Title: On the Validity of the Conjugate Pairing Rule for Lyapunov
Exponents
Authors: F. Bonetto, E.G.D. Cohen, C. Pugh
Abstract: For Hamiltonian systems subject to an external potential,
which in the presence of a thermostat will reach a nonequilibrium
stationary state, Dettmann and Morriss proved a strong conjugate
pairing rule (SCPR) for pairs of Lyapunov exponents in the case of
isokinetic (IK) stationary states which have a given kinetic energy.
This SCPR holds for all initial phases of the system, all times t and
all numbers of particles N. This proof was generalized by Wojtkovski
and Liverani to include hard interparticle potentials. A geometrical
reformulation of those results is presented. The present paper proves
numerically, using periodic orbits for the Lorentz gas, that SCPR
cannot hold for isoenergetic (IE) stationary states, which have a
given total internal energy. In that case strong evidence is obtained
for CPR to hold for large N and t, where it can be conjectured that
the larger N, the smaller t will be. This suffices for statistical
mechanics.
Keywords: dynamical systems, Lyapunov exponents, symmetry, pairing
rule
Adresses:
F. Bonetto: Mathematics Department, Bush Campus, Rutgers University,
Highland Park, 0NJ 08903
tel. 732-445-3512
e-mail bonetto@onsager.rutgers.edu
E.G.D. Cohen: box 28, Rockefeller University, 1230 York Avenue, New York,
NY 10021
tel: 212-327-8855
e-mail: egdc@rockvax.rockefeller.edu
C.Pugh: Mathematics Department, University of California
Berkeley California
Archived in: mp_arc@math.utexas.edu #98-165
chao-dyn@xyz.lanl.gov #9803011