Lyapunov instability of dense Lennard-Jones fluids

We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems in three dimensions with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential. A new algorithm is discussed which incorporates ideas from control theory and constrained nonequilibrium molecular dynamics. Equilibrium and nonequilibrium steady states are examined. The latter are generated by the application of an external field F₄ through which equal numbers of particles are accelerated in opposite directions, and by thermostatting the system using Nos\'e-Hoover or Gauss mechanics. In equilibrium (F₄=0) the Lyapunov spectra are symmetrical and may be understood in terms of a simple Debye model for vibrational modes in solids. For nonequilibrium steady states (F₄0) the Lyapunov spectra are not symmetrical and indicate a collapse of the phase-space density onto an attracting fractal subspace with an associated loss in dimensionality proportional to the square of the applied field. Because of this attractor's vanishing volume in phase space and the instability of the corresponding repellor it is not possible to observe trajectories violating the second law of thermodynamics in spite of the time-reversal invariance of the equations of motion. Thus Nos\'e-Hoover mechanics, of which Gauss's isokinetic mechanics is a special case, resolves the reversibility paradox first stated by J. Loschmidt Sitzungsber. kais. Akad. Wiss. Wien 2. Abt. 73, 128 (1876) for nonequilibrium steady-state systems.