Conjugate-pairing rule and thermal-transport coefficients

Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer-range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation-like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state.

Extensivity and additivity of the Kolmogorov-Sinai entropy for simple fluids

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the "dynamical chemical potential," the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

Stable and unstable periodic orbits in the one-dimensional lattice ϕ^{4} theory

Periodic orbits for the classical ϕ^{4} theory on the one-dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, free and fixed boundary conditions. Through the process, we investigate which normal modes of the linear theory can or cannot be extended to the full nonlinear theory and why. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance, and Lyapunov spectra for these orbits. The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic intersite potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the ϕ^{4} theory, with some concrete examples.

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.

Chaotic properties of isokinetic-isobaric atomic systems under planar shear and elongational flows

An investigation of the chaotic properties of nonequilibrium atomic systems under planar shear and planar elongational flows is carried out for a constant pressure and temperature ensemble, with the combined use of a Gaussian thermostat and a Nosé-Hoover integral feedback mechanism for pressure conservation. A comparison with Lyapunov spectra of atomic systems under the same flows and at constant volume and temperature shows that, regardless of whether the underlying algorithm describing the flow is symplectic, the degrees of freedom associated with the barostat have no overall influence on chaoticity and the general conjugate pairing properties are independent of the ensemble. Finally, the dimension of the strange attractor onto which the phase space collapses is found not to be significantly altered by the presence of the Nosé-Hoover barostatting mechanism.

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.

Numerical study of the steady state fluctuation relations far from equilibrium

A thermostatted dynamical model with five degrees of freedom is used to test the fluctuation relation of Evans and Searles (Omega-FR) and that of Gallavotti and Cohen (Lambda-FR). In the absence of an external driving field, the model generates a time-independent ergodic equilibrium state with two conjugate pairs of Lyapunov exponents. Each conjugate pair sums to zero. The fluctuation relations are tested numerically both near and far from equilibrium. As expected from previous work, near equilibrium the Omega-FR is verified by the simulation data while the Lambda-FR is not confirmed by the data. Far from equilibrium where a positive exponent in one of these conjugate pairs becomes negative, we test a conjecture regarding the Lambda-FR [Bonetto et al., Physica D 105, 226 (1997); Giuliani et al., J. Stat. Phys. 119, 909 (2005)]. It was conjectured that when the number of nontrivial Lyapunov exponents that are positive becomes less than the number of such negative exponents, then the form of the Lambda-FR needs to be corrected. We show that there is no evidence for this conjecture in the empirical data. In fact, when the correction factor differs from unity, the corrected form of Lambda-FR is less accurate than the uncorrected Lambda-FR. Also as the field increases the uncorrected Lambda-FR appears to be satisfied with increasing accuracy. The reason for this observation is likely to be that as the field increases, the argument of the Lambda-FR more and more accurately approximates the argument of the Omega-FR. Since the Omega-FR works for arbitrary field strengths, the uncorrected Lambda-FR appears to become ever more accurate as the field increases. The final piece of evidence against the conjecture is that when the smallest positive exponent changes sign, the conjecture predicts a discontinuous change in the "correction factor" for Lambda-FR. We see no evidence for a discontinuity at this field strength.

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.

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.

Pairing of Lyapunov exponents for a hard-sphere gas under shear in the thermodynamic limit

We consider a dilute gas of hard spheres under shear. We use one of the predominant models to study this system, namely, the so-called SLLOD equations of motion, with an isokinetic Gaussian thermostat in between collisions, to get a stationary total peculiar kinetic energy. Based on the previously obtained result that in the nonequilibrium steady state and in the case the number of particles N becomes large, the coefficient of dynamical friction representing the isokinetic Gaussian thermostat for the SLLOD dynamics fluctuates with 1/sqrt[N] fluctuations around a fixed value, we show on analytical grounds that for a hard sphere gas at small shear rate and with a large number of spheres, the conjugate pairing of the Lyapunov exponents is expected to be violated at the fourth power of the constant shear rate in the bulk.

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.

Stepwise structure of Lyapunov spectra for many-particle systems using a random matrix dynamics

The structure of the Lyapunov spectra for the many-particle systems with a random interaction between the particles is discussed. The dynamics of the tangent space is expressed as a master equation, which leads to a formula that connects the positive Lyapunov exponents and the time correlations of the particle interaction matrix. Applying this formula to one- and two-dimensional models we investigate the stepwise structure of the Lyapunov spectra that appear in the region of small positive Lyapunov exponents. Long range interactions lead to a clear separation of the Lyapunov spectra into a part exhibiting stepwise structure and a part changing smoothly. The part of the Lyapunov spectrum containing the stepwise structure is clearly distinguished by a wave-like structure in the eigenstates of the particle interaction matrix. The two-dimensional model has the same step widths as found numerically in a deterministic chaotic system of many hard disks.

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.

Lyapunov spectrum of granular gases

We calculate and study the Lyapunov spectrum of a granular gas maintained in a steady state by an isokinetic thermostat. Considering restitution coefficients greater than unity allows us to show that the spectra change smoothly and continuously at equilibrium. The shearing instability of the granular gas, however, provokes an abrupt change in the structure of the spectrum. The relationship between various physically relevant quantities and the energy dissipation rate differs from previously studied nonequilibrium steady states.

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.

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.