Fluctuations and asymmetry via local Lyapunov instability in the time-reversible doubly thermostated harmonic oscillator

Forward and backward trajectories from time-symmetric equations of motion can have time-asymmetric stability properties, and exhibit time-asymmetric fluctuations. Away from equilibrium this symmetry breaking is the mechanical equivalent of the second law of thermodynamics. Strange attractor states obeying the second law are time-reversed versions of (unobservable) repeller states which violate that law. Here, we consider both the equilibrium and the nonequilibrium cases for a simple deterministically thermostated oscillator. At equilibrium the extended phase-space distribution is a smooth Gaussian function. Away from equilibrium the distribution is instead a fractal strange attractor. In both cases we illustrate local time-symmetry breaking. We also quantify the forward–backward fluctuation asymmetry for the thermostated oscillator.