Harmonizing Quantum Gravity with Thermodynamics: Determination of the EUP Parameter by Prigogine’s Law

In 1974, Stephen Hawking made the groundbreaking discovery that black holes emit thermal radiation, characterized by a specific temperature now known as the Hawking temperature. While his original derivation is intricate, retrieving the exact expressions for black hole temperature and entropy in a simpler, more intuitive way without losing the core physical principles behind Hawking's assumptions is possible. This is obtained by employing the Heisenberg uncertainty principle which is known to be connected to the vacuum fluctuation. This exercise allows us to easily perform more complex calculations involving the effects of quantum gravity. This work aims to answer the following question: \textit{Is it possible to reconcile Prigogine's second law of thermodynamics for open systems and the second law of black hole dynamics with Hawking radiation}? Due to quantum gravity effects, the Heisenberg uncertainty principle has been extended to the Generalized Uncertainty Principle (GUP) and successively to the Extended Uncertainty Principle (EUP). The expression for the EUP parameter is obtained by conjecturing that Prigogine's second law of thermodynamics and the second law of black holes are not violated by Hawking thermal radiation mechanism. The modified expression for the entropy of a Schwarzschild black hole is also derived.