Equilibrium and nonequilibrium Lyapunov spectra for dense fluids and solids

The Lyapunov exponents describe the time-averaged rates of expansion and contraction of a Lagrangian hypersphere made up of comoving phase-space points. The principal axes of such a hypersphere grow, or shrink, exponentially fast with time. The corresponding set of phase-space growth and decay rates is called the ``Lyapunov spectrum.'' Lyapunov spectra are determined here for a variety of two- and three-dimensional fluids and solids, both at equilibrium and in nonequilibrium steady states. The nonequilibrium states are all boundary-driven shear flows, in which a single boundary degree of freedom is maintained at a constant temperature, using a Nos\'e-Hoover thermostat. Even far-from-equilibrium Lyapunov spectra deviate logarithmically from equilibrium ones. Our nonequilibrium spectra, corresponding to planar-Couette-flow Reynolds numbers ranging from 13 to 84, resemble some recent approximate model calculations based on Navier-Stokes hydrodynamics. We calculate the Kaplan-Yorke fractal dimensionality for the nonequilibrium phase-space flows associated with our strange attractors. The far-from-equilibrium dimensionality may exceed the number of additional phase-space dimensions required to describe the time dependence of the shear-flow boundary.