  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.
