Emergence of cosmic space in Tsallis modified gravity from equilibrium and non-equilibrium thermodynamic perspective

In this paper, we obtain the law of emergence with Tsallis entropy from the thermodynamic laws. We first derive the law of emergence from the equilibrium description of the unified first law and Clausius relation. However, it has been shown that considering Tsallis entropy as the horizon entropy, the Clausius relation $\delta Q=T dS$ does not hold due to non-equilibrium thermodynamics and is replaced by the entropy-balance relation $dS=\frac{\delta Q}{T}+d_{i}S$, where $d_iS$ is the additional entropy produced due to the irreversible thermodynamic process. Hence, we derive the law of emergence from the non-equilibrium description of thermodynamic laws. The comparison between the law of emergence in both cases shows that the law of emergence from the non-equilibrium approach describes the expansion of the universe in terms of the areal volume, unlike the effective volume in the equilibrium case. We have further shown that the law of emergence also satisfies the condition of the maximization of entropy for a de Sitter universe; thus, the entropy of the universe evolves to a bounded value in the asymptotic future.