On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

F. Bonetto; E. G. D. Cohen; C. Pugh

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On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

F. Bonetto·E. G. D. Cohen·C. Pugh

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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.

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F. Bonetto, E. G. D. Cohen, C. Pugh. 1998-03-09. On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents. https://arxiv.org/abs/chao-dyn/9803011

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