Lyapunov instability for a periodic Lorentz gas thermostated by deterministic scattering

Abstract

In recent work a deterministic and time-reversible boundary thermostat called thermostating by deterministic scattering has been introduced for the periodic Lorentz gas [Phys. Rev. Lett. 84, 4268 (2000)]. Here we assess the nonlinear properties of this new dynamical system by numerically calculating its Lyapunov exponents. Based on a revised method for computing Lyapunov exponents, which employs periodic orthonormalization with a constraint, we present results for the Lyapunov exponents and related quantities in equilibrium and nonequilibrium. Finally, we check whether we obtain the same relations between quantities characterizing the microscopic chaotic dynamics and quantities characterizing macroscopic transport as obtained for conventional deterministic and time-reversible bulk thermostats.

Thermostating by Deterministic Scattering: Construction of Nonequilibrium Steady States

“…In particular, one can inquire about the universality of relations between these quantities as found for specific thermostats. On the basis of this Letter, these issues have been studied in recent work [23][24][25][26]: For a system of hard disks under temperature gradient and shear thermostated by our method, Ref. [23] shows that, in general, no identity between phase space contraction and entropy production exists.…”

“…Furthermore, no conjugate pairing rule between Lyapunov exponents holds for systems thermostated by deterministic scattering [26]. A detailed comparison of our thermostat to different types of Nosé-Hoover thermostats in the driven periodic Lorentz gas revealed different kinds of bifurcation scenarios depending on the specific way of thermostating [24,25]. To obtain universal characterizations of NSS generated by different thermostating schemes thus remains an open question.…”

Symmetric Jacobian for local Lyapunov exponents and Lyapunov stability analysis revisited

“…Both Oseledec and RK methods will be illustrated for the Lorenz [14] and the Rössler [15] model by comparing them with each other and by also applying them to shorter time intervals for transient chaos. It is one of the main points of this contribution to show that the method using a set orthogonal to the flow is more adequate to describe the chaotic behaviour than the Oseledec method using general directions, thus explaining why Rateitschak and Klages [10] found much improved results introducing this method for complex chaotic behaviour.…”

“…This is performed by adding a simple constraint to the Oseledec frame as already done in Ref. [10]: The first vector remains fixed along the flow direction dx, and any further vectors are subsequently orthogonalised after each rotation, which implies that only the divergence and not any partial acceleration are assessed at every point on the trajectory. This avoids the additional complexity introduced by the Oseledec method, and the local exponents are in accord with the local divergence of the trajectories.…”

“…Since in many cases the average acceleration is zero, the integration for long times cancels this contribution, thus only the value for divergence remains. These shortcomings can easily be avoided by using a method introduced by Rateitschak and Klages (RK) [10]: Select as an initial direction u the direction of dx, thus u is parallel to the trajectory. Certainly, this direction will remain parallel to the trajectory for all times.…”