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

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.

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.