Conjugate Pairing Rule Research Articles

Planar mixed flow and chaos: Lyapunov exponents and the conjugate-pairing rule

In this work we characterize the chaotic properties of atomic fluids subjected to planar mixed flow, which is a linear combination of planar shear and elongational flows, in a constant temperature thermodynamic ensemble. With the use of a recently developed nonequilibrium molecular dynamics algorithm, compatible and reproducible periodic boundary conditions are realized so that Lyapunov spectra analysis can be carried out for the first time. Previous studies on planar shear and elongational flows have shown that Lyapunov spectra organize in different ways, depending on the character of the defining equations of the system. Interestingly, planar mixed flow gives rise to chaotic spectra that, on one hand, contain elements common to those of shear and elongational flows but also show peculiar, unique traits. In particular, the influence of the constituent flows in regards to the conjugate-pairing rule (CPR) is analyzed. CPR is observed in homogeneously thermostated systems whose adiabatic (or unthermostated) equations of motion are symplectic. We show that the component associated with the shear tends to selectively excite some of those degrees, and is responsible for violations in the rule.

Lyapunov spectra and conjugate-pairing rule for confined atomic fluids

In this work we present nonequilibrium molecular dynamics simulation results for the Lyapunov spectra of atomic fluids confined in narrow channels of the order of a few atomic diameters. We show the effect that realistic walls have on the Lyapunov spectra. All the degrees of freedom of the confined system have been considered. Two different types of flow have been simulated: planar Couette flow and planar Poiseuille flow. Several studies exist on the former for homogeneous flows, so a direct comparison with previous results is performed. An important outcome of this work is the demonstration of how the spectrum reflects the presence of two different dynamics in the system: one for the unthermostatted fluid atoms and the other one for the thermostatted and tethered wall atoms. In particular the Lyapunov spectrum of the whole system does not satisfy the conjugate-pairing rule. Two regions are instead distinguishable, one with negative pairs' sum and one with a sum close to zero. To locate the different contributions to the spectrum of the system, we computed "approximate" Lyapunov exponents belonging to the phase space generated by the thermostatted area and the unthermostatted area alone. To achieve this, we evolved Lyapunov vectors projected into a reduced dimensional phase space. We finally observe that the phase-space compression due to the thermostat remains confined into the wall region and does not significantly affect the purely Newtonian fluid region.

Boundary condition independence of molecular dynamics simulations of planar elongational flow

The simulation of liquid systems in a nonequilibrium steady state under planar elongational flow (PEF) for indefinite time is possible only with the use of the so-called Kraynik-Reinelt (KR) periodic boundary conditions (PBCs) on the simulation cell. These conditions admit a vast range of implementation parameters, which regulate how the unit lattice is deformed under elongation and periodically remapped onto itself. Clearly, nonequilibrium properties of homogeneous systems in a steady state have to be independent of the boundary conditions imposed on the unit cell. In order to confirm the independence of measurable properties of a system under PEF from the particular set of periodic boundary conditions, we compute the Lyapunov spectra, apply the conjugate pairing rule, and carefully analyze the so-called unpaired exponents for an atomic fluid of various sizes and state points. We further compute the elongational viscosity for various implementations of boundary conditions. All our results confirm the independence from KR PBCs for the dynamics of phase-space trajectories and for the transport coefficients.

Chaotic properties of planar elongational flow and planar shear flow: Lyapunov exponents, conjugate-pairing rule, and phase space contraction

The simulation of planar elongational flow in a nonequilibrium steady state for arbitrarily long times has recently been made possible, combining the SLLOD algorithm with periodic boundary conditions for the simulation box. We address the fundamental questions regarding the chaotic behavior of this type of flow, comparing its chaotic properties with those of the well-established SLLOD algorithm for planar shear flow. The spectra of Lyapunov exponents are analyzed for a number of state points where the energy dissipation is the same for both flows, simulating a nonequilibrium steady state for isoenergetic and isokinetic constrained dynamics. We test the conjugate-pairing rule and confirm its validity for planar elongation flow, as is expected from the Hamiltonian nature of the adiabatic equations of motion. Remarks about the chaoticity of the convective part of the flows, the link between Lyapunov exponents and viscosity, and phase space contraction for both flows complete the study.

New observations regarding deterministic, time-reversible thermostats and Gauss’s principle of least constraint

Deterministic thermostats are frequently employed in nonequilibrium molecular dynamics simulations in order to remove the heat produced irreversibly over the course of such simulations. The simplest thermostat is the Gaussian thermostat, which satisfies Gauss's principle of least constraint and fixes the peculiar kinetic energy. There are of course infinitely many ways to thermostat systems, e.g., by fixing sigma(i)/p(i)/mu+l. In the present paper we provide, for the first time, convincing arguments as to why the conventional Gaussian isokinetic thermostat (mu = 1) is unique in this class. We show that this thermostat minimizes the phase space compression and is the only thermostat for which the conjugate pairing rule holds. Moreover, it is shown that for finite sized systems in the absence of an applied dissipative field, all other thermostats (mu not = 1) perform work on the system in the same manner as a dissipative field while simultaneously removing the dissipative heat so generated. All other thermostats (mu not = 1) are thus autodissipative. Among all mu, thermostats, only the mu = 1 Gaussian thermostat permits an equilibrium state.

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

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.

Lyapunov exponent pairing for a thermostatted hard-sphere gas under shear in the thermodynamic limit.

We demonstrate why for a sheared gas of hard spheres, described by the SLLOD equations with an isokinetic Gaussian thermostat in between collisions, deviations of the conjugate pairing rule for the Lyapunov spectrum are to be expected, employing a previous result that for a large number of particles N, the isokinetic Gaussian thermostat is equivalent to a constant friction thermostat, up to 1/sqrt[N] fluctuations. We also show that these deviations are at most of the order of the fourth power in the shear rate.

Conjugate pairing of Lyapunov exponents for isokinetic shear flow algorithms.

Previous numerical calculations of the Lyapunov exponents for the eight particle isokinetic SLLOD algorithm for shear viscosity are extended to higher shear rates and a more careful error analysis presented. These calculations imply that within error bars, the conjugate pairing rule is satisfied for this system. The shift in the unpaired exponent appears to be unconnected with the shift in the other conjugate pairs. This distinguishes one degree of freedom from all others in the system.

Breaking conjugate pairing in thermostated billiards by a magnetic field

We demonstrate that in the thermostated three-dimensional Lorentz gas, the symmetry of the Lyapunov spectrum can be broken by adding to the system an external magnetic field not perpendicular to the electric field. For perpendicular field vectors, there is a Hamiltonian reformulation of the dynamics and the conjugate pairing rule still holds. This indicates that symmetric Lyapunov spectra have nothing to do with time-reversal symmetry or reversibility; instead, it seems to be related to the existence of a Hamiltonian connection.

Lyapunov instability for a hard-disk fluid in equilibrium and nonequilibrium thermostated by deterministic scattering

We compute the full Lyapunov spectra for a hard-disk fluid under temperature gradient and under shear. The Lyapunov exponents are calculated using a recently developed formalism for systems with elastic hard collisions. The system is thermalized by deterministic and time-reversible scattering at the boundary, whereas the bulk dynamics remains Hamiltonian. This thermostating mechanism allows for energy fluctuations around a mean value which is reflected by only two vanishing Lyapunov exponents in equilibrium and nonequilibrium. In nonequilibrium steady states the phase-space volume is contracted on average, leading to a negative sum of the Lyapunov exponents. Since the system is driven inhomogeneously we do not expect the conjugate pairing rule to hold, which is indeed shown to be the case. Finally, the Kaplan–Yorke dimension and the Kolmogorov–Sinai entropy are calculated from the Lyapunov spectra.

The conjugate-pairing rule for non-Hamiltonian systems.

In systems that satisfy the Conjugate Pairing Rule (CPR), the spectrum of Lyapunov exponents is symmetric. The sum of each conjugate pair of exponents is identical. Since in dissipative systems the sum of all the exponents is the entropy production divided by Boltzmann's constant, the calculation of transport coefficients from the Lyapunov exponents is greatly simplified in systems that satisfy CPR. Sufficient conditions for CPR are well known: the underlying adiabatic dynamics should be symplectic. However, the necessary conditions for CPR are not known. In this paper we report on the results of computer simulations which shed light on the necessary conditions for the CPR to hold. We provide, for the first time, convincing evidence that the standard molecular dynamics algorithm for calculating shear viscosity violates the CPR, even in the thermodynamic limit. In spite of this it appears that the sum of the maximal exponents is equal to the entropy production per degree of freedom. Thus it appears that the shear viscosity can still be calculated using the standard viscosity algorithm by summing the maximal pair of exponents.(c) 1998 American Institute of Physics.

Thermostats: Analysis and application.

Gaussian isokinetic and isoenergetic deterministic thermostats are reviewed in the correct historical context with their later justification using Gauss' principle of least constraint. The Nose-Hoover thermostat for simulating the canonical ensemble is also developed. For some model systems the Lyapunov exponents satisfy the conjugate pairing rule and a Hamiltonian formulation is obtained. We prove the conjugate pairing rule for nonequilibrium systems where the force is derivable from a potential. The generalized symplectic structure and Hamiltonian formulation is discussed. The application of such thermostats to the Lorentz gas is considered in some detail. The periodic orbit expansion methods are used to calculate averages and to categorize the generic transitions in the structure of the attractor. We prove that the conductivity in the nonequilibrium Lorentz gas is non-negative. (c) 1998 American Institute of Physics.

Non-equilibrium molecular dynamics integrators using Maple

The conjugate pairing rule states that for Hamiltonian systems to which a Gaussian or Nosé–Hoover thermostat is applied, the spectrum of Lyapunov exponents should exhibit a simple symmetry: that the sum of conjugate pairs of exponents should be a constant independent of the pair index. This symmetry is important for it enables one to calculate the entropy production and thereby the non-equilibrium transport coefficient, by summing just two exponents, say the maximal exponents. However, the conjugate pairing rule has never been proved for the standard computer simulation algorithm for shear flow namely the Sllod algorithm. We use the symbolic algebra program, Maple, to examine the validity of the conjugate pairing rule for the Sllod algorithm. We do this by using the multiple precision facility of Maple to carry out computer simulations for Sllod to very high accuracy. We also use the symbolic algebra capability of Maple to prove the conjugate pairing rule for ideal gases subject to thermostatted shear flow and also for unthermostatted shear flow of fluids at arbitrary density.

On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

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 Wojtkowski 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.

Lyapunov Instability of the Boundary-Driven Chernov–Lebowitz Model for Stationary Shear Flow

We report on the computation of full Lyapunov spectra of the boundary-driven Chernov–Lebowitz model for stationary planar shear flow. The Lyapunov exponents are calculated with a recently developed formalism for systems with elastic hard collisions. Although the Chernov–Lebowitz model is strictly energy conserving, any phase-space volume is subjected to a contraction due to the reflection rules of the hard disks colliding with the walls. Consequently, the sum of Lyapunov exponents is negative. As expected for an inhomogeneously driven system, the Lyapunov spectra do not obey the conjugate pairing rule. The external driving makes the system less chaotic, which is reflected in a decrease of the Kolmogorov–Sinai entropy if the driving is increased.

Symplectic properties of algorithms and simulation methods

Symplectic algorithms are investigated for their phase space conserving properties for thermo-statted Hamiltonians commonly used in equilibrium molecular dynamics. Corresponding algorithms can be generated for the dissipative Sllod equations of motion for Couette flow in two dimensions. This study focuses on the verification of the conjugate pairing rule (CPR) for such systems. For thermostatted Hamiltonian dynamics, adiabatic and thermostatted Dolls algorithms, the CPR is satisfied at each time step during a simulation unlike the Sllod case in which there are small yet finite deviations from the CPR.

Hamiltonian reformulation and pairing of Lyapunov exponents for Nosé-Hoover dynamics

The Nos\'eHamiltonian is adapted, leading to a derivation of the Nos\'e-Hoover equations of motion which does not involve time transformations, and in which the degree of freedom corresponding to the external reservoir is treated on the same footing as those of the rest of the system. In this form it is possible to prove the conjugate pairing rule for Lyapunov exponents of this system.

Lyapunov Spectrum and the Conjugate Pairing Rule for a Thermostatted Random Lorentz Gas: Kinetic Theory

We calculate all four nonzero Lyapunov exponents for a three-dimensional, dilute, random Lorentz gas by combining dynamical systems and Boltzmann equation methods. In the presence of an external field and a Gaussian thermostat the Lyapunov exponents, calculated up to second order in the applied field, satisfy a conjugate pairing rule. Agreement of the results obtained here with those of computer simulations of Dellago and Posch [following Letter, Phys. Rev. Lett. 78, 211 (1997)].

Lyapunov instability in a system of hard disks in equilibrium and nonequilibrium steady states.

We present a generalization of Benettin's classical algorithm for the calculation of full Lyapunov spectra to the case of dynamical systems where the smooth streaming is interrupted by a differentiable map at discrete times. With this formalism we derive the transformation rules for the offset vectors in tangent space for a system of hard particles in equilibrium and nonequilibrium steady states. In particular, we study the color conductivity of a system of hard disks carrying color charges subjected to an external color field. Full Lyapunov spectra are obtained numerically for equilibrium systems of 64 and 144 hard disks. Furthermore, the maximum Lyapunov exponent and the Kolmogorov-Sinai entropy are studied over a wide range of densities. Both mimic the collision rate very well. In the low density regime the maximum Lyapunov exponent is found to follow the relation ${\ensuremath{\lambda}}_{1}$\ensuremath{\propto}-\ensuremath{\rho} ln\ensuremath{\rho}, as conjectured by Krylov. Full Lyapunov spectra are also reported for nonequilibrium steady-state systems of 64 hard disks, which carry color charges and are externally perturbed by an applied color field. The simulations cover a wide range of densities and fields. From a careful study of small three- and four-particle systems the validity of the conjugate pairing rule is established numerically with an error less than 0.1%. Also the number of vanishing Lyapunov exponents due to the conserved quantities---center of mass, linear momentum, and kinetic energy---is discussed in some detail. \textcopyright{} 1996 The American Physical Society.

Conjugate pairing in the three-dimensional periodic Lorentz gas.

We numerically evaluate the Lyapunov exponents for the three-dimensional periodic Lorentz gas in an electric field, with a Gaussian thermostat. For small values of the field, the dynamics appears to be ergodic. For larger fields there are chaotic and periodic windows. The Lyapunov exponents are found to obey the conjugate pairing rule in both chaotic and stable regions, indicating that conjugate pairing can occur for small systems, and is not restricted to the large N limit.