Ulam’s method is a popular discretization scheme for stochastic operators that involves the construction of a transition probability matrix controlling a Markov chain on a set of cells covering some domain. We consider an application to satellite-tracked undrogued surface-ocean drifting buoy trajectories obtained from the National Oceanic and Atmospheric Administration Global Drifter Program dataset. Motivated by the motion of Sargassum in the tropical Atlantic, we apply Transition Path Theory (TPT) to drifters originating off the west coast of Africa to the Gulf of Mexico. We find that the most common case of a regular covering by equal longitude–latitude side cells can lead to a large instability in the computed transition times as a function of the number of cells used. We propose a different covering based on a clustering of the trajectory data that is stable against the number of cells in the covering. We also propose a generalization of the standard transition time statistic of TPT that can be used to construct a partition of the domain of interest into weakly dynamically connected regions.
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Augmented transition path theory for sequences of events
Transition path theory provides a statistical description of the dynamics of a reaction in terms of local spatial quantities. In its original formulation, it is limited to reactions that consist of trajectories flowing from a reactant set A to a product set B. We extend the basic concepts and principles of transition path theory to reactions in which trajectories exhibit a specified sequence of events and illustrate the utility of this generalization on examples.
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- Award ID(s):
- 2054306
- NSF-PAR ID:
- 10444872
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 157
- Issue:
- 9
- ISSN:
- 0021-9606
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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By analyzing a time-homogeneous Markov chain constructed using trajectories of undrogued drifting buoys from the NOAA Global Drifter Program, we find that probability density can distribute in a manner that resembles very closely the recently observed recurrent belt of high Sargassum concentration in the tropical Atlantic between 5 and 10°N, coined the Great Atlantic Sargassum Belt ( GASB). A spectral analysis of the associated transition matrix further unveils a forward attracting almost-invariant set in the northwestern Gulf of Mexico with a corresponding basin of attraction weakly connected with the Sargasso Sea but including the nutrient-rich regions around the Amazon and Orinoco rivers mouths and also the upwelling system off the northern coast of West Africa. This represents a data-based inference of potential remote sources of Sargassum recurrently invading the Intra-Americas Seas (IAS). By further applying Transition Path Theory (TPT) to the data-derived Markov chain model, two potential pathways for Sargassum into the IAS from the upwelling system off the coast of Africa are revealed. One TPT-inferred pathway takes place along the GASB. The second pathway is more southern and slower, first going through the Gulf of Guinea, then across the tropical Atlantic toward the mouth of the Amazon River, and finally along the northeastern South American margin. The existence of such a southern TPT-inferred pathway may have consequences for bloom stimulation by nutrients from river runoff.more » « less
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Transition path theory computes statistics from ensembles of reactive trajectories. A common strategy for sampling reactive trajectories is to control the branching and pruning of trajectories so as to enhance the sampling of low probability segments. However, it can be challenging to apply transition path theory to data from such methods because determining whether configurations and trajectory segments are part of reactive trajectories requires looking backward and forward in time. Here, we show how this issue can be overcome efficiently by introducing simple data structures. We illustrate the approach in the context of nonequilibrium umbrella sampling, but the strategy is general and can be used to obtain transition path theory statistics from other methods that sample segments of unbiased trajectories.more » « less
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Abstract Dynamic motion often controls selectivity in reactions featuring two consecutive potential‐energy transition states. Here we report density functional theory (DFT)‐based direct dynamics trajectories and machine learning classification analysis for cyclopentadienone dimerization and a N 2 extrusion reaction leading to semibullvalene. These reactions have consecutive transition states, and there is dynamic selectivity that determines which of two possible C‐C bonds is formed after the first transition state. For cyclopentadienone dimerization with a bispericyclic first transition state, machine learning analysis using transition‐state based features provided >90% trajectory classification accuracy, but only using AdaBoost and random forest algorithms. Many other relatively sophisticated machine learning algorithms showed poor accuracy despite the obvious motion responsible for selectivity. Feature importance analysis confirmed that the sigmatropic rearrangement vibrational motion in the bispericyclic transition state provides prediction of which of the second C‐C bonds is dynamically formed. For the reaction leading to semibullvalene, machine learning analysis provides solid accuracy for classifying trajectories and predicting which C‐C bond is formed and which C‐C bond is broken immediately after N 2 ejection. Like the cyclopentadienone dimerization reaction, machine learning feature importance analysis showed that the sigmatropic rearrangement vibrational motion in the N 2 extrusion transition state determines which C‐C bond is formed and which is broken. Surprisingly, machine learning struggles to predict which trajectories undergo a subsequent [3,3] sigmatropic rearrangement process, which isomerizes equivalent forms of semibullvalene.more » « less
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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.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0098587
