A Hamiltonian Demon

Abstract

This paper describes a deterministic process, henceforth referred to as a demon, for altering the equilibrium distribution of a billiards-like dynamical system. The system consists of particles whose energies are integer valued, and where all energy-exchange mechanisms preserve this quantization. As a result, time and positions are continuous, while velocities are discrete. The demon is a localized, velocity-dependent potential whose effect is to simply alter the direction of an incident particle. It is shown how the demon can steer the system away from equilibrium to create a temperature differential, a density differential, a circulation, and a large-scale oscillation. It is further shown how the system – including the demon – can be captured by a time-independent Hamiltonian, where the set of states whose energies are integer valued is an invariant set. In addition, given any finite time horizon, the novel behavior extends to continuous energies in a neighborhood of the invariant set. Since the set of states where energy is quantized has zero Lebesgue measure, the results are not inconsistent with statistical characterizations of the second law of thermodynamics.