial^2{S}}{\partial U^2}$ and $\alpha$ for a unified description of cosmic expansion ($a(t) = a_{0}\exp [H_{0}t -\frac{H_{1}}{(n-1)t^{n-1}}]$).
• Figure 5: Scale factor for an universe exhibitng ekpyrotic bounce, $a(t) = \left[ 1+a_0t^2 \right]^n\exp\left( \frac{1}{\beta-1}(t_s-t)^{1-\beta} \right)$.