Master equation approach to the conjugate pairing rule of Lyapunov spectra for many-particle thermostated systems.

Abstract

The master equation approach to Lyapunov spectra for many-particle systems is applied to nonequilibrium thermostated systems to discuss the conjugate pairing rule. We consider iso-kinetic thermostated systems with a shear flow sustained by an external restriction, in which particle interactions are expressed as a Gaussian white randomness. Positive Lyapunov exponents are calculated by using the Fokker-Planck equation to describe the tangent vector dynamics. We introduce another Fokker-Planck equation to describe the time-reversed tangent vector dynamics, which leads to the calculation of the negative Lyapunov exponents. Using the Lyapunov exponents provided by these two Fokker-Planck equations we show the conjugate pairing rule is satisfied for thermostated systems with a shear flow in the thermodynamic limit which allow us to replace the friction coefficient with a constant number. We also give an explicit form to connect the Lyapunov exponents with the time correlation of the interaction matrix in a thermostated system with a color field.