Irreversible statistical mechanics from reversible motion: Q2R automata

Abstract

Q2R cellular automata are a nice pedagogical example for addressing a century-old question discussed at the 20th IUPAP Statistical Physics Conference in Paris last July: How is thermodynamic irreversibility, as seen in entropy increase, compatible with microscopic reversibility, as in Newton's equations of motion? The Boltzmann–Zermelo argument says that the times for returning to the low-entropy initial state of a large system are far longer than the age of the universe. How can we test this assertion? Molecular dynamics in its usual form is inadequate to address this question, since arithmetic rounding errors as well as discretization of space and time preclude return to the exact initial configuration. Monte Carlo simulations of Ising models avoid such errors but require random numbers and are therefore not reversible in the usual sense. The Q2R update rules of cellular automata for microcanonical Ising models are reversible and the system returns exactly to the initial configuration after an exponentially long time. Computer simulation of Q2R cellular automata help us to understand some very old fundamental problems. This property will be reviewed, as well well as unexplained critical exponents obtained in large-scale simulations.