Non-equilibrium thermodynamics of an expanding Swiss-Cheese braneworld cosmology with a matter bounce

Abstract

We investigate the non-equilibrium thermodynamics of a Swiss-cheese (SC) braneworld universe. By employing Hayward's unified first law and the Clausius relation at the apparent horizon, we first show that the standard equilibrium treatment of the effective SC braneworld fluid, together with the usual Hawking temperature, reproduces the Bekenstein-Hawking area law without any braneworld correction to the entropy. This demonstrates that the brane quadratic energy density corrections cannot, by themselves, generate a modified entropy area relation within the equilibrium framework. We therefore formulate a non-equilibrium thermodynamic description in which deviations from the Bekenstein-Hawking entropy are accompanied by an internal entropy production term. For a general horizon entropy functional, we derive the corresponding entropy production rate and then specialize to a logarithmic and inverse area quantum-corrected entropy. The resulting irreversible contribution is expressed in terms of the Hubble parameter and its derivative. We apply the formalism to a nonsingular matter-bounce solution in the SC braneworld. During the expanding phase, the total entropy production remains positive satisfying the GSLT. Moreover, the late-time behavior of the second derivative of the total entropy becomes negative, indicating a decelerating entropy growth and an asymptotic approach toward thermodynamic equilibrium. Far from the bounce, the equation of state parameter $w \approx -1$, indicating a dark energy-dominated regime. Significant departures from $w = -1$ are confined to the bounce vicinity, where high energy braneworld corrections drive the nonsingular transition. Thus, the model naturally transitions from a high energy bounce into an asymptotic dark energy phase during late-time expansion.