The Lyapunov exponent is widely used as the quantity to express the dynamical instability and the amount of the information of dynamical systems. In general there is a Lyapunov exponent for each degree of freedom in the dynamical valuables, and the sorted set of such Lyapunov exponents, known as the Lyapunov spectrum, has been the subject of study in many-particle systems. Some algorithms for numerical computations of Lyapunov spectra are well known (eg. the algorithm due to Benettin et al. and constraint methods), and a recent rapid development of computers has made it possible to calculate Lyapunov spectra for larger systems. However an analytical calculation of full Lyapunov spectra for many-particle systems is still a difficult task at present, although some analytical methods for a part of the Lyapunov spectrum, such as the largest Lyapunov exponent, etc., have been proposed and applied to many-particle systems (eg. the kinetic approach and the geometric approach).

The master equation approach is one of the methods that can be used to calculate the full Lyapunov spectra for many-particle systems. This method is applied to systems with random particle interactions, and uses a master equation to describe the tangent space dynamics. Under the assumption that the random interactions are expressed by a Gaussian white randomness, the master equation is simply attributed to a Fokker-Planck equation, and leads to a direct connection between the Lyapunov exponents and the time correlation of the matrix specifying the particle interactions. We apply this method to the following two topics.

• Stepwise structure of Lyapunov spectra

  We consider one- and two-dimensional models using the master equation approach, in which a stepwise structure of the Lyapunov spectra appears in the region of small positive Lyapunov exponents. Long range interactions lead to a clear separation of the Lyapunov spectra into a part exhibiting the stepwise structure and a part changing smoothly. In these models a wave-like structure is found in the eigenstates of the particle interaction matrix.

• Conjugate pairing rule for non-equilibrium thermostatted systems with a shear field

  We consider iso-kinetic thermostatted systems with a shear flow sustained by an external boundary condition. An "anti-Fokker-Planck equation" to describe the time-reversed tangent vector dynamics is introduced and used to calculate the negative Lyapunov exponents under the assumption of Gaussian white particle interactions. We show that the conjugate pairing rule is satisfied for thermostatted systems with a shear field in the thermodynamic limit.

[1] T. Taniguchi and G. P. Morriss, Phys. Rev. E 65, (2002) 056202.

[2] T. Taniguchi and G. P. Morriss, nlin.CD/0204003.