Transition state theory for activated systems with driven anharmonic barriers

Classical Fisher information for differentiable dynamical systems

Fisher information is a lower bound on the uncertainty in the statistical estimation of classical and quantum mechanical parameters. While some deterministic dynamical systems are not subject to random fluctuations, they do still have a form of uncertainty. Infinitesimal perturbations to the initial conditions can grow exponentially in time, a signature of deterministic chaos. As a measure of this uncertainty, we introduce another classical information, specifically for the deterministic dynamics of isolated, closed, or open classical systems not subject to noise. This classical measure of information is defined with Lyapunov vectors in tangent space, making it less akin to the classical Fisher information and more akin to the quantum Fisher information defined with wavevectors in Hilbert space. Our analysis of the local state space structure and linear stability leads to upper and lower bounds on this information, giving it an interpretation as the net stretching action of the flow. Numerical calculations of this information for illustrative mechanical examples show that it depends directly on the phase space curvature and speed of the flow.

Energy transport between heat baths with oscillating temperatures

Energy transport is a fundamental physical process that plays a prominent role in the function and performance of myriad systems and technologies. Recent experimental measurements have shown that subjecting a macroscale system to a time-periodic temperature gradient can increase thermal conductivity in comparison to a static temperature gradient. Here, we theoretically examine this mechanism in a nanoscale model by applying a stochastic Langevin framework to describe the energy transport properties of a particle connecting two heat baths with different temperatures, where the temperature difference between baths is oscillating in time. Analytical expressions for the energy flux of each heat bath and for the system itself are derived for the case of a free particle and a particle in a harmonic potential. We find that dynamical effects in the energy flux induced by temperature oscillations give rise to complex energy transport hysteresis effects. The presented results suggest that applying time-periodic temperature modulations is a potential route to control energy storage and release in molecular devices and nanosystems.

Phase space geometry of isolated to condensed chemical reactions

The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the design and control of chemical reactions.

Chaos in conservative discrete-time systems subjected to parameter drift

Based on the example of a paradigmatic area preserving low-dimensional mapping subjected to different scenarios of parameter drifts, we illustrate that the dynamics can best be understood by following ensembles of initial conditions corresponding to the tori of the initial system. When such ensembles are followed, snapshot tori are obtained, which change their location and shape. Within a time-dependent snapshot chaotic sea, we demonstrate the existence of snapshot stable and unstable foliations. Two easily visualizable conditions for torus breakup are found: one in relation to a discontinuity of the map and the other to a specific snapshot stable manifold, indicating that points of the torus are going to become subjected to strong stretching. In a more general setup, the latter can be formulated in terms of the so-called stable pseudo-foliation, which is shown to be able to extend beyond the instantaneous chaotic sea. The average distance of nearby point pairs initiated on an original torus crosses over into an exponential growth when the snapshot torus breaks up according to the second condition. As a consequence of the strongly non-monotonous change of phase portraits in maps, the exponential regime is found to split up into shorter periods characterized by different finite-time Lyapunov exponents. In scenarios with plateau ending, the divided phase space of the plateau might lead to the Lyapunov exponent averaged over the ensemble of a torus being much smaller than that of the stationary map of the plateau.

Application of Floquet theory to dynamical systems with memory

We extend the recently developed generalized Floquet theory [Phys. Rev. Lett. 110, 170602 (2013)] to systems with infinite memory, i.e., a dependence on the whole previous history. In particular, we show that a lower asymptotic bound exists for the Floquet exponents associated to such cases. As examples, we analyze the cases of an ideal 1D system, a Brownian particle, and a circuit resonator with an ideal transmission line. All these examples show the usefulness of this new approach to the study of dynamical systems with memory, which are ubiquitous in science and technology.

Upside/Downside statistical mechanics of nonequilibrium Brownian motion. II. Heat transfer and energy partitioning of a free particle

The energy partitioning during activation and relaxation events under steady-state conditions for a Brownian particle driven by multiple thermal reservoirs of different local temperatures is investigated. Specifically, we apply the formalism derived in Paper I [G. T. Craven and A. Nitzan, J. Chem. Phys. 148, 044101 (2018)] to examine the thermal transport properties of two sub-ensembles of Brownian processes, distinguished at any given time by the specification that all the trajectories in each group have, at that time, energy either above (upside) or below (downside) a preselected energy threshold. Dynamical properties describing energy accumulation and release during activation/relaxation events and relations for upside/downside energy partitioning between thermal reservoirs are derived. The implications for heat transport induced by upside and downside events are discussed.

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.

Upside/Downside statistical mechanics of nonequilibrium Brownian motion. I. Distributions, moments, and correlation functions of a free particle

Statistical properties of Brownian motion that arise by analyzing, separately, trajectories over which the system energy increases (upside) or decreases (downside) with respect to a threshold energy level are derived. This selective analysis is applied to examine transport properties of a nonequilibrium Brownian process that is coupled to multiple thermal sources characterized by different temperatures. Distributions, moments, and correlation functions of a free particle that occur during upside and downside events are investigated for energy activation and energy relaxation processes and also for positive and negative energy fluctuations from the average energy. The presented results are sufficiently general and can be applied without modification to the standard Brownian motion. This article focuses on the mathematical basis of this selective analysis. In subsequent articles in this series, we apply this general formalism to processes in which heat transfer between thermal reservoirs is mediated by activated rate processes that take place in a system bridging them.

Nonexponential kinetics of ion pair dissociation in electrofreezing water

Temporally- or spatially-heterogeneous environments can participate in many kinetic processes, from chemical reactions and self-assembly to the forced dissociation of biomolecules. Here, we simulate the molecular dynamics of a model ion pair forced to dissociate in an explicit, aqueous solution. Triggering dissociation with an external electric field causes the surrounding water to electrofreeze and the ion pair population to decay nonexponentially. To further probe the role of the aqueous environment in the kinetics, we also simulate dissociation events under a purely mechanical force on the ion pair. In this case, regardless of whether the surrounding water is a liquid or already electrofrozen, the ion pair population decays exponentially with a well-defined rate constant that is specific to the medium and applied force. These simulation data, and the rate parameters we extract, suggest the disordered kinetics in an electrofreezing medium are a result of the comparable time scales of two concurrent processes, electrofreezing and dissociation.

Lagrangian descriptors of driven chemical reaction manifolds

The persistence of a transition state structure in systems driven by time-dependent environments allows the application of modern reaction rate theories to solution-phase and nonequilibrium chemical reactions. However, identifying this structure is problematic in driven systems and has been limited by theories built on series expansion about a saddle point. Recently, it has been shown that to obtain formally exact rates for reactions in thermal environments, a transition state trajectory must be constructed. Here, using optimized Lagrangian descriptors [G. T. Craven and R. Hernandez, Phys. Rev. Lett. 115, 148301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.148301], we obtain this so-called distinguished trajectory and the associated moving reaction manifolds on model energy surfaces subject to various driving and dissipative conditions. In particular, we demonstrate that this is exact for harmonic barriers in one dimension and this verification gives impetus to the application of Lagrangian descriptor-based methods in diverse classes of chemical reactions. The development of these objects is paramount in the theory of reaction dynamics as the transition state structure and its underlying network of manifolds directly dictate reactivity and selectivity.