Calculation of Poincar\'e recurrence time and irreversibility in Statistical Mechanics

One of the important questions in statistical mechanics is how irreversibility (time's arrow) occurs when Newton equations of motion are time reversal invariant. Boltzmann's argument is that the observed irreversibility is due to the very large number of particles. One objection to Boltzmann was based on Poincar\'e's recursion theorem: a classical hamiltonian confined system returns after some time, called today Poincar\'e recurrence time (PRT), close to its initial configuration. Boltzmann's reply was that for N = 10²³ particles, PRT is very large and exceeds the age of the universe. In this paper we compute for the first time, a typical recurrence time $ T(N)$ for a gas of N particles. We find that T(N) ~ Nᶻ exp (y N) and determine the exponents y and z for different values of the particle density and kinetic energy (temperature). We find that $ T(N) $ exceeds the age of the Universe for a relatively small number of particles, much smaller than 10²³. Our results support Boltzmann's argument on the origin of irreversibility in statistical mechanics.