Jason R. Green

Relationship between dynamical entropy and energy dissipation far from thermodynamic equilibrium

Significance The dissipation of energy occurs naturally in systems as diverse as those in biology and as common as those responsible for the weather. Systems that dissipate energy while self-assembling also show promise as a synthetic route to responsive nanoscale materials. Optimizing the energy efficiency of these syntheses requires reducing the theoretical description to only those variables that are essential to an accurate prediction of the energy lost as heat. Here, with computer simulations and a model of assembly, we establish a methodology for identifying those essential variables. With this minimal description of the system we also find a linear relationship between the dynamical entropy of the self-assembling system and the energy dissipated as heat during assembly. Connections between microscopic dynamical observables and macroscopic nonequilibrium (NE) properties have been pursued in statistical physics since Boltzmann, Gibbs, and Maxwell. The simulations we describe here establish a relationship between the Kolmogorov–Sinai entropy and the energy dissipated as heat from a NE system to its environment. First, we show that the Kolmogorov–Sinai or dynamical entropy can be separated into system and bath components and that the entropy of the system characterizes the dynamics of energy dissipation. Second, we find that the average change in the system dynamical entropy is linearly related to the average change in the energy dissipated to the bath. The constant energy and time scales of the bath fix the dynamical relationship between these two quantities. These results provide a link between microscopic dynamical variables and the macroscopic energetics of NE processes.

Measuring disorder in irreversible decay processes

Rate coefficients can fluctuate in statically and dynamically disordered kinetics. Here, we relate the rate coefficient for an irreversibly decaying population to the Fisher information. From this relationship we define kinetic versions of statistical-length squared and divergence that measure cumulative fluctuations in the rate coefficient. We show the difference between these kinetic quantities measures the amount of disorder, and is zero when the rate coefficient is temporally and spatially unique.

Order and disorder in irreversible decay processes

Dynamical disorder motivates fluctuating rate coefficients in phenomenological, mass-action rate equations. The reaction order in these rate equations is the fixed exponent controlling the dependence of the rate on the number of species. Here, we clarify the relationship between these notions of (dis)order in irreversible decay, n A → B, n = 1, 2, 3, …, by extending a theoretical measure of fluctuations in the rate coefficient. The measure, Jn-Ln (2)≥0, is the magnitude of the inequality between Jn, the time-integrated square of the rate coefficient multiplied by the time interval of interest, and Ln (2), the square of the time-integrated rate coefficient. Applying the inequality to empirical models for non-exponential relaxation, we demonstrate that it quantifies the cumulative deviation in a rate coefficient from a constant, and so the degree of dynamical disorder. The equality is a bound satisfied by traditional kinetics where a single rate constant is sufficient. For these models, we show how increasing the reaction order can increase or decrease dynamical disorder and how, in either case, the inequality Jn-Ln (2)≥0 can indicate the ability to deduce the reaction order in dynamically disordered kinetics.

Characterizing molecular motion in H2O and H3O+ with dynamical instability statistics.

Sets of finite-time Lyapunov exponents characterize the stability and instability of classically chaotic dynamical trajectories. Here we show that their sample distributions can contain subpopulations identifying different types of dynamics. In small isolated molecules these dynamics correspond to distinct elementary motions, such as isomerizations. Exponents are calculated from constant total energy molecular dynamics simulations of H(2)O and H(3)O(+), modelled with a classical, reactive, all-atom potential. Over a range of total energy, exponent distributions for these systems reveal that phase space exploration is more chaotic near saddles corresponding to isomerization and less chaotic near potential energy minima. This finding contrasts with previous results for Lennard-Jones clusters, and is explained in terms of the potential energy landscape.

Chaotic dynamics near steep transition states

Classical molecular motion near potential energy saddles can be more or less chaotic relative to motion near minima. The relative degree of chaos depends on the extent of coupling between the degrees of freedom and on the curvature of the potential energy landscape. Here, we explore these effects using constant energy molecular dynamics simulations and independent criteria associated with locally chaotic behavior – namely, the constancy of the local mode action and the magnitude of finite-time Lyapunov exponents. These criteria reconcile the chaotic basins and relatively ordered saddles of the Lennard-Jones trimer, with the chaotic saddles and ordered basins for reactive, all-atom H2O described by the Garofalini H2O potential. By modifying the Garofalini and Lennard-Jones models we separate the compounding effects of nonlinear three-body interactions and steep reaction path curvature on the local degree of chaos near saddles and minima.

Effects of temperature and mass conservation on the typical chemical sequences of hydrogen oxidation

Macroscopic properties of reacting mixtures are necessary to design synthetic strategies, determine yield, and improve the energy and atom efficiency of many chemical processes. The set of time-ordered sequences of chemical species are one representation of the evolution from reactants to products. However, only a fraction of the possible sequences is typical, having the majority of the joint probability and characterizing the succession of chemical nonequilibrium states. Here, we extend a variational measure of typicality and apply it to atomistic simulations of a model for hydrogen oxidation over a range of temperatures. We demonstrate an information-theoretic methodology to identify typical sequences under the constraints of mass conservation. Including these constraints leads to an improved ability to learn the chemical sequence mechanism from experimentally accessible data. From these typical sequences, we show that two quantities defining the variational typical set of sequences-the joint entropy rate and the topological entropy rate-increase linearly with temperature. These results suggest that, away from explosion limits, data over a narrow range of thermodynamic parameters could be sufficient to extrapolate these typical features of combustion chemistry to other conditions.

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.

Nonequilibrium phase coexistence and criticality near the second explosion limit of hydrogen combustion

While hydrogen is a promising source of clean energy, the safety and optimization of hydrogen technologies rely on controlling ignition through explosion limits: pressure-temperature boundaries separating explosive behavior from comparatively slow burning. Here, we show that the emergent nonequilibrium chemistry of combustible mixtures can exhibit the quantitative features of a phase transition. With stochastic simulations of the chemical kinetics for a model mechanism of hydrogen combustion, we show that the boundaries marking explosive domains of kinetic behavior are nonequilibrium critical points. Near the pressure of the second explosion limit, these critical points terminate the transient coexistence of dynamical phases-one that autoignites and another that progresses slowly. Below the critical point temperature, the chemistry of these phases is indistinguishable. In the large system limit, the pseudo-critical temperature converges to the temperature of the second explosion limit derived from mass-action kinetics.

Extending the length and time scales of Gram-Schmidt Lyapunov vector computations

Abstract Lyapunov vectors have found growing interest recently due to their ability to characterize systems out of thermodynamic equilibrium. The computation of orthogonal Gram–Schmidt vectors requires multiplication and QR decomposition of large matrices, which grow as N 2 (with the particle count). This expense has limited such calculations to relatively small systems and short time scales. Here, we detail two implementations of an algorithm for computing Gram–Schmidt vectors. The first is a distributed-memory message-passing method using Scalapack. The second uses the newly-released MAGMA library for GPUs. We compare the performance of both codes for Lennard–Jones fluids from N = 100 to 1300 between Intel Nahalem/Infiniband DDR and NVIDIA C2050 architectures. To our best knowledge, these are the largest systems for which the Gram–Schmidt Lyapunov vectors have been computed, and the first time their calculation has been GPU-accelerated. We conclude that Lyapunov vector calculations can be significantly extended in length and time by leveraging the power of GPU-accelerated linear algebra.