Ergodicity of One-dimensional Oscillators with a Signum Thermostat

Gibbs' canonical ensemble describes the exponential equilibrium distribution $f(q, p, T) \propto e^{-H(q,p)/kT}$ for an ergodic Hamiltonian system interacting with a 'heat bath' at temperature T. The simplest deterministic heat bath can be represented by a single 'thermostat variable' $\zeta$. Ideally, this thermostat controls the kinetic energy so as to give the canonical distribution of the coordinates and momenta {q, p}. The most elegant thermostats are time-reversible and include the extra variable(s) needed to extract or inject energy. This paper describes a single-variable 'signum thermostat.' It is a limiting case of a recently proposed 'logistic thermostat.' It has a single adjustable parameter and can access all of Gibbs' microstates for a wide variety of one-dimensional oscillators.