Ensemble Of Trajectories Research Articles

Nonergodicity of d-dimensional generalized Lévy walks and their relation to other space-time coupled models.

We investigate the nonergodicity of the generalized Lévy walk introduced by Shlesinger etal. [Phys. Rev. Lett. 58, 1100 (1987)PRLTAO0031-900710.1103/PhysRevLett.58.1100] with respect to the squared displacements. We present detailed analytical derivations of our previous findings outlined in a recent letter [Phys. Rev. Lett. 120, 104501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.104501], give detailed interpretations, and in particular emphasize three surprising results. First, we find that the mean-squared displacements can diverge for a certain range of parameter values. Second, we show that an ensemble of trajectories can spread subdiffusively, whereas individual time-averaged squared displacements show superdiffusion. Third, we recognize that the fluctuations of the time-averaged squared displacements can become so large that the ergodicity breaking parameter diverges, what we call infinitely strong ergodicity breaking. This phenomenon can also occur for paramter values where the lag-time dependence of the mean-squared displacements is linear indicating normal diffusion. In order to numerically determine the full distribution of time-averaged squared displacements, we use importance sampling. For an embedding of our findings into existing results in the literature, we define a more general model which we call variable speed generalized Lévy walk and which includes well-known models from the literature as special cases such as the space-time coupled Lévy flight or the anomalous Drude model. We discuss and interpret our findings regarding the generalized Lévy walk in detail and compare them with the nonergodicity of the other space-time coupled models following from the more general model.

A partially linearized spin-mapping approach for simulating nonlinear optical spectra.

We present a partially linearized method based on spin-mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin partially linearized density matrix method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is, indeed, the case by calculating the nonlinear optical response functions relevant for the pump-probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna-Matthews-Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiments.

Equilibria and oscillations in a reversible mechanical system

The method of stochastic molecular modeling, developed by the authors for calculating the transport coefficients of rarefied gas in a free medium, is generalized to describe transport processes in confined conditions. The phase trajectories of the studied molecular system are simulated stochastically, and the simulation of the dynamics of the molecule is split into processes. First, its shift in the configuration space is realized, and then a possible collision with other molecules is played out. The calculation of all observables, in particular, the transport coefficients is carried out by averaging over an ensemble of independent phase trajectories. The interaction of gas molecules with the boundary is described by mirror or mirror-diffuse laws. The efficiency of the algorithm is demonstrated by calculating the selfdiffusion coefficient of argon in the nanochannel. The accuracy of modeling is investigated, its dependence on the number of particles and phase trajectories used for averaging. The viscosity of rarefied gases in the nanochannel has been systematically studied. It is shown that it is non-isotropic, and its difference along and across the channel is determined by the interaction of gas molecules with the channel walls. By changing the material of the walls, it is possible to significantly change the viscosity of the gas, and it can be several times greater than in volume, or less. The indicated anisotropy of viscosity is recorded not only in nano-, but also in microchannels.

Second-order analysis of Fokker–Planck ensemble optimal control problems

Ensemble optimal control problems governed by a Fokker-Planck equation with space-time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker-Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first- and second-order optimality conditions is presented. For this purpose, the well-posedness of the Fokker-Planck equation, and new estimates concerning an inhomogeneous Fokker-Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control-to-state ma p appearing in the first-and second-order analysis.

Spectral density of individual trajectories of an active Brownian particle

We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion (BM) as exhibited, for example, by the dynamics of a chemically-active Janus colloid. We evaluate the standardly-defined spectral density, i.e. the STSD averaged over a statistical ensemble of trajectories in the limit of an infinitely long observation time T, and also go beyond the standard analysis by considering the coefficient of variation γ of the distribution of the STSD. Moreover, we analyse the finite-T behaviour of the STSD and γ, determine the cross-correlations between spatial components of the STSD, and address the effects of translational diffusion on the functional forms of spectral densities. The exact expressions that we obtain unveil many distinctive features of active BM compared to its passive counterpart, which allow to distinguish between these two classes based solely on the spectral content of individual trajectories.

The Mouse Action Recognition System (MARS) software pipeline for automated analysis of social behaviors in mice.

The study of naturalistic social behavior requires quantification of animals' interactions. This is generally done through manual annotation-a highly time-consuming and tedious process. Recent advances in computer vision enable tracking the pose (posture) of freely behaving animals. However, automatically and accurately classifying complex social behaviors remains technically challenging. We introduce the Mouse Action Recognition System (MARS), an automated pipeline for pose estimation and behavior quantification in pairs of freely interacting mice. We compare MARS's annotations to human annotations and find that MARS's pose estimation and behavior classification achieve human-level performance. We also release the pose and annotation datasets used to train MARS to serve as community benchmarks and resources. Finally, we introduce the Behavior Ensemble and Neural Trajectory Observatory (BENTO), a graphical user interface for analysis of multimodal neuroscience datasets. Together, MARS and BENTO provide an end-to-end pipeline for behavior data extraction and analysis in a package that is user-friendly and easily modifiable.

Efficient recurrent neural network methods for anomalously diffusing single particle short and noisy trajectories

Anomalous diffusion occurs at very different scales in nature, from atomic systems to motions in cell organelles, biological tissues or ecology, and also in artificial materials, such as cement. Being able to accurately measure the anomalous exponent associated to a given particle trajectory, thus determining whether the particle subdiffuses, superdiffuses or performs normal diffusion, is of key importance to understand the diffusion process. Also it is often important to trustingly identify the model behind the trajectory, as it this gives a large amount of information on the system dynamics. Both aspects are particularly difficult when the input data are short and noisy trajectories. It is even more difficult if one cannot guarantee that the trajectories output in experiments are homogeneous, hindering the statistical methods based on ensembles of trajectories. We present a data-driven method able to infer the anomalous exponent and to identify the type of anomalous diffusion process behind single, noisy and short trajectories, with good accuracy. This model was used in our participation in the anomalous diffusion (AnDi) challenge. A combination of convolutional and recurrent neural networks was used to achieve state-of-the-art results when compared to methods participating in the AnDi challenge, ranking top 4 in both classification and diffusion exponent regression.

A gentle introduction to the non-equilibrium physics of trajectories: Theory, algorithms, and biomolecular applications

Despite the importance of non-equilibrium statistical mechanics in modern physics and related fields, the topic is often omitted from undergraduate and core-graduate curricula. Key aspects of non-equilibrium physics, however, can be understood with a minimum of formalism based on a rigorous trajectory picture. The fundamental object is the ensemble of trajectories, a set of independent time-evolving systems, which easily can be visualized or simulated (e.g., for protein folding) and which can be analyzed rigorously in analogy to an ensemble of static system configurations. The trajectory picture provides a straightforward basis for understanding first-passage times, "mechanisms" in complex systems, and fundamental constraints on the apparent reversibility of complex processes. Trajectories make concrete the physics underlying the diffusion and Fokker-Planck partial differential equations. Last but not least, trajectory ensembles underpin some of the most important algorithms that have provided significant advances in biomolecular studies of protein conformational and binding processes.

Entropic Path Sampling: Computational Protocol to Evaluate Entropic Profile along a Reaction Path.

Fleeting intermediates constitute dynamically stepwise mechanisms. They have been characterized in molecular dynamics trajectories, but whether these intermediates form a free energy minimum to become entropic intermediates remains elusively defined. We developed a computational protocol known as entropic path sampling to evaluate the entropic variation of reacting species along a reaction path based on an ensemble of trajectories. Using cyclopentadiene dimerization as a model reaction, we observed an entropy maximum along the reaction path which originates from an enhanced conformational flexibility as the reacting species enter into a flat energy region. As the reacting species further approach product formation, unfavorable entropic restriction fails to offset the potential energy drop, resulting in no free energy minimum along the post-TS pathway. Our results show that cyclopentadiene dimerization involves an entropy maximum that leads to dynamic intermediates with elongated lifetimes, but the reaction does not involve entropic intermediates.

A Lagrangian Perspective on Stable Water Isotopes During the West African Monsoon

AbstractWe present a Lagrangian framework for identifying mechanisms that control the isotopic composition of mid‐tropospheric water vapor in the Sahel region during the West African Monsoon 2016. In this region mixing between contrasting air masses, strong convective activity, as well as surface and rain evaporation lead to high variability in the distribution of stable water isotopologues. Using backward trajectories based on high‐resolution isotope‐enabled model data, we obtain information not only about the source regions of Sahelian air masses, but also about the evolution of O and its isotopologue HDO (expressed as D) along the pathways of individual air parcels. We sort the full trajectory ensemble into groups with similar transport pathways and hydro‐meteorological properties, such as precipitation and relative humidity, and investigate the evolution of the corresponding paired {O, } distributions. The use of idealized process curves in the {O, } phase space allows us to attribute isotopic changes to contributions from (a) air mass mixing, (b) Rayleigh condensation during convection, and (c) microphysical processes depleting the vapor beyond the Rayleigh prediction, i.e., partial rain evaporation in unsaturated and isotopic equilibration in saturated conditions. Different combinations of these processes along the trajectory ensembles are found to determine the final isotopic composition in the Sahelian troposphere during the monsoon. The presented Lagrangian framework is a powerful tool for interpreting tropospheric water vapor distributions. In the future, it will be applied to satellite observations of {O, } over Africa and other regions in order to better quantify characteristics of the hydrological cycle.

Reinforcement learning of rare diffusive dynamics.

We present a method to probe rare molecular dynamics trajectories directly using reinforcement learning. We consider trajectories that are conditioned to transition between regions of configuration space in finite time, such as those relevant in the study of reactive events, and trajectories exhibiting rare fluctuations of time-integrated quantities in the long time limit, such as those relevant in the calculation of large deviation functions. In both cases, reinforcement learning techniques are used to optimize an added force that minimizes the Kullback-Leibler divergence between the conditioned trajectory ensemble and a driven one. Under the optimized added force, the system evolves the rare fluctuation as a typical one, affording a variational estimate of its likelihood in the original trajectory ensemble. Low variance gradients employing value functions are proposed to increase the convergence of the optimal force. The method we develop employing these gradients leads to efficient and accurate estimates of both the optimal force and the likelihood of the rare event for a variety of model systems.

Forecasting Development of COVID-19 Epidemic in European Union Using Entropy-Randomized Approach

The paper is devoted to the forecasting of the COVID-19 epidemic by the novel method of randomized machine learning. This method is based on the idea of estimation of probability distributions of model parameters and noises on real data. Entropy-optimal distributions correspond to the state of maximum uncertainty which allows the resulting forecasts to be used as forecasts of the most "negative" scenario of the process under study. The resulting estimates of parameters and noises, which are probability distributions, must be generated, thus obtaining an ensemble of trajectories that considered to be analyzed by statistical methods. In this work, for the purposes of such an analysis, the mean and median trajectories over the ensemble are calculated, as well as the trajectory corresponding to the mean over distribution values of the model parameters. The proposed approach is used to predict the total number of infected people using a three-parameter logistic growth model. The conducted experiment is based on real COVID-19 epidemic data in several countries of the European Union. The main goal of the experiment is to demonstrate an entropy-randomized approach for predicting the epidemic process based on real data near the peak. The significant uncertainty contained in the available real data is modeled by an additive noise within 30%, which is used both at the training and predicting stages. To tune the hyperparameters of the model, the scheme is used to configure them according to a testing dataset with subsequent retraining of the model. It is shown that with the same datasets, the proposed approach makes it possible to predict the development of the epidemic more efficiently in comparison with the standard approach based on the least-squares method.

Dynamical phase transition in the activity-biased fully-connected random field Ising model: connection with glass-forming systems

We analyse biased ensembles of trajectories for the random-field Ising model on a fully-connected lattice, which is described exactly by mean-field theory. By coupling the activity of the system to a dynamical biasing field, we find a range of dynamical phase transitions, including spontaneous symmetry breaking into ordered states. For weak bias, the phase behaviour is controlled by extrema of the free energy, which may be local minima or saddle points. For large bias, the system tends to states of extremal activity, which may differ strongly from free energy minima. We discuss connections of these results to random first-order transition theory of glasses, which motivates an extension of the analysis to random-field Ising models where the dynamical activity is not symmetric under magnetisation reversal.

Floquet Phases of Matter via Classical Prethermalization.

We demonstrate that the prethermal regime of periodically driven (Floquet), classical many-body systems can host nonequilibrium phases of matter. In particular, we show that there exists an effective Hamiltonian that captures the dynamics of ensembles of classical trajectories despite the breakdown of this description at the single trajectory level. In addition, we prove that the effective Hamiltonian can host emergent symmetries protected by the discrete time-translation symmetry of the drive. The spontaneous breaking of such an emergent symmetry leads to a subharmonic response, characteristic of time crystalline order, that survives to exponentially late times in the frequency of the drive. To this end, we numerically demonstrate the existence of classical prethermal time crystals in systems with different dimensionalities and ranges of interaction. Extensions to higher order and fractional time crystals are also discussed.

Determination of kinetic properties in unimolecular dissociation of complex systems from graph theory based analysis of an ensemble of reactive trajectories.

In this paper, we report how graph theory can be used to analyze an ensemble of independent molecular trajectories, which can react during the simulation time-length, and obtain structural and kinetic information. This method is totally general and here is applied to the prototypical case of gas phase fragmentation of protonated cyclo-di-glycine. This methodology allows us to analyze the whole set of trajectories in an automatic computer-based way without the need of visual inspection but by getting all the needed information. In particular, we not only determine the appearance of different products and intermediates but also characterize the corresponding kinetics. The use of colored graph and canonical labeling allows for the correct characterization of the chemical species involved. In the present case, the simulations consist of an ensemble of unimolecular fragmentation trajectories at constant energy such that from the rate constants at different energies, the threshold energy can also be obtained for both global and specific pathways. This approach allows for the characterization of ion-molecule complexes, likely through a roaming mechanism, by properly taking into account the elusive nature of such species. Finally, it is possible to directly obtain the theoretical mass spectrum of the fragmenting species if the reacting system is an ion as in the specific example.

Nonparametric Analysis of Nonequilibrium Simulations.

We extend the nonparametric framework of reaction coordinate optimization to nonequilibrium ensembles of (short) trajectories. For example, we show how, starting from such an ensemble, one can obtain an equilibrium free-energy profile along the committor, which can be used to determine important properties of the dynamics exactly. A new adaptive sampling approach, the transition-state ensemble enrichment, is suggested, which samples the configuration space by "growing" committor segments toward each other starting from the boundary states. This framework is suggested as a general tool, alternative to the Markov state models, for a rigorous and accurate analysis of simulations of large biomolecular systems, as it has the following attractive properties. It is immune to the curse of dimensionality, does not require system-specific information, can approximate arbitrary reaction coordinates with high accuracy, and has sensitive and rigorous criteria to test optimality and convergence. The approaches are illustrated on a 50-dimensional model system and a realistic protein folding trajectory.

Laser-subcycle control of electronic excitation across system boundaries

We report on the results of a joint experimental and numerical study on the sub-cycle laser field-driven electron dynamics that underlie the population of highly excited electronic states in multiply ionized argon dimers by electron recapture processes. Our experiments using few-cycle laser pulses with a known carrier-envelope phase (CEP) in combination with reaction microscopy reveal a distinct CEP-dependence of the electron emission and recapture process and, furthermore, a small but significant CEP-offset to the scenario in which no excited argon dimers are produced. With the help of classical ensemble trajectory simulations we trace down these different CEP-dependencies to subtle differences in the laser-driven sub-cycle electron trajectory dynamics that involve in both cases the transfer of an electron from one argon ion across the system boundary to the neighboring ion and its transient capture on this ion.

The dynamical significance of valley-ridge inflection points

In this paper we demonstrate that valley-ridge inflection (VRI) points of a potential energy surface (PES) have a dynamical influence on the fate of trajectories of the underlying Hamiltonian system. These points have attracted the attention of chemists in the past decades when studying selectivity problems in organic chemical reactions whose energy landscape exhibits a post-transition-state bifurcation in the region between two sequential saddles without an intervening energy minimum. To address the dynamical significance of valley-ridge inflection points, we construct a symmetric potential energy function that allows us to move the location of the VRI point while keeping the locations and energies of the critical points fixed. In this setup, we carry out a parametric study of the dynamical behavior of ensembles of trajectories in terms of the energy of the chemical system and the position of the VRI point. Our analysis reveals that the location of the VRI point controls the fraction of trajectories that recross the high energy saddle region of the PES without entering either of the potential wells that are separated by the low energy saddle.

Simulation of collision-induced absorption spectra based on classical trajectories and ab initio potential and induced dipole surfaces. II. CO2-Ar rototranslational band including true dimer contribution.

This paper presents further development of the new semi-classical trajectory-based formalism described in Paper I [Chistikov et al., J. Chem. Phys. 151, 194106 (2019)]. We report the results of simulation and analysis of the low-frequency collision-induced absorption (CIA) in CO2-Ar, including its true dimer component. Our consideration relies on the use of ab initio intermolecular potential energy and induced dipole surfaces for CO2-Ar calculated in an assumption of a rigid CO2 structure using the CCSD(T) method. The theory, the details of which are reported in Paper I [Chistikov et al., J. Chem. Phys. 151, 194106 (2019)], permits taking into account the effect of unbound and quasi-bound classical trajectories on the CIA in the range of a rototranslational band. This theory is largely extended by trajectory-based simulation of the true bound dimer absorption in the present paper. The spectra are obtained from a statistical average over a vast ensemble of classical trajectories restricted by properly chosen domains in the phase space. Rigorous classical theory is developed for two low-order spectral moments interpreted as the Boltzmann-weighted average of the respective dipole functions. These spectral moments were then used to check the accuracy of our trajectory-based spectra, for which both spectral moments can be evaluated independently in terms of specific integrals over the trajectory-based calculated spectral profiles. Good agreement between the spectral moments calculated as integrals over the frequency domain or the phase space largely supports the reliability of our simulated CIA spectra, which conform with the available microwave and far-infrared observations.

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes.

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: “Recent Advances In Single-Particle Tracking: Experiment and Analysis” edited by Janusz Szwabiński and Aleksander Weron.