Conjugate Pairing Rule Research Articles

Extremum properties of the Gaussian thermostat

We use nonequilibrium molecular dynamics computer simulation to explore a number of properties of the Gaussian thermostat. We show for a nonequilibrium system at a fixed state point, that within an infinite family of thermostats the Gaussian thermostat appears to: (i) minimise the largest Lyapunov exponent; (ii) maximise the smallest Lyapunov exponent; (iii) minimise the magnitude of the phase space compression; (iv) maximise the sum of the positive Lyapunov exponents and (v) maximise both the Kaplan-Yorke and the Mori dimensions of the system. We have recently proved, that among this family of thermostats, the Gaussian thermostat alone satisfies the conjugate pairing rule.

Statistical mechanics of systems far from equilibrium, as studied by molecular dynamics

Abstract Recently it has been shown that even far from equilibrium, bulk thermal transport coefficients can be calculated from the maximal, phase space, Lyapunov exponents using the conjugate pairing rule. We point out that the Lyapunov exponents themselves can be expressed in terms of “Green-Kubo” relations.

Field-dependent conductivity and diffusion in a two-dimensional Lorentz gas

The conductivity and diffusion of a color-charged two-dimensional thermostatted Lorentz gas in a color field is studied by a variety of methods. In this gas, point particles move through a regular triangular array of soft scatterers, where, in the presence of a field, a nonequilibrium stationary state is reached by coupling to a Gaussian thermostat. The zero-field conductivity and diffusion coefficient are computed with equilibrium molecular dynamics dynamics from the Green-Kubo formula and the Einstein relation. Their values are consistent and approach those obtained by Machta and Zwanzig in the limit of hard (disk) scatterers. The field-dependent conductivity is obtained from its constitutive relation, from the coupling constant to the thermostat, and by using the recently derived conjugate pairing rule of Evans, Cohen, and Morriss, from the two maximal Lyapunov exponents of the Lorentz gas in the stationary state. All these methods give consistent results. Finally, elements of the field-dependent diffusion tensor have been computed. At zero field, they are consistent with the zero-field conductivity, but they vanish beyond a critical field strength, suggesting a dynamical phase transition at the critical field; the conductivity appears to remain finite, approaching a constant value for large field strengths.

The Gaussian thermostat, phase space compression and the conjugate pairing rule

We recently proposed a possible variational principle for ergodic, non-equilibrium steady states (NESS) far from equilibrium (1991, Phys. Rev. Lett., 67, 2597). We provided computer simulation results in support of the hypothesis that in NESS with fixed internal energy, volume, number of particles and applied external field, the decrease of the volume of phase space which is accessible to the system is a local minimum with respect to variations in endogenous variables. This hypothesis is a microscopic and nonlinear generalization of the principle of minimum entropy production which is valid only for NESS close to equilibrium where both the local thermodynamic equilibrium and the Onsager reciprocal relations are valid. In the present paper, we point out that for a wide class of possible thermostatting mechanisms, the Gaussian thermostat minimizes the phase space compression. With respect to our variational principle, the Gaussian thermostat is therefore the optimal thermostat. Further, for this class of po...

Conjugate-pairing rule and thermal-transport coefficients.

It has recently been shown by Evans, Cohen, and Morriss [Phys. Rev. A 42, 5990 (1990)] that bulk thermal-transport coefficients can be calculated from the maximal Lyapunov exponents by using the conjugate-pairing rule. In the present paper we use computer simulation to explore the validity of the rule under conditions which are more general than those used in the derivation by Evans, Cohen, and Morriss. We show that the existence of a IsteadyP state for dissipative dynamics and the satisfaction of the condition known as the adiabatic incompressibility of phase space (i.e., area preserving in the absence of a thermostat) are insufficient for the rule to hold. We also show that the linear field-induced shift of the Lyapunov exponents with respect to the exponent index, is among the nonequilibrium molecular-dynamics algorithms, apparently peculiar to the SLLOD algorithm (so named because of its close relationship to the Dolls tensor algorithm) for shear flow.