A goal of CHAMPS is to bring together developments in mathematics with problems in chemistry that reveal a new and unique insight. The method of Lagrangian descriptors is an excellent example of such a synergy.

Lagrangian descriptors are a trajectory diagnostic for revealing phase space structures in dynamical systems. The method was originally developed in the context of Lagrangian transport studies in fluid dynamics (Madrid and Mancho 2009) but the wide applicability of the method has recently been recognized in chemistry, see (Craven and Hernandez 2015, Junginger and Hernandez 2016, Feldmaier, Junginger et al. 2017, Junginger, Duvenbeck et al. 2017, Junginger, Main et al. 2017).

The method is very compelling since it is straightforward to implement computationally, the interpretation is evident, and it provides a “high resolution” method for exploring high dimensional phase space with low dimensional slices. It also applies to both Hamiltonian and non-Hamiltonian systems (Lopesino, Balibrea-Iniesta et al. 2017) as well as to systems with arbitrary, even stochastic, time-dependence (Balibrea-Iniesta, Lopesino et al. 2016). Moreover, Lagrangian descriptors can be applied directly to data sets, without the need of an explicit dynamical system (Mendoza, Mancho et al. 2014).

Briefly, Lagrangian descriptors are implemented as follows. Each point in a chosen grid of initial conditions for trajectories in phase space is assigned a value according to the arclength of the trajectory starting at that initial condition after integration for a fixed, finite time, both backward and forward in time (all initial conditions in the grid are integrated for the same time). The idea is that the influence of phase space structures on trajectories will result in differences in arclength of nearby trajectories near a phase space structure. This has been quantified in terms of the notion of “singular structures’ in the Lagrangian descriptor plots, which are easy to recognize visually (Mancho, Wiggins et al. 2013, Lopesino, Balibrea-Iniesta et al. 2017).

Trajectories are the “primitive objects” that are used to explore phase space structure. In fact, phase space structure is “built” from trajectories. For high dimensional phase space this approach is problematic and prone to issues of interpretation since a tightly grouped set of initial conditions may result in trajectories that become “lost” with respect to each other in phase space. The method of Lagrangian descriptors turns this problem on its head by emphasizing the initial conditions of trajectories, rather than the precise location of their futures and pasts, after a specified amount of time.  A low dimensional “slice” of phase space can be selected and sampled with a grid of initial conditions of high resolution. Since the phase space structure is encoded in the initial conditions of the trajectories, no resolution is lost as the trajectories evolve in time.

We remark that the original Lagrangian trajectory diagnostic is the arclength. This has been modified in (Lopesino, Balibrea et al. 2015, Lopesino, Balibrea-Iniesta et al. 2017) where, effectively, a different type of norm is used as a trajectory diagnostic. This has been shown to have many advantages over the arclength. For example, it allows for a rigorous analysis of the notion of “singular structures” in certain cases and the relation of this notion to invariant manifolds. It also allows a natural decomposition of the Lagrangian descriptor in a way that isolates distinct dynamical effects. This was utilized in (Demian and Wiggins 2017) in order to show that a Lagrangian descriptor could be used to detect the Lyapunov periodic orbits in the two degree-of-freedom Henon-Heiles Hamiltonian system.

This figure is from (Mendoza, Mancho et al. 2014) and shows the results of the  use of the Lagrangian descriptor for revealing flow structures in the Gulf Stream using a geophysical  fluid dynamics data set.

The use and further development of Lagrangian descriptors is a topic underlying many of the themes of CHAMPS: from understanding the role of phase space structure in high dimensional systems to dimensional reduction and as well as for machine learning from data sets.

References

Balibrea-Iniesta, F., et al. (2016). “Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems.” International Journal of Bifurcation and Chaos 26(13): 1630036.

Craven, G. T. and R. Hernandez (2015). “Lagrangian descriptors of thermalized transition states on time-varying energy surfaces.” Physical review letters 115(14): 148301.

Demian, A. S. and S. Wiggins (2017). “Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors.” International Journal of Bifurcation and Chaos 27(14): 1750225.

Feldmaier, M., et al. (2017). “Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors.” Chemical Physics Letters 687: 194-199.

Junginger, A., et al. (2017). “Chemical dynamics between wells across a time-dependent barrier: Self-similarity in the Lagrangian descriptor and reactive basins.” The Journal of chemical physics 147(6): 064101.

Junginger, A. and R. Hernandez (2016). “Lagrangian descriptors in dissipative systems.” Physical Chemistry Chemical Physics 18(44): 30282-30287.

Junginger, A., et al. (2017). “Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points.” Physical Review A 95(3): 032130.

Lopesino, C., et al. (2015). “Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps.” Communications in Nonlinear Science and Numerical Simulation 27(1): 40-51.

Lopesino, C., et al. (2017). “A theoretical framework for lagrangian descriptors.” International Journal of Bifurcation and Chaos 27(01): 1730001.

Madrid, J. J. and A. M. Mancho (2009). “Distinguished trajectories in time dependent vector fields.” Chaos: An Interdisciplinary Journal of Nonlinear Science 19(1): 013111.

Mancho, A. M., et al. (2013). “Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems.” Communications in Nonlinear Science and Numerical Simulation 18(12): 3530-3557.

Mendoza, C., et al. (2014). “Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets.” Nonlinear Processes in Geophysics 21(3): 677-689.

 

 

A.S. Demian and S. Wiggins, Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors, International Journal of Bifurcation and Chaos, 27 (4), 1750225 (2017).

http://www.worldscientific.com/doi/abs/10.1142/S021812741750225X

 The purpose of this paper is to apply Lagrangian Descriptors, a concept used to describe phase space structure, to autonomous Hamiltonian systems with two degrees of freedom in order to detect periodic solutions. We propose a method for Hamiltonian systems with saddle-center equilibrium and apply this approach to the classical Hénon–Heiles system. The method was successful in locating the unstable Lyapunov orbits in phase space.

B.K. Carpenter, G.S. Ezra, S. C. Farantos, Z. C. Kramer, and S. Wiggins. Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points, Regular and Chaotic Dynamics, 23(1), 60-79 (2018).

https://link.springer.com/article/10.1134/S1560354718010069

 

In this paper we analyze a two degree of freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the “roaming mechanism” whose reaction dynamics are of current interest in the chemistry community.

B.K. Carpenter, G.S. Ezra,  S. C. Farantos, Z. C. Kramer, and S. Wiggins. Empirical Classification of Trajectory Data: An Opportunity for the Use of Machine Learning in Molecular Dynamics. J. Phys. Chem. B. DOI: 10.1021/acs.jpcb.7b08707

Publication Date (Web): October 2, 2017.
https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.7b08707

This paper uses trajectory data and machine learning approaches to “learn” phase space structures. Classical Hamiltonian trajectories are initiated at random points in phase space on a fixed energy shell of a model two degree of freedom potential, consisting of two interacting minima in an otherwise flat energy plane of infinite extent.  Below the energy of the plane, the dynamics are demonstrably chaotic.  However, most of the work in this paper involves trajectories at a fixed energy that is 1% above that of the plane, in which regime the dynamics exhibit behavior characteristic of chaotic scattering.  The trajectories are analyzed without reference to the potential, as if they had been generated in a typical direct molecular dynamics simulation.  The questions addressed are whether one can recover useful information about the structures controlling the dynamics in phase space from the trajectory data alone, and whether, despite the at least partially chaotic nature of the dynamics, one can make statistically meaningful predictions of trajectory outcomes from initial conditions.  It is found that key unstable periodic orbits, which can be identified on the analytical potential, appear by simple classification of the trajectories, and that the specific roles of these periodic orbits in controlling the dynamics are also readily discerned from the trajectory data alone.  Two different approaches to predicting trajectory outcomes from initial conditions are evaluated, and it is shown that the more successful of them has ~90% success.  The results are compared with those from a simple neural network, which has higher predictive success (97%) but requires the information obtained from the “by-hand” analysis to achieve that level.  Finally, the dynamics, which occur partly on the very flat region of the potential, show characteristics of the much-studied phenomenon called “roaming.” On this potential, it is found that roaming trajectories are effectively “failed” periodic orbits, and that angular momentum can be identified as a key controlling factor, despite the fact that it is not a strictly conserved quantity.  It is also noteworthy that roaming on this potential occurs in the absence of a “roaming saddle,” which has previously been hypothesized to be a necessary feature for roaming to occur.

The Champs (Chemistry and mathematics in phase space) Kick off Meeting took place 15th & 16th January 2018.  This two-day conference was held at The Watershed in Bristol and launched the Chemistry and Mathematics in Phase Space project. 70 people from all over the world attended the event which had a stimulating set of talks from eminent speakers.

EVENT Schedule inc.talks V.3

It brought together an international group of distinguished speakers who gave first class talks to a large audience on a wide range of areas in Mathematics and Chemistry.

The conference dinner was held in the Bristol Museum and Art Gallery

The kick off meeting was the first organized activity of Champs bringing together chemists and mathematicians. The success of this meeting reinforces our optimism that this is an opportune time for such an interdisciplinary collaboration of chemists and mathematicians and we expect that this will be the first of many such successful meetings  of Champs related topics.

 

Watch as the Voronoi screen is carefully fixed in place on the glass atrium at the new home for the School of Mathematics. For more information on the Voronoi screen, and the Fry building please visit: http://www.bristol.ac.uk/maths/fry-bu…