Time 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. One objection to irreversibility is based on Poincar\'e's recursion theorem: a classical hamiltonian confined system returns after some time, so-called Poincar\'e recurrence time (PRT), close to its initial configuration. Boltzmann's reply was that for a $N \sim 10^{23}$ macroscopic number of particles, PRT is very large and exceeds the age of the universe. In this paper we compute for the first time, using molecular dynamics, a typical recurrence time $T(N)$ for a realistic case of a gas of $N$ particles. We find that $T(N) \sim N^z \exp (y N)$ and determine the exponents $y$ and $z$ for different values of the particle density and temperature. We also compute $y$ analytically using Boltzmann's hypotheses. We find an excellent agreement with the numerical results. This agreement validates Boltzmann's hypotheses which are not yet mathematically proven. We establish that that $T(N)$ exceeds the age of the Universe for a relatively small number of particles, much smaller than $10^{23}$.