Gravity can significantly modify classical and quantum Poincare recurrence theorems

Abstract

Poincare recurrence theorem states that any finite system will come arbitrary close to its initial state after some very long but finite time. At the statistical level, this by itself does not represent a paradox, but apparently violates the second law of thermodynamics, which may lead to some confusing conclusions for macroscopic systems. However, this statement does not take gravity into account. If two particles with a given center of mass energy come at the distance shorter than the Schwarzschild diameter apart, according to classical gravity they will form a black hole. In the classical case, a black hole once formed will always grow and effectively quench the Poincare recurrence. We derive the condition under which the classical black hole production rate is higher than the classical Poincare recurrence rate. In the quantum case, if the temperature of the black hole is lower than the temperature of the surrounding gas, such a black hole cannot disappear via Hawking evaporation. We derive the condition which gives us a critical temperature above which the black hole production is faster than quantum Poincare recurrence time. However, in quantum case, the quantum Poincare recurrence theorem can be applied to the black hole states too. The presence of the black hole can make the recurrence time longer or shorter, depending on whether the presence of the black hole increases or decreases the total entropy. We derive the temperature below which the produced black hole increases the entropy of the whole system (gas particles plus a black hole). Finally, if evolution of the system is fast enough, then newly formed black holes will merge and accrete particles until one large black hole dominates the system. We give the temperature above which the presence of black holes increase the entropy of the whole system and prolongs the Poincare recurrence time.