Time-Symmetry Breaking in Hamiltonian Mechanics

摘要

Hamiltonian trajectories are strictly time-reversible. Any time series ofHamiltonian coordinates {q} satisfying Hamilton's motion equations willlikewise satisfy them when played "backwards", with the corresponding momentachanging signs : {+p} --> {-p}. Here we adopt Levesque and Verlet's preciselybit-reversible motion algorithm to ensure that the trajectory reversibility isexact, with the forward and backward sets of coordinates identical.Nevertheless, the associated instantaneous Lyapunov instability, or "sensitivedependence on initial conditions" of "chaotic" (or "Lyapunov unstable")bit-reversible coordinate trajectories can still exhibit an exponentiallygrowing time-symmetry-breaking irreversibility. Surprisingly, the positive andnegative exponents, as well as the forward and backward Lyapunov spectra, areusually not closely related, and so give four differing topological measures of"local" chaos. We have demonstrated this symmetry breaking for fluidshockwaves, for free expansions, and for chaotic molecular collisions. Here weillustrate and discuss this time-symmetry breaking for threestatistical-mechanical systems, [1] a minimal (but still chaotic) one-body"cell model" with a four-dimensional phase space; [2] relatively smallcolliding crystallites, for which the whole Lyapunov spectrum is accessible;[3] a near-continuum inelastic collision of two larger 400-particle balls. Inthe last two of these pedagogical problems the two colliding bodies coalesce.The particles most prone to Lyapunov instability are dramatically different inthe two time directions. Thus this Lyapunov-based symmetry breaking furnishesan interesting Arrow of Time.