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Dense‐fluid Lyapunov spectra via constrained molecular dynamics
The Journal of Chemical Physics
Abstract
A Lagrange‐multiplier method for finding the complete spectrum of Lyapunov exponents, which describe the spreading and mixing of many‐body phase‐space trajectories, is developed and applied here to simple two‐ and three‐dimensional equilibrium fluids with short‐range repulsive forces. The numerical values of the Lyapunov exponents converge well, in computer simulations of 103 to 105 time steps, and are insensitive both to the initial conditions and to the numerical accuracy of the trajectory integration.
... Since the pioneering work of Hoover et al. [2], many investigations of the chaotic properties of the many-particle systems have been carried out. Our own goal is to understand the relation between the irreversible macroscopic behavior of the atomic systems and the underlying microscopic theory with time-reversal symmetry. ...
... The particles interact with each other through a short-range repulsive pair potential. In this work, to optimize the numerical processes, we adopt the short-range pair potential introduced in Ref. [2], ...
... truncated at the cutoff radius, r = 1, where the first three derivatives vanish. The smooth truncation at short range minimizes the errors associated with numerical integration [2]. The repulsive part of this potential resembles that of the rare-gas system. ...
We study changes in the chaotic properties of a many-body system undergoing a solid-fluid phase transition. To do this, we compute the temperature dependence of the largest Lyapunov exponents for both two- and three-dimensional periodic systems of N-particles for various densities. The particles interact through a soft-core potential. The two-dimensional system exhibits an apparent second-order phase transition as indicated by a -shaped peak in the specific heat. The first derivative of with respect to the temperature shows a peak at the same temperature. The three-dimensional system shows jumps, in both system energy and , at the same temperature, suggesting a first-order phase transition. Relaxation phenomena in the phase-transition region are analyzed by using the local time averages. Comment: 16 pages, REVTeX, 10 eps figures, epsfig.sty
... As we can see in Fig. 13 where representative RPs around threshold 20 are plotted, the RPs are similar, presenting a rich structure that permits to extract a more detailed idea about the system dynamics. This may be due to the fact that MD systems are in fact far higher dimensional (see Hoover et al., 1987 andNayak et al., 1995) than the Lorenz system which is a low dimensional system. ...
In this work, a topological criterion is proposed for selecting a recurrence threshold for constructing a recurrence plot of a time series. It is based on a metric structure of the set of the recurrence plots that is defined by the recurrence plot deviation distance among recurrence plots, introduced in a previous paper by the authors. In this process for a range of threshold values, the corresponding recurrence plots are constructed. Then, a value of the threshold is considered to be optimal when the image of its recurrence plot remains close to images of the other recurrence plots constructed for close values of thresholds based on the topological criterion introduced in the present work. The results are applied both to time series emanating from the Lorenz dynamical system and to Molecular Dynamic simulations.
... As an international collaboration of uniformly nice people (cf. Moran, Hoover, and Bestiale 2016; Hoover, Posch, and Bestiale 1987; Hoover et al. 1988; see Tartamelia 2014 for an explanation) , lacking access to a croquet field (cf. Hassell and May 1974) , and not identifying any excellent pun to be made from ordering our names (cf. ...
The rhetoric of " excellence " is pervasive across the academy. It is used to refer to research outputs as well as researchers, theory and education, individuals and organisations, from art history to zoology. But what does " excellence " mean? Does it in fact mean anything at all? And is the pervasive narrative of excellence and competition a good thing? Drawing on a range of sources we interrogate " excellence " as a concept and find that it has no intrinsic meaning as used in the academy. Rather it functions as a linguistic interchange mechanism or boundary object. To investigate whether linguistic function is useful we examine how excellence rhetoric combines with narratives of scarcity and competition and show that hypercompetition that arises leads to a performance of " excellence " that is completely at odds with the qualities of good research. We trace the roots of issues in reproducibility, fraud, as well as diversity to the stories we tell ourselves as researchers and offer an alternative rhetoric based on soundness. " Excellence " is not excellent, it is a pernicious and dangerous rhetoric that undermines the very foundations of good research and scholarship.
... This is algorithmically achieved with the help of a set of perturbation vectors (±) g (Γ); = 1, . . . , D, which are either periodically re-orthonormalized with a Gram-Schmidt (GS) procedure (equivalent to a QRdecomposition) [27], or are continuously constrained to stay orthonormal with a triangular matrix of Lagrange multipliers [2,[28][29][30][31]. For historical reasons we refer to them as GS-vectors. ...
We use molecular dynamics simulations to compute the Lyapunov spectra of many-particle systems resembling simple fluids in thermal equilibrium and in non-equilibrium stationary states. Here we review some of the most interesting results and point to open questions.
... The complete spectrum calculation involves the solution of 6N × 6N + 6N coupled differential equations. There are a number of reports on various aspects of Lyapunov exponents in clusters such as the distribution of Lyapunov exponents [14,24], effects of three-body interactions [16,17] and the case of dense fluids [18][19][20]. ...
Classical dynamics of clusters can be characterized in terms of the regularity of the motion along the constant energy surfaces. In the case of rotating clusters, this characterization strongly depends on angular momentum and the partitioning of the kinetic energy between the rotational and vibrational degrees of freedom. Here we review the tools for such an analysis with special emphasis on the changes in Lyapunov exponents.
... For a D−dimensional dynamical system there exist D Lyapunov vectors e (α) (t) ∈ R D , α = 1, . . . , D which obey the equations of motion [14,15] ...
We introduce static and dynamic correlation functions for the spatial densities of Lyapunov vector fluctuations. They enable us to show, for the first time, the existence of hydrodynamic Lyapunov modes in chaotic many-particle systems with soft core interactions, which indicates universality of this phenomenon. Our investigations for Lennard-Jones fluids yield, in addition to the Lyapunov exponent - wave vector dispersion, the collective dynamic excitations of a given Lyapunov vector. In the limit of purely translational modes the static and dynamic structure factor are recovered.
... Also known for more than twenty years is the algorithm for numerical computation of Lyapunov exponents due to Benettin and others [10, 11]. Later, a constraint method was introduced [12]. More recently, the effects of the hard collisions have been properly taken into account [3, 13] . ...
The Lyapunov spectrum corresponding to a periodic orbit for a two dimensional
many particle system with hard core interactions is discussed. Noting that the
matrix to describe the tangent space dynamics has the block cyclic structure,
the calculation of the Lyapunov spectrum is attributed to the eigenvalue
problem of a 16x16 reduced matrices regardless of the number of particles. We
show that there is the thermodynamic limit of the Lyapunov spectrum in this
periodic orbit. The Lyapunov spectrum has a step structure, which is explained
by using symmetries of the reduced matrices.
This is our current research perspective on models providing insight into statistical mechanics. It is necessarily personal, emphasizing our own interest in simulation as it developed from the National Laboratories’ work to the worldwide explosion of computation of today. We contrast the past and present in atomistic simulations, emphasizing those simple models that best achieve reproducibility and promote understanding. Few-body models with pair forces have led to today’s “realistic” simulations with billions of atoms and molecules. Rapid advances in computer technology have led to change. Theoretical formalisms have largely been replaced by simulations incorporating ingenious algorithm development. We choose to study particularly simple, yet relevant, models directed toward understanding general principles. Simplicity remains a worthy goal, as does relevance. We discuss hard-particle virial series, melting, thermostatted oscillators with and without heat conduction, chaotic dynamics, fractals, the connection of Lyapunov spectra to thermodynamics, and finally simple linear maps. Along the way, we mention directions in which additional modeling could provide more clarity and yet more interesting developments in the future.
This is our current research perspective on models providing insight into statistical mechanics. It is necessarily personal, emphasizing our own interest in simulation as it developed from the National Laboratories' work to the worldwide explosion of computation of today. We contrast the past and present in atomistic simulations, emphasizing those simple models which best achieve reproducibility and promote understanding. Few-body models with pair forces have led to today's ``realistic'' simulations with billions of atoms and molecules. Rapid advances in computer technology have led to change. Theoretical formalisms have largely been replaced by simulations incorporating ingenious algorithm development. We choose to study particularly simple, yet relevant, models directed toward understanding general principles. Simplicity remains a worthy goal, as does relevance. We discuss hard-particle virial series, melting, thermostatted oscillators with and without heat conduction, chaotic dynamics, fractals, the connection of Lyapunov spectra to thermodynamics, and finally simple linear maps. Along the way we mention directions in which additional modelling could provide more clarity and yet more interesting developments in the future.
The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original for- mulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or Filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit time-scale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram{Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation.
The rhetoric of “excellence” is pervasive across the academy. It is used to refer to research outputs as well as researchers, theory and education, individuals and organizations, from art history to zoology. But does “excellence” actually mean anything? Does this pervasive narrative of “excellence” do any good? Drawing on a range of sources we interrogate “excellence” as a concept and find that it has no intrinsic meaning in academia. Rather it functions as a linguistic interchange mechanism. To investigate whether this linguistic function is useful we examine how the rhetoric of excellence combines with narratives of scarcity and competition to show that the hyper-competition that arises from the performance of “excellence” is completely at odds with the qualities of good research. We trace the roots of issues in reproducibility, fraud, and homophily to this rhetoric. But we also show that this rhetoric is an internal, and not primarily an external, imposition. We conclude by proposing an alternative rhetoric based on soundness and capacity-building. In the final analysis, it turns out that that “excellence” is not excellent. Used in its current unqualified form it is a pernicious and dangerous rhetoric that undermines the very foundations of good research and scholarship. This article is published as part of a collection on the future of research assessment.
The rhetoric of “excellence” is pervasive across the academy. It is used to refer to research outputs as well as researchers, theory and education, individuals and organisations, from art history to zoology. But does “excellence” actually mean anything? Does this pervasive narrative of “excellence” do any good? Drawing on a range of sources we interrogate “excellence” as a concept and find that it has no intrinsic meaning in academia. Rather it functions as a linguistic interchange mechanism. To investigate whether this linguistic function is useful we examine how the rhetoric of excellence combines with narratives of scarcity and competition to show that the hypercompetition that arises from the performance of “excellence” is completely at odds with the qualities of good research. We trace the roots of issues in reproducibility, fraud, and homophily to this rhetoric. But we also show that this rhetoric is an internal, and not primarily an external, imposition. We conclude by proposing an alternative rhetoric based on soundness and capacity-building. In the final analysis, it turns out that that “excellence” is not excellent. Used in its current unqualified form it is a pernicious and dangerous rhetoric that undermines the very foundations of good research and scholarship.
The rhetoric of “excellence” is pervasive across the academy. It is used to refer to research outputs as well as researchers, theory and education, individuals and organisations, from art history to zoology. But what does “excellence” mean? Does it in fact mean anything at all? And is the pervasive narrative of excellence and competition a good thing? Drawing on a range of sources we interrogate “excellence” as a concept and find that it has no intrinsic meaning as used in the academy. Rather it functions as a linguistic interchange mechanism or boundary object. To investigate whether linguistic function is useful we examine how excellence rhetoric combines with narratives of scarcity and competition and show that hypercompetition that arises leads to a performance of “excellence” that is completely at odds with the qualities of good research. We trace the roots of issues in reproducibility, fraud, as well as diversity to the stories we tell ourselves as researchers and offer an alternative rhetoric based on soundness. “Excellence” is not excellent, it is a pernicious and dangerous rhetoric that undermines the very foundations of good research and scholarship.Submitted for peer review. A version that can be commented on is available at the link:
We study the numerical instability in calculating the full Lyapunov spectrum, especially the negative Lyapunov exponents, of a dense fluid in equilibrium. We incorporate 2D constant offset vectors associated with the center-of-mass motion into the numerical algorithm. These vectors constrain the other 2D(N - 1) offset vectors to stay in the phase space subspace. The symmetric spectrum implied by time-reversal symmetry of the motion equation is obtained using a modified computing algorithm.
Molecular dynamics simulations were performed for soft- and hard-sphere systems, for number densities ranging from 0.5 to 1.0, and the Kolmogorov-Sinai entropy (KS entropy) and self-diffusion coefficients were calculated. It is found that the KS entropy, when expressed in terms of average collision frequency, is uniquely related to the self-diffusion coefficient by a simple scaling law. The dependence of the KS entropy on average collision frequency and number density was also explored. Numerical results show that the scaling laws proposed by Dzugutov, and by Beijeren, Dorfman, Posch, and Dellago, can be applied to both soft- and hard-sphere systems by changing to more generalized forms.
Raman isotropic band profiles of the ν2 mode of acetonitrile in binary mixtures with methanol have been studied over the whole concentration range and between 196 and 330 K. Attempts have been made to understand the spectral behaviour in terms of variations in vibrational dephasing as a function of environment and in terms of rapid chemical exchange between complexed and non-complexed acetonitrile molecules. If exchange dynamics are assumed to be important it is found that the dissociation rate constant (k21) for this reaction is of the order of 1011 s−1. This rate seems unrealistically high although similar rates have been obtained for other hydrogen-bonded systems. Nevertheless, the band shape changes dramatically across the temperature range and this demonstrates clearly that a “merging” band profile does not necessarily prove that exchange dynamic processes are important. Bandwidth and frequency shifts across the concentration range could be attributed to increases in exchange rate as the amount of methanol increases or the temperature increases. However, the most probable explanation is that there is a change in vibrational dephasing rate due to environmental fluctuations. We clearly demonstrate that even at 0.001 molar fraction of CH3CN in CH3OH a finite number of CH3CN molecules are “free” (on the vibrational timescale) from the hydrogen-bonded interaction. An explanation for this rather surprising behaviour has been sought (and found) in terms of multiple hydrogen-bonding equilibria in this system. The effect has been shown to be associated with extensive methanol aggregation. An equilibrium model has been devised which predicts accurately the relative intensities of the two ν(CN) bands and the unusual behaviour in binary mixtures of this type.
We report far-infrared and Raman data for solutions of pyridine and iodine in different solvent environments. The ν(ï-I) band shows drastic frequency and band width variations due to variations in the dynamic relaxation processes with surrounding environment. We explore the contribution of chemical (site-site) exchange to these band shapes and for most situations find it to be small, at least within the classical (two-body) framework of equilibrium between ‘complexed’ and uncomplexed iodine. However, if excess iodine is used with a weakly complexing solvent (benzene or dioxan) the behavioural origin is less clear and site lifetimes may become low enough to cause effects in addition to those due to vibrational relaxation induced by environmental potential fluctuations.
We describe group theory statistical mechanics, GTSM, which enables us to predict new non-vanishing time correlation functions in fluids at steady state subjected to planar couette flow. These are by symmetry trivially zero at equilibrium. An ensemble average is treated using the rules of group theory in the laboratory XYZ frame and in the molecule-fixed xyz frame of the point group character tables. In this paper we determine the effect of couette flow on a range of ensemble averages by establishing the symmetry of the strain rate tensor in terms of the irreducible representations of the Rh(3) rotation reflection group in the XYZ frame. This symmetry, D(0)g + D(1)g + D(2)g, is the same as the pressure tensor, P and consists of an antisymmetric vorticity term, D(1)g and a symmetric strain rate component of symmetry D(0)g + D(2)g. This allows non-zero ensemble averages of the same symmetry in the XYZ frame. Depending on the number of off-diagonal elements in the strain rate tensor, up to six new off-diagonal elements of microscopic time-autocorrelation functions of type, appear by GTSM in the XYZ frame. We confirm this theory for monatomic fluids using molecular dynamics computer simulation. The SLLOD equations of motion for couette dvx/dZ flow were implemented. We calculated non-vanishing peculiar quantity autocorrelation functions, ACF, of the generic form, , (R is the position of a molecule) and for the Lennard-Jones fluid. The new correlation functions are highly structured and generally have a finite negative value at t = 0. They can exhibit time reversal dissymmetry, especially at low density.
Lyapunov spectra are measured for a three-dimensional many-body dense fluid, not only at equilibrium, but also in the presence of an isoenergetic nonequilibrium field generating a pair of equal and opposite currents. The Lyapunov spectra bear a strong resemblance to the Debye spectrum of solid-state physics.
We have calculated the Kolmogorov entropy for three- and seven-particle clusters bound by Lennard-Jones potentials and for three-particle clusters bound by Morse potentials of various ranges. We have used two quite different methods, one of which is new, which give consistent results. We find that all of these systems are classically chaotic over a wide range of energies surrounding the estimated quantum-mechanical zero point energies. Furthermore, for the three-particle clusters, we can rationalize the variation in the degree of chaos with total energy in terms of the local structure of the clusters’ potential energy surface.
Simulations by isothermal and isoergic molecular dynamics show that vibrationally-cold clusters, initially at some high-energy point on their potential surfaces, relax monotonically down to low-lying minima where they are trapped. The most significant point regarding this relaxation is that species with sawtoothlike and staircaselike potentials show the same qualitative behavior, equilibrating their vibrations after each major saddle crossing. This result justifies the use of transition state kinetics for constructing coarse-grained master equations to describe the well-to-well flow of population distributions on complex potential energy surfaces, even in cases such as (KCl)32 and some protein models which have staircase topographies in which large drops in potential energy from one well to the next might suggest nonthermal kinetics could occur.
The dynamics of triatomic clusters is investigated employing two-body Lennard-Jones and three-body Axilrod-Teller potential functions. Lyapunov exponents are calculated for the total energy range of -2.70ε
• Received 26 April 1996
DOI:https://doi.org/10.1103/PhysRevA.55.538
©1997 American Physical Society
We develop a molecular dynamic method to evaluate the full Lyapunov spectrum for two-dimensonal fluids composed of rigid diatomic molecules. The Lyapunov spectra are obtained for 18 rigid diatomic molecules for various bond lengths d(10-3<~d<~1.0) in two dimensions with periodic boundary conditions, and interacting with Hoover and Weeks-Chandler-Anderson short-range repulsive forces. The general trends and characteristic features of the Lyapunov spectra are examined for both potentials. Our results are compared with those obtained from I. Borzsák et al. [Phys. Rev. E 53, 3694 (1996)], whose model uses the Lagrange multiplier method.
Molecular dynamics simulations were performed for soft- and hard-sphere systems, for number densities ranging from 0.5 to 1.0, and the Kolmogorov-Sinai entropy (KS entropy) and self-diffusion coefficients were calculated. It is found that the KS entropy, when expressed in terms of average collision frequency, is uniquely related to the self-diffusion coefficient by a simple scaling law. The dependence of the KS entropy on average collision frequency and number density was also explored. Numerical results show that the scaling laws proposed by Dzugutov, and by Beijeren, Dorfman, Posch, and Dellago, can be applied to both soft- and hard-sphere systems by changing to more generalized forms.
The Grassberger–Procaccia method has been employed to study the transitions which occur as a classical Ar3 cluster, modeled by pairwise Lennard‐Jones potentials, passes from a rigid, solid‐like form to a nonrigid, liquid‐like form with increasing energy. Power spectra and lower bounds on the fractal dimensions and K entropies are presented at several energies along the caloric curve for the Ar3 cluster. In addition, the full spectrum of Liapunov exponents has been computed at these same energies to get an accurate value of the K entropy. Chaotic behavior, though relatively small, is observed even at low energies where the power spectrum displays largely normal‐mode structure. The degree of chaotic behavior increases with energy at energies where some degree of regularity is observed in the spectrum. However, at energies that just allow the system to pass into and across saddle regions separating local potential minima, the phase space appears to be separable into a region within the equilateral triangle potential well where the behavior is highly chaotic, and a region of lower dimensions and less chaos around the saddle of the linear configuration. Dimensions from approximately three to eight are observed. A clear separability of time scales for establishment of different extents of ergodicity permits the determination of fractal dimensions of the manifold on which the phase points moves, for time scales of physical, i.e., observable significance. We believe this to be the first evaluation of the dimensionality of the space on which the phase point moves, for a Hamiltonian system displaying this range of dimensions.
Clusters provide a particularly convenient laboratory for studying general characteristics of multidimensional potential surfaces and for relating the forms of those surfaces to the dynamics exhibited by the objects whose properties are governed by these surfaces. The multidimensional potentials of systems such as atomic van der Waals clusters and the binary clusters of alkali halide molecules can be represented very reliably, and now molecular van der Waals clusters of such species as CCl4 can be represented with considerable confidence as well. The extent of validity has been established for classical mechanical simulations, so that credence can now be given to the results of classical molecular dynamics simulations of clusters of second-row and heavier species. These illustrate finite-system counterparts of various kinds of phase changes, isomerization, and evolution of chaotic and ergodic behavior, as well as the effects on multidimensional potential surfaces of the character of the forces between pairs of particles.
By analyzing time-reversed trajectories from irreversible dissipative systems, we effectively reverse the order and signs of all the Lyapunov exponents. This reversal makes it possible to obtain the most negative Lyapunov exponents relatively easily. We illustrate the validity of this idea by studying the Lorenz model of Rayleigh-Bénard instability.
We performed molecular dynamics simulations of soft spherical particles over wide ranges of densities and temperatures corresponding to fluids, glasses and crystalline solids, and calculated the full Lyapunov spectra of these systems. For either phase corresponding exponents essentially scale with the square-root of the temperature in accordance with kinetic theory. The density dependence is more pronounced and less systematic. The shape of the spectrum of a glass is different from that of a crystalline solid at the same density and temperature and resembles the spectrum of the initial dense liquid-like phase. For dilute gases the sum of the positive exponents approaches zero and is increasingly dominated by the largest exponent. Although systematic changes of the Lyapunov spectra were observed, it seems that the spectral shape does not uniquely determine the phase of the system.
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.
We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems in three dimensions with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential. A new algorithm is discussed which incorporates ideas from control theory and constrained nonequilibrium molecular dynamics. Equilibrium and nonequilibrium steady states are examined. The latter are generated by the application of an external field Fe through which equal numbers of particles are accelerated in opposite directions, and by thermostatting the system using Nosé-Hoover or Gauss mechanics. In equilibrium (Fe=0) the Lyapunov spectra are symmetrical and may be understood in terms of a simple Debye model for vibrational modes in solids. For nonequilibrium steady states (Fe!=0) the Lyapunov spectra are not symmetrical and indicate a collapse of the phase-space density onto an attracting fractal subspace with an associated loss in dimensionality proportional to the square of the applied field. Because of this attractor's vanishing volume in phase space and the instability of the corresponding repellor it is not possible to observe trajectories violating the second law of thermodynamics in spite of the time-reversal invariance of the equations of motion. Thus Nosé-Hoover mechanics, of which Gauss's isokinetic mechanics is a special case, resolves the reversibility paradox first stated by J. Loschmidt [Sitzungsber. kais. Akad. Wiss. Wien 2. Abt. 73, 128 (1876)] for nonequilibrium steady-state systems.
Ergodic behavior in liquids, supercooled liquids, and glasses is examined with a focus on the time scale needed to obtain ergodicity. A measure, d(t), which is based on the time-averaged energies of the individual particles and which is referred to as the ``energy metric,'' is introduced to probe the approach to ergodic behavior. We suggest that d(t) obeys a dynamical scaling law for long but finite times and that it can be used to characterize the degree of stochasticity in measure preserving systems with large numbers of degrees of freedom. Examination of d(t) indicates that the configuration space is explored by a ``diffusive'' process in the space of the dynamical energy variables used in constructing the energy metric. The characteristic diffusion constant associated with this process is argued to be analogous to the well-known maximal Lyapunov exponent which is often used to characterize stochasticity in systems with few degrees of freedom. Based on the long-time behavior of d(t) it is shown that ergodicity is effectively broken in the glassy state. In addition to broken ergodicity, the possibility that a subtle symmetry is broken as the liquid-to-glass transition takes place is examined. It is suggested that a ``discrete'' symmetry, to be referred to as the statistical symmetry, is broken in the glassy phase. This is illustrated by analyzing the distribution of the energy of the particles. Based on this, we expect long-time dynamics and structural relaxation in glasses to be dominated by fluctuations in domains of finite length within which the particles are highly correlated. This is in accord with the ideas of Adams and Gibbs. All of the above arguments are illustrated with the aid of molecular-dynamics simulations of soft-sphere mixtures.
The Lyapunov exponents describe the time-averaged rates of expansion and contraction of a Lagrangian hypersphere made up of comoving phase-space points. The principal axes of such a hypersphere grow, or shrink, exponentially fast with time. The corresponding set of phase-space growth and decay rates is called the ''Lyapunov spectrum.'' Lyapunov spectra are determined here for a variety of two- and three-dimensional fluids and solids, both at equilibrium and in nonequilibrium steady states. The nonequilibrium states are all boundary-driven shear flows, in which a single boundary degree of freedom is maintained at a constant temperature, using a Nose-Hoover thermostat. Even far-from-equilibrium Lyapunov spectra deviate logarithmically from equilibrium ones. Our nonequilibrium spectra, corresponding to planar-Couette-flow Reynolds numbers ranging from 13 to 84, resemble some recent approximate model calculations based on Navier-Stokes hydrodynamics. We calculate the Kaplan-Yorke fractal dimensionality for the nonequilibrium phase-space flows associated with our strange attractors. The far-from-equilibrium dimensionality may exceed the number of additional phase-space dimensions required to describe the time dependence of the shear-flow boundary.
We investigate both dynamic and time-averaged chaos in a series of problems ranging from a simple pendulum to many-body chains and strings of particles in a gravitational field. Chain and spring systems typically display time-averaged Lyapunov spectra resembling those of one- and two-dimensional statistical-mechanical systems, but with details depending strongly on the nature of the links and the choice of canonical coordinates.
The dynamical properties of two measures to probe ergodic behavior in Hamiltonian systems with a large number of degrees of freedom are investigated. The measures, namely, the energy metric d(t) and the fluctuation metric Omega(t) are both based on the time-averaged energies of the individual particles comprising the system. The energy metric d(t) is obtained by following the dynamics of the system using two independent configurations, whereas Omega(t) is expressed in terms of the properties of a single trajectory. Both measures obey a dynamical scaling law for long but finite times. The scaling law for d(t), which was previously established numerically, and for Omega(t) is derived for systems that are effectively ergodic. The scaling function suggests that the configuration space is explored by a ``diffusive'' process in the space of the variables used to construct d(t) and Omega(t). Furthermore, a single parameter, namely DE and DOmega, characterizes the time scales needed for effective ergodicity to be obtained. These ideas are used to study the behavior of supercooled and glassy states of soft-sphere binary alloys using microcanonical molecular dynamics. The scaling forms for d(t) and Omega(t) are explicitly demonstrated. The temperature dependence of the numerically computed diffusion constants (DE and DOmega) reveals that the nature of phase-space exploration changes both near the fluid-solid transition and near the liquid-to-glass transition. The changes in DE and DOmega with temperature are well described by a Vogel-Fulcher equation close to the glass transition temperature. In addition, the distribution of the time-averaged individual particle energies P(e;t), moments of which are related to Omega(t), is shown to be broad in the glassy state. We argue that the time dependence of P(e;t) can be used to probe local structural relaxations in glasses.
We study the Lyapunov instability of a three-dimensional fluid composed of rigid diatomic molecules by molecular dynamics simulation. We use center-of-mass coordinates and angular variables for the configurational space variables. The spectra of Lyapunov exponents are obtained for 32 rigid diatomic molecules interacting through the Weeks-Chandler-Andersen potential for various bond lengths and densities. We show the general trends and characteristic features of the spectra of the Lyapunov exponents, and discuss the different contributions between translational and rotational degrees of freedom depending on the density and the bond length from the calculation of the projection of a certain subspace of the tangent space vectors.
The dynamical instability of many-body systems is best characterized through the time-dependent local Lyapunov spectrum [lambda(j)], its associated comoving eigenvectors [delta(j)], and the "global" time-averaged spectrum [<lambda(j)>]. We study the fluctuations of the local spectra as well as the convergence rates and correlation functions associated with the delta vectors as functions of j and system size N. All the number dependences can be described by simple power laws. The various powers depend on the thermodynamic state and force law as well as system dimensionality.
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.
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.
The relationship between the Kolmogorov-Sinai entropy, h(KS) and the self-diffusion coefficient D is studied for two classical simple fluid systems with purely repulsive potentials (one system with a Wayne-Chandler-Anderson potential and the other with a hard-sphere potential). Numerical simulation data for h(KS) and D, normalized by the average collision frequency nu and the diameter of the particle sigma as natural units of time and distance, reveal that, in the region spanning from normal liquid up to near solidification (0.50< or =rho< or =0.93), the Kolmogorov-Sinai entropy has a power law dependeney on the self-diffusion coefficient of the form h(KS)/nu proportional, variant (D/sigma(2)nu)(eta), in which eta is independent of density and temperature.
Lyapunov instability of a "diatomic" system of coupled map lattices is studied and the dynamics of Lyapunov modes (LMs) is compared with phonon dynamics. Similar to the phonon case mass differences between neighboring sites induce gaps in the Lyapunov spectrum and LMs split into two types correspondingly. An unexpected finding is that contrary to the phonon case a nontrivial threshold value for the mass difference is required for the occurrence of the spectral gap and the splitting of LMs. A possible origin of such a nontrivial threshold value of mass differences is suggested.
We use a constant driving forceF
d
together with a Gaussian thermostatting constraint forceF
d
to simulate a nonequilibrium steady-state current (particle velocity) in a periodic, two-dimensional, classical Lorentz gas. The ratio of the average particle velocity to the driving force (field strength) is the Lorentz-gas conductivity. A regular Galton-board lattice of fixed particles is arranged in a dense triangular-lattice structure. The moving scatterer particle travels through the lattice at constant kinetic energy, making elastic hard-disk collisions with the fixed particles. At low field strengths the nonequilibrium conductivity is statistically indistinguishable from the equilibrium Green-Kubo estimate of Machta and Zwanzig. The low-field conductivity varies smoothly, but in a complicated way, with field strength. For moderate fields the conductivity generally decreases nearly linearly with field, but is nearly discontinuous at certain values where interesting stable cycles of collisions occur. As the field is increased, the phase-space probability density drops in apparent fractal dimensionality from 3 to 1. We compare the nonlinear conductivity with similar zero-density results from the two-particle Boltzmann equation. We also tabulate the variation of the kinetic pressure as a function of the field strength,
We show that Nosromane-bar mechanics provides a link between computer simulations of nonequilibrium processes and real-world experiments. Reversible Nose-bar equations of motion, when used to constrain non- equilibrium boundary regions, generate stable dissipative behavior within an adjoining bulk sample governed by Newton's equations of motion. Thus, irreversible behavior consistent with the second law of thermodynamics arises from completely reversible microscopic motion. Loschmidt's reversibility paradox is surmounted by this Nose-bar-Newton system, because the steady-state nonequilibrium probability density in the many-body phase space is confined to a zero-volume attractor.
Based on the Lyapunov characteristic exponents, the ergodic property of
dissipative dynamical systems with a few degrees of freedom is studied
numerically by employing, as an example, the Lorenz system. The Lorenz
system shows the spectra of (+,0,-) type concerning the 1-dimensional
Lyapunov exponents, and the exponents take the same values for orbits
starting from almost of all initial points on the attractor.
This result suggests that the ergodic property for general dynamical
systems not necessarily belonging to the category of the axiom-A may
also be characterized in the framework of the spectra of the Lyapunov
characteristic exponents.
DOI:https://doi.org/10.1103/PhysRevLett.11.241
Lyapunov spectra are measured for a three-dimensional many-body dense fluid, not only at equilibrium, but also in the presence of an isoenergetic nonequilibrium field generating a pair of equal and opposite currents. The Lyapunov spectra bear a strong resemblance to the Debye spectrum of solid-state physics.
Benettin, Calgani and Strelcyn studied the dynamical separation of neighboring phase-space trajectories, determining the corresponding Lyapunov exponents by discrete rescaling of the intertrajectory separation. We incorporate rescaling directly into the equations of motion, preventing Lyapunov instability by using an effective constraint force.
- W G Moran
- S Hoover
- Bestiale
Moran, W. G. Hoover, and S. Bestiale, 1. Stat. Phys. 48, 709 (1987).
- W G Evans
- Hoover
Evans and W. G. Hoover, Annu. Rev. Fluid Mech. 18,243 (1986).
See formula 25.5.10 on p. 896. 'I, Shimada and I
4Handbook 0/ Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by M. Abramowitz and I. A. Stegun, National Bureau of Standards Applied Mathematics Series (National Bureau of Standards, Washington, D.C., 1964), Vol. 55. See formula 25.5.10 on p. 896.
'I, Shimada and I, Nagashima, Prog. Theor. Pbys. 61,1605 (1979).
- L Holian
- W G Hoover
- H A Posch
L. Holian, W. G. Hoover, and H. A. Posch, Pbys. Rev. Lett. 59, 10
( 1987).