On Exponential Long-Time Evolution in Statistical Mechanics

Penrose & Coveney (1994) recently introduced an invertible discrete-time dynamical system called the pastry-cook's transformation, for which they constructed a `canonical' non-equilibrium ensemble. In the present paper, we apply the Brussels formalism of non-equilibrium statistical mechanics to this system. The use of the formalism is justified rigorously, and the operators which arise in the theory are calculated exactly. The set of ensembles for which the theory is valid is a Banach space of functions satisfying a certain smoothness condition. This condition ensures that ensembles show a decay towards equilibrium, in agreement with the time asymmetry observed in thermodynamics. We also calculate the decay of time correlation functions using Ruelle's theory of dynamical resonances. We find that all three methods furnish essentially the same description of the exponential decay to equilibrium in this system.