Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions
TThe phase space structure and the escape timedynamics in a Van der Waals model forexothermic reactions
Francisco Gonzalez Montoya a and Stephen Wiggins aa School of Mathematics, University of Bristol, Bristol, BS8 1UG,United Kingdom [email protected]@bristol.ac.uk
Abstract
We study the phase space structures that control the transport in aclassical Hamiltonian model for a chemical reaction. This model has beenproposed to study the yield of products in an ultracold exothermic reac-tion [1]. In the considered model, two elements determine the evolution ofthe system: a Van der Waals force and short-range force associated withthe many-body interactions. In the previous work has been used smallrandom periodic changes in the direction of the momentum to simulatethe short-range many-body interactions. In the present work, randomGaussian bumps have been added to the Van der Waals potential energysimulate the short-range effects between the particles in the system. Wecompare both variants of the model and explain their differences similar-ities and differences from a phase space perspective. In order to visualizethe structures that direct the dynamics in the phase space, we constructa natural Lagrangian descriptor for Hamiltonian systems based on theMaupertuis action S = (cid:82) q f q i p · d q . a r X i v : . [ n li n . C D ] J un Introduction
Recent progress in experimental techniques allows to study chemistry in certaincold systems [2–4]. A current experiment of interest studies the products gen-erated by the collision complex between two cold potassium-rubidium dimers,2KRb → [K Rb ] (cid:70) → K + Rb . Two ultra cold dimers KRb meet at 300 nK, interact, and form a collisioncomplex [K Rb ] (cid:70) . The energy of the particles in this collision complex isaround 4000 K. After this stage the final products, K and Rb , are generated.The final energy of this products is expected to be around 14 K [2, 5–9].The considerably large difference of energy between the atoms in the colli-sion complex and the final products is an important property to consider in themodelling of the system. A classical model has been proposed in [1] to calcu-late the lifetime of the collision complex, or equivalently the yield of the finalproduct in this cold chemical reaction. The justification for this model is basedon semiclassical considerations. This model is an option to avoid the directquantum calculations that require a large number of eigenstates to describe thedynamics of this kind of systems with a deep potential well [10].The basic idea in the construction of the model is that the long-range inter-action in the system determines the escape of the particles when the energiesare close to the threshold energy necessary to escape. For energies very close tothe threshold, only the particles with enough momentum in the radial directioncan escape from the potential well, which is referred to as the “cauldron” in [1].The short-range interactions are related to the collisions between molecules inthe collision complex.In the present work, we study the dynamics of this classical model fromthe phase space perspective. The phase space approach has been developedand applied recently to understand better the chemical reaction dynamics indifferent systems [11–16]. In Section 2, we describe the two variants of themodel and their basic properties. Section 3 contains the construction of theLagrangian descriptors based on the Maupertuis action S . In section 4, westudy the phase space of the integrable and the perturbed systems. Also, wecompare the phase space structures involved in the dynamics of the trajectoriesthat escape in the two variant of the model: the perturbed system and thekicked system. This study of the phase space helps us to understand the escapetime of particles from the potential well in the in section 5. Finally, we presentconclusions and remarks. 2 Model
In this section, we study the basic features of the 2 degree of freedom classicalmodel proposed in [1] to estimate the yield of the final product in the chemicalreaction described in section 1. This model is inspired from previous workby Wannier [17] where the ionization of electrons due to the collision betweenelectrons in a helium atom is considered. With this model, Wanier obtained athreshold law for the yield of ionized electrons as a function of the energy. Inboth systems, the justification of the use of a classical approach is based on asemiclassical WKB analysis [1, 17].An important characteristic of both models is that the total force in thesystem has two terms to define the escape process of the particles to infinity. Thefirst force is a long-range interaction that determines the asymptotic motion.The second force has short-range interaction and in combination with the firstforce generates a complicated dynamics in the region close to the origin.In the model for a cold chemical reaction, the system is considered as a two-body problem. The force related to the asymptotic motion is a Van der Waalsforce. The potential energy associated with this force is V ( r ) = − C ( β | r | + α ) , (1)where r = ( x, y ) is the position from the origin, and the numerical value of theconstants in this example are C = 16130 au, β = 2 . α = 110 au,see figure 1. The potential energy V ( r ) is negative and goes to zero when | r | goes to infinity. This potential energy has rotational symmetry and gives riseto integrable dynamics. For negative values of the total energy E , the phasespace is bounded, and the particles are confined. For E > V ( r ). The energy E is in Kelvinunits. The cauldron is very deep, its minimum is around − V ( r ), there exists an effec-tive potential energy V eff ( r ) parametrized by the z component of the angularmomentum, L z . The threshold energy of the escape is determined by the max-imum of the effective potential at the critical radius. Related to this radius3igure 2: Van der Waals effective potential energy V eff ( r ). The maximum of V eff ( r ) determine the radio of the hyperbolic periodic orbit γ . The numericalvalue of the z component of the angular momentum is L z (cid:39)
108 au.exists a circular unstable periodic orbit that projects to a circle in the configu-ration space. This periodic orbit γ is a normally hyperbolic invariant manifold(NHIM), almost any trajectory close to the orbit moves away from the orbit atan exponential rate after some time, only the trajectories in its stable manifoldconverge to the orbit γ . In the following sections, we explain the role of thisfamily of hyperbolic periodic orbits in the dynamics of the system.Around the minimum of the potential V ( r ), the short-range force acts. In [1]there are two proposals for the force to break the rotational symmetry andmimic the many-body interaction. The first proposal consists of adding to thepotential V ( r ) some random Gaussian bumps scattered inside the region witha radius r < V ( r ) = V ( r ) + n (cid:88) i =1 Ae − B | r − r i | , (2)where A = 0 . B = 10 au are the coefficients that define the Gaussianbumps, and r i are the position of their centres. The figure shows a plot of thepotential energy V ( r ) in colour scale.The second variant of the short-range force has been proposed to simplifythe numerical calculations of the trajectories. The basic idea is to generatetrajectories “similar” to the trajectories for the perturbed system without theinclusion of the random bumps in the potential energy that generate instabilitiesin the numerical calculations. The alternative explored in [1] is the use of smallperiodic random changes in the direction of the momentum of the particles.This changes in the momentum are only possible if the particles are in the sameregion where the bumps are in the other variant of the model, r < V ( r ). The values of A =0 .
002 au and B = 10 au has been chosen in this plot so that the randombumps appreciably break the symmetry. For the numerical calculations weused A = 0 . B = 10 au.for the trajectories under the perturbed potential V ( r ). This method is calledrandom momentum kicks.The figure 4 shows the trajectories with the same initial conditions for thethree cases and its corresponding relative changes in the energy. The numericalcalculations of the trajectories are done with a Taylor polynomial integratororder 21 implemented in the language Julia [22–24]. Using this integrator therelative changes of the energies in the three cases have a similar order of mag-nitude. The blue trajectory corresponds to a particle under the influence of theVan der Waals potential V ( r ). This trajectory is bounded and quasiperiodic.The red trajectory corresponds to a particle under the perturbed potential V ( r ).This red trajectory has a more complicated behaviour generated by the Gaus-sian bumps before to escape to the asymptotic region. The orange trajectoryis obtained with the random kicks method, afther some time close to the originthe particle escape to infinity. 5 a) Integrable trajectory. (b) E ∆ E for the integrable trajectory.(c) Nonintegrable trajectory. (d) E ∆ E for the nonintegrable trajectory.(e) Kicked trajectory. (f) E ∆ E for the kicked trajectory. Figure 4: On the left side the trajectories in the configuration space for the threecases considered: integrable, perturbed nonintegrable, and kicked. On the rightside, their corresponding relative changes of the energy E ∆ E . The time t is inatomic units au. Their initial energies angular momentum are equal in the threecases, E = 100 K and L z (cid:39)
108 au. The green dots on the plot of the trajectorycorresponding to the perturbed system indicate the centres of the Gaussianbumps in the potential V ( r ). For these values of E and L z the particle cannot escape in the integrable case, see V eff ( r ) in figure 2. However, the short-range interactions present in the nonintegrable and kicked system change thedynamics and the particle can escape to infinity.6 Lagrangian Descriptors and the Maupertuisaction S S to reveal the phase space structures.The action S is an essential quantity in the study of Hamiltonian systems andplays a fundamental role in semiclassical approximations. In this section, weexplain why is possible to construct a Lagrangian descriptor based on S . First,let us consider the definition of the Lagrangian descriptor proposed in [26] andbasic ideas about the phase space structures that determine the dynamics.The general definition of the Lagrangian descriptor is as follows. Consider asystem of ordinary differential equations d x dt = v ( x ) , x ∈ R n , t ∈ R (3)where v ( x ) ∈ C r ( r ≥
1) in x and continuous in time t . The definition ofLagrangian descriptor depends on the initial condition x = x ( t ), on the timeinterval [ t + τ − , t + τ + ], and takes the form, M ( x , t , τ + , τ − ) = M + ( x , t , τ + ) + M − ( x , t , τ − )= (cid:90) t + τ + t F ( x ( t )) dt + (cid:90) t t + τ − F ( x ( t )) dt, (4)where a F is a positive function on the solutions x ( t ), x ( t ) = x and τ + (cid:62) τ − (cid:54) τ + and τ − can changebetween different initial conditions and allows us to stop the integration once atrajectory leave a specific region in the phase space. In this way it is possibleto reveal only the phase space structures contained in a defined region.The phase space of a 2 degree of freedom Hamiltonian system has 4 dimen-sions. Considering the conservation of energy, it is possible to represent thedynamics of the system in the 3-dimensional constant energy level set. In this3-dimensional manifold, we can visualise the dynamics and identify the essentialphase space structures that direct the dynamics.The periodic orbits are basic objects for understanding the dynamics of thesystem in the constant energy level set. Around a stable periodic orbit, theKAM-tori confine the trajectories in a bounded region defined by the tori. Incontrast, the dynamics in a neighbourhood of an unstable hyperbolic periodicorbit has different behaviour, the trajectories around the orbit diverge fromthe orbit after some time. There are two invariant surfaces under the flowassociated to a hyperbolic periodic orbit, the stable and unstable manifold ofthe hyperbolic periodic orbit. Its stable and unstable manifolds intersect inthe hyperbolic periodic orbit and direct the trajectories in the neighbourhood.The definition of the stable and unstable manifolds W s/u ( γ ) of the hyperbolicperiodic orbit γ is the following, W s/u ( γ ) = { x | x ( t ) → γ, t → ±∞} . (5)The stable manifold W s ( γ ) is the set of trajectories that converge to theperiodic orbit γ as the time t goes to ∞ . The definition of the unstable manifold7 u ( γ ) is similar. The unstable manifold it is the set of trajectories convergingto the periodic orbit as the time t goes to −∞ .In a 2 degree of freedom Hamiltonian system, the invariant manifolds W s/u ( γ )are 2-dimensional surfaces. These surfaces form impenetrable barriers that di-rect the dynamics in the 3-dimensional constant energy level set [27, 28]. An-other important property of the stable and unstable manifolds related to thechaotic dynamics is that, if a stable manifold and an unstable manifold inter-sect transversally at one place, then there are an infinite number of transversalintersections between them. The structure generated by the union of the stableand unstable manifolds is called a tangle and defines a set of tubes that directthe dynamics in the constant energy level set [29, 30]. A remarkable propertyof the dynamics in the constant energy level set is that the trajectories in atube never cross the boundaries of a tube. This fact is a consequence of theuniqueness of the solutions of the ordinary differential equations.The Lagrangian descriptors are appropriate tools to reveal the phase spacestructure, especially, to find stable and unstable manifolds of periodic orbits [31–33]. To understand the basic idea that holds up the detection, let us consider thebehaviour of the trajectories in a neighbourhood of a stable manifold W s ( γ ). Allthe trajectories in W s ( γ ) converge to the periodic orbit γ , and the trajectoriesin a small neighbourhood of W s ( γ ) have similar behaviour just for a finiteinterval of time. After this interval of time, the trajectories move apart fromthe unstable hyperbolic periodic orbit γ following the unstable manifold W u ( γ ).This different behaviour of the trajectories generates the singularities in theLagrangian descriptors and other chaotic indicators, like scattering functions[34–36].The stationary action principle developed by Leibniz, Euler, and Maupertuisestablishes that the action S of a Hamiltonian system defined as S = (cid:90) q f q i p · d q (6)has an extreme value on the trajectory of the system. The quantities p and q are the generalized momenta the generalized coordinates of the system. It ispossible to construct a natural Lagrangian descriptor for Hamiltonian systemsunder the following considerations.Let us consider a system such that the kinetic energy is a quadratic functionof the generalized velocities ˙ q then T = d q dt · M ( q ) · d q dt , (7)where M ( q ) is the mass tensor and it is a function only of the generalizedcoordinates q . For such systems, exist an identity between the kinetic energy,the generalized momenta, and the generalized velocities,2 T = p · ˙ q , (8)provided that the potential energy V ( q ) is not a function of ˙ q . By defining adistance ds in the space of generalized coordinates ds = d q · M ( q ) · d q , (9)8ne recognizes the mass tensor M ( q ) as a metric tensor. The kinetic energy canbe written as T = 12 (cid:18) dsdt (cid:19) (10)or, equivalently, 2 T dt = p · d q = √ T ds. (11)Hence, the action S can be expressed as S = (cid:90) q f q i p · d q = (cid:90) q f q i (cid:112) E − V ( q )) ds = (cid:90) t f t i T dt. (12)Therefore, the quantity p · d q and its integral, the action S , are positive quan-tities along any trajectory in the phase space and can, therefore, be used toconstruct a Lagrangian descriptor to study the phase space for this type ofHamiltonian systems.The Lagrangian descriptor M S based on the action S evaluated at times τ − , τ + and the point x = x ( t ) = ( q , p ) on the trajectory x ( t ) = ( q ( t ) , p ( t ))is defined as M S ( x , t , τ + , τ − ) = S + ( x , t , τ + ) + S − ( x , t , τ − )= (cid:90) q + q p · d q + (cid:90) q q − p · d q = (cid:90) t + τ + t T ( x ( t )) dt + (cid:90) t t + τ − T ( x ( t )) dt. (13)9 Dynamics and phase space
In order to understand the dynamics of the perturbed and kicked systems, itis convenient to begin by analysing the phase space of the integrable system.The phase space structures in the integrable system are easy to visualise andserve as a reference to study the structures in the other two cases. We use theLagrangian descriptor based on the action M S , constructed in the section 3,and the Poincare map as tools to visualise the relevant structures in the phasespace to understand the dynamics. The integrable system has rotational symmetry. Therefore a natural choice ofinitial conditions to analyze the dynamics is a set that considers this symmetry.The figure 5 shows the Lagrangian descriptor M S evaluated in the canonicalconjugate plane y – p y at x = 0 and p x > E >
0, the values of the kinetic energy are smaller inthe asymptotic region than close to the minimum of V ( r ). The kinetic energyof the particles that escape to infinity converges to its minimum possible value,the total energy E , see figure 1. The integral of the kinetic energy with respectto the time is proportional to the action S . Then, for any large enough finiteinterval of time, the Lagrangian descriptor is smaller for unbounded trajecto-ries than for the trapped trajectories. The trapped trajectories form integrableislands around the stable periodic orbit corresponding to the minimum of the V eff ( r ).In the integrable case, the boundary of the stable islands is defined by thestable and unstable manifolds of the hyperbolic periodic orbit γ associatedto the maximum in the effective potential V eff ( r ). The hyperbolic periodicorbit γ corresponds to the point with p y = 0 and the maximal values of y in the green-yellow region. The symmetric point with respect to the p y axiscorresponds to the analogous periodic orbit with − L z , for simplicity we consideronly the orbit γ in the following argumentation. At this point, two lines thatextend to the asymptotic region intersect, see figures 5(d) and 10 correspondingto E = 100 K. Those lines, where the value of the Lagrangian descriptor have anabrupt peak, are the intersections of the stable and unstable manifolds W s/u ( γ )with the set of initial conditions. These invariant manifolds have dimension 2and divide the constant energy level set.10 a) E = 0 . E = 1 K(c) E = 10 K (d) E = 100 K Figure 5: Lagrangian descriptor M S with initial conditions on the plane y – p y at x = 0, and p x > E . The value ofthe integration times are τ + , − τ − = 2 × au. Next, we analyse the phase space of the perturbed system defined by the poten-tial energy V ( r ). To compare the results with the integrable case, it is convenientto consider the same kind of initial conditions. In the figure 6 there are plots ofthe Lagrangian descriptor evaluated in the plane y – p y at x = 0 and p x > V ( r ). However, the blue regions associated with thetrajectories that escape to the asymptotic region are similar in both cases.11 a) E = 0 . E = 1 K(c) E = 10 K (d) E = 100 K Figure 6: Lagrangian descriptor M S with initial conditions on the plane y – p y and p x > E . The value of the integrationtimes are τ + , − τ − = 2 × au.In order to appreciate better the details in the Lagrangian descriptor forthe nointegrable case, the figure 7 shows a magnifications of the region with y > γ , see Lagrangian descriptor plot in figure 8. The orbit γ isthe deformation of the original hyperbolic periodic orbit γ generated by theperturbation.The temporal irregular behaviour of the trajectories around the KAM is-lands is an example of a phenomenon called transient chaos [37–39]. This com-plicated transient behaviour is common in open Hamiltonian systems. Somerecent studies of the phase space structures of open Hamiltonian systems withtwo and three degrees of freedom are in [34–36, 40, 41].12 a) E = 0 . E = 0 . E = 1 K (d) E = 1 K(e) E = 10 K (f) E = 10 K(g) E = 100 K (h) E = 100 K Figure 7: Magnification of the Lagrangian descriptor M S plots for the non-integrable system in the figure 6 and their corresponding Poincare maps. Thevalue of the integration times for the plots are τ + , − τ − = 1 × au. Theplots on the right side show only the intersections of the trajectories with thePoincare plane that remain in the domain after some time. In this manner iseasy to distinguish the regions that escape fast (the external large white region),the transient chaotic sea generated by the homoclinic tangle of γ , (the regionwhere the intersections form a irregular pattern), and the stable KAM islands(the region where the iteration form closed curves).13igure 8: Magnification of the Lagrangian descriptor M S plot for the nonin-tegrable system for E = 100 K in figure 7(g). The periodic orbit γ intersectthe plane in the corner of the blue triangle with low values of M S , close to thepoint (10 , γ in the integrable system. The Gaussian perturbations decay very fast andits contribution to the potential energy V ( r ) is small in the neighbourhood ofthe periodic orbit γ . However, the stable and unstable manifolds of γ intersecttransversally and form a chaotic homoclinic tangle. These manifolds determinethe entry and exit from the region around the KAM islands. The size of theexit lobes is small compared with the transcient chaotic sea around the KAMislands. The value of the integration times for this plot are τ + , − τ − = 2 × au. 14 .3 Comparison between the dynamics of the perturbednonintegrable system and the kicked system In the nonintegrable system, the Gaussian perturbations change the dynamicsaround the origin. The KAM islands are surrounded by the transient chaotic seagenerated by the homoclinic tangle of the periodic orbit γ . To appreciate moredetails about the dynamics around the KAM islands, we consider the Poincaremap of red trajectory in figure 4(c). The Poincare map and the Lagrangiandescriptor M S as a background are in figure 7. The Lagrangian descriptorreveals the complicated structure of the tangle between the stable and unstablemanifolds W s/u ( γ ). The size of the lobes where the trajectories escape to infinityis small compared to the transient chaotic sea generated by the homoclinictangle of the periodic orbit γ . Then, the volume that escapes from the transientchaotic sea is small in each iteration of the Poincare map and the unboundedtrajectories in the transient chaotic sea intersect the Poincare section manytimes before escaping to the asymptotic region.Figure 9: Phase space structure of the nonintegrable system for E = 100 K.The red points are the intersections of the trajectory in the figure 4(c) with thePoincare surface. The corresponding Lagrangian descriptor M S plot for thenonintegrable system is at the background.As is mentioned before in the section 2, the evolution rule of a trajectory inthe kicked system is the combination of evolution under the influence of potentialenergy of the integrable system V ( r ) and time-periodic random changes inthe direction of the momentum if the trajectory is in the region close to theminimum of the potential energy V ( r ), r <
5. To visualize the dynamicsgenerated by the kicks let us consider the Poincare map associated with theorange trajectory in figure 4(e) and the phase space structures in the integrablesystem generated by V ( r ) for the same value of E as a background. Figure10 shows the Poincare map of these orange kicked trajectory, some invariantclosed curves of the integrable system, and the Lagrangian descriptor of theintegrable system. The random change in the momentum direction is in theinterval [ − π/ , π/ E = 100 K in the figure 4(c). Thistrajectory crosses different invariant closed invariant curves corresponding tothe integrable system, the invariant curves are in black. The correspondingLagrangian descriptor plot for the integrable system is at the background.Figure 11: The orange points are the iterations of the Poincare map for anotherthe kicked trajectory with E = 100 K. This trajectory spends more time inregion defined by stable island corresponding to the integrable system that thetrajectory in the figure 10. The corresponding Lagrangian descriptor plot forthe integrable system is at the background.16 Escape times from the cauldron
In systems with unbounded phase space, a relevant quantity to study is thenumber of particles that remain in one particular region of the phase space asa function of time, different examples with mixed face space has been studiedin [42, 43]. In the present work, a natural region to consider is the regioncontained inside the radius of the most external hyperbolic periodic orbit in thephase space.For the hyperbolic periodic orbits and its generalization in more dimensions,the NHIMs, exist a natural surface to study the transport through bottlenecks inphase space. This surface is called the dividing surface associated to the periodicorbit and plays an important role in the transition state theory in phase spaceproposed by Wigner for systems with two dimensions in [44] and extended forsystems with more dimensions in [11]. The algorithm to construct a dividingsurface of periodic orbits is basically the same that for a NHIM. The procedureconsist of three simple steps: • Project the periodic orbit (NHIM) in the configuration space. • For each point r in the projection, construct the circumference (sphere)in momentum plane (space) using the equation (cid:88) i p i m i = E − V ( r ) (14) • Take the union of all these circumferences (spheres) in the phase space toconstruct the dividing surface.The dividing surface associated to a periodic orbit (NHIM) has three im-portant properties in the to study the transport in the phase space and thechemical reaction dynamics: • The periodic orbit (NHIM) and its correspondig orbit (NHIM) with op-posite momentum are contained in their dividing surface. • This two periodic orbits (NHIMs) are the boundaries in the dividing sur-face between the regions where the trajectories enter into the phase spaceregion contained by the dividing surface and trajectories that left the sameregion. • The flux through the dividing surface is minimal. That is, if the dividingsurface of the periodic orbit (NHIM) is deformed, the flux through itincreases.For periodic orbits associated with saddle points in the potential energy, thecorresponding dividing surfaces are spheres. In the present model, the projectionof the hyperbolic periodic orbits γ and γ in the configuration space encircle thepotential well then their corresponding dividing surfaces are torus in the phasespace. Another recent example of system with torus genus 1 and 2 as dividingsurfaces is in [19]. The intersection of the dividing surface with the plane p y – y is two vertical segment lines. One segment intersects the periodic orbit γ andthe other one intersects the periodic orbit with opposite momentum, see figure17. All the trajectories that start in the potential well and escape to infinity needto cross the dividing surface.Let us denote by R the region in the constant energy level set delimitedby the dividing surface. The intersection of the region R with the phane p y – y is the region between the two vertical line segments corresponding to theintersection of the dividing surface with the same plane. The procedure tocalculate the number of particles in this region as a function of time N ( t ) is thefollowing. A random homogeneous distribution of initial conditions with energy E is taken in the region R . Their corresponding trajectories are integrated untilsome maximum time and the number of trajectories that remain in the region R until the time t is recorded. The numerical results for the integrable, perturbednonintegrable, and kicked systems are in figure 12.Figure 12: Number of particles in the region R as a funcion of time N ( t ) fordifferent values of the energy E . The time t is in au. The blue, orange, andred lines correspond to the integrable, nonintegrable and the kicked systemsrespectively.The results show that the behaviour is very similar for the three cases atthe beginning. This similarity is related with the blue region shown on theLagrangian descriptor plots in the figures 5 and 6. For the three systems, thedynamics of the trajectories in the blue regions is very similar, except for thetrajectories in the small lobes in the perturbed nonintegrable system. However,for larger times the differences in the curves for N ( t ) are clear. All the trajec-tories in the kicked system escape from R and go to infinity, then N ( t ) goes tozero in a finite time in this case. For the integrable and nonintegrable systems N ( t ) converge to a constant proportional to the volume of the islands containedin the region R . 18 Conclutions and remarks
In Hamiltonian systems such that the kinetic energy is a quadratic functionof the generalized velocities and potential energy independent of the velocities,it is possible to construct a Lagrangian descriptor M S based on the classicalaction S . The Lagrangian descriptors are useful tools to reveal the structures inphase space that determine the dynamics, like KAM islands and tangles betweenstable and unstable manifolds of hyperbolic periodic orbits.In the symmetric case, the most external periodic orbit γ is associated withthe maximum value of the effective potential V eff ( r ). The hyperbolic periodicorbit γ is deformed into the hyperbolic periodic orbit γ when the potentialenergy lost the symmetry and the system becomes nonintegrable. The periodicorbit γ encircles the potential well. Then, the dividing surface for these twodegrees of freedom systems is a torus. This is a general property of this typeHamiltonian systems close to a system with rotational symmetry.The two variants of the model have some phase space regions where theirdynamics are similar. Then, the escape times of particles that travel only inthese regions are similar in both models as well. However, if the particles travelin the regions where the two models have different rules for the dynamics, theescape time could be very different. Those facts are reflected in the similarbehaviour of the number of particles N ( t ) in the region R for short times andits discrepancy for long times.In the perturbed nonintegrable version of the model, the addition Gaussianbump in the potential energy breaks the rotational symmetry of the system.The dynamics of the particles with initial conditions in the transient chaoticsea around de KAM islands is determined by the homoclinic tangle betweenthe stable and unstable manifolds W s/u ( γ ). In this example, the exit lobes inthe nonintegrable perturbed case are very small compared with the transientchaotic sea around de KAM islands. Then, the escape time for particles withinial conditions on the transient chaotic sea is larger than for the other parti-cles with initial conditions outside the transient chaotic sea. This scenario iscommon in open Hamiltonian systems with 2 degrees of freedom and rotationalsymmetry when a small perturbation breaks the symmetry and the energy is alittle above the threshold energy. The scenario is a consequence of the existenceand persistence of the homoclinic tangles of the hyperbolic periodic orbit γ andKAM stable islands under perturbations.The trajectories in the stable islands remain in the islands all the time forthe integrable and perturbed nonintegrable systems. In the case of the kickedsystem, the momentum kicks make the trajectories jump from one invariantcurve to another one. Then, the particles escape from the region defined by theinvariant tori of the integrable case to infinity after some finite time. Therefore,the dynamics of the two variant of the model have very different behaviour forlong times. We acknowledge the support of EPSRC Grant no. EP/P021123/1. S W ac-knowledges the support of the Office of Naval Research (Grant No. N00014-01-1-0769). 19 eferences [1] Micheline B. Soley and Eric J. Heller. Classical approach to collision com-plexes in ultracold chemical reactions.
Phys. Rev. A , 98:052702, Nov 2018.[2] S. Ospelkaus, K.-K. Ni, D. Wang, M. H. G. de Miranda, B. Neyen-huis, G. Qu´em´ener, P. S. Julienne, J. L. Bohn, D. S. Jin, and J. Ye.Quantum-state controlled chemical reactions of ultracold potassium-rubidium molecules.
Science , 327(5967):853–857, 2010.[3] Roman V. Krems. Molecules near absolute zero and external field con-trol of atomic and molecular dynamics.
International Reviews in PhysicalChemistry , 24(1):99–118, 2005.[4] R. V. Krems. Cold controlled chemistry.
Phys. Chem. Chem. Phys. ,10:4079–4092, 2008.[5] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, B. Neyenhuis,J. J. Zirbel, S. Kotochigova, P. S. Julienne, D. S. Jin, and J. Ye. A highphase-space-density gas of polar molecules.
Science , 322(5899):231–235,2008.[6] Jeremy M. Hutson. Ultracold chemistry.
Science , 327(5967):788–789, 2010.[7] Kk Ni, S Ospelkaus, D Wang, G Qu´em´ener, Brian Neyenhuis, M Miranda,J Bohn, Jun Ye, and D. Jin. Dipolar collisions of polar molecules in thequantum regime.
Nature , 464:1324–8, 04 2010.[8] M. Miranda, A. Chotia, Brian Neyenhuis, D. Wang, G. Quemener, S. Os-pelkaus, J. Bohn, Jun Ye, and D. Jin. Controlling the quantum stereody-namics of ultracold bimolecular reactions.
Nature Physics , 7, 10 2010.[9] Luigi De Marco, Giacomo Valtolina, Kyle Matsuda, William G. Tobias,Jacob P. Covey, and Jun Ye. A degenerate fermi gas of polar molecules.
Science , 363(6429):853–856, 2019.[10] W. E. Bies, L. Kaplan, and E. J. Heller. Scarring effects on tunneling inchaotic double-well potentials.
Phys. Rev. E , 64:016204, Jun 2001.[11] H Waalkens, R Schubert, and S Wiggins. Wigner’s dynamical trasitionstate theory in phase space: classical and quantum.
Nonlinearity , 21:R1–R118, 2008.[12] Hiroshi Teramoto, Mikito Toda, Masahiko Takahashi, Hirohiko Kono, andTamiki Komatsuzaki. Mechanism and experimental observability of globalswitching between reactive and nonreactive coordinates at high total ener-gies.
Phys. Rev. Lett. , 115:093003, Aug 2015.[13] Martin Tsch¨ope, Matthias Feldmaier, J¨org Main, and Rigoberto Hernan-dez. Neural network approach for the dynamics on the normally hyperbolicinvariant manifold of periodically driven systems.
Phys. Rev. E , 101:022219,Feb 2020. 2014] Matthias Feldmaier, Philippe Schraft, Robin Bardakcioglu, Johannes Reiff,Melissa Lober, Martin Tsch¨ope, Andrej Junginger, J¨org Main, ThomasBartsch, and Rigoberto Hernandez. Invariant manifolds and rate con-stants in driven chemical reactions.
The Journal of Physical ChemistryB , 123(9):2070–2086, 2019. PMID: 30730733.[15] M. Agaoglou, V.J. Garc´ıa-Garrido, M. Katsanikas, and S. Wiggins. Thephase space mechanism for selectivity in a symmetric potential energy sur-face with a post-transition-state bifurcation.
Chemical Physics Letters ,754:137610, 2020.[16] V´ıctor J. Garc´ıa-Garrido, Makrina Agaoglou, and Stephen Wiggins. Ex-ploring isomerization dynamics on a potential energy surface with an index-2 saddle using lagrangian descriptors.
Communications in Nonlinear Sci-ence and Numerical Simulation , 89:105331, 2020.[17] Gregory H. Wannier. The threshold law for single ionization of atoms orions by electrons.
Phys. Rev. , 90:817–825, Jun 1953.[18] Elfi Kraka and Dieter Cremer. Computational analysis of the mechanismof chemical reactions in terms of reaction phases: Hidden intermediatesand hidden transition states.
Accounts of chemical research , 43:591–601,03 2010.[19] Francisco Gonzalez Montoya and Stephen Wiggins. Revealing roamingon the double morse potential energy surface with lagrangian descriptors.
Journal of Physics A: Mathematical and Theoretical , 53(23):235702, may2020.[20] Perttu J. J. Luukko, Byron Drury, Anna Klales, Lev Kaplan, Eric J. Heller,and Esa R¨as¨anen. Strong quantum scarring by local impurities.
ScientificReports , 6(37656):6, November 2016.[21] Eric J. Heller.
The Semiclassical Way to Dynamics and Spectroscopy .Princeton University Press, 2018.[22] Jorge A P´erez-Hern´andez and Luis Benet. PerezHz/TaylorIntegration.jl:TaylorIntegration v0.4.1. https://doi.org/10.5281/zenodo.2562352 ,feb 2019.[23] Luis Benet and David Sanders. Taylorseries.jl: Taylor expansions in oneand several variables in julia.
Journal of Open Source Software , 4(36):1043,2019.[24] Luis Benet and David P. Sanders. Juliadiff/taylorseries.jl: Joss paper. https://doi.org/10.5281/zenodo.2601941 , April 2019.[25] JA Jim´enez Madrid and Ana M Mancho. Distinguished trajectories in timedependent vector fields.
Chaos: An Interdisciplinary Journal of NonlinearScience , 19(1):013111, 2009.[26] Carlos Lopesino, Francisco Balibrea-Iniesta, V´ıctor Garc´ıa Garrido,Stephen Wiggins, and A Mancho. A Theoretical Framework for LagrangianDescriptors.
International Journal of Bifurcation and Chaos , 27:1730001,2017. 2127] Z Kov´acs and L Wiesenfeld. Topological aspects of chaotic scattering inhigher dimensions.
Phys. Rev. E , 63(5):56207, apr 2001.[28] S Wiggins, L Wiesenfeld, C Jaff´e, and T Uzer. Impenetrable Barriers inPhase-Space.
Phys. Rev. Lett. , 86(24):5478–5481, jun 2001.[29] AM Ozorio De Almeida, N De Leon, Manish A Mehta, and C Clay Marston.Geometry and dynamics of stable and unstable cylinders in hamiltoniansystems.
Physica D: Nonlinear Phenomena , 46(2):265–285, 1990.[30] N De Leon, Manish A Mehta, and Robert Q Topper. Cylindrical manifoldsin phase space as mediators of chemical reaction dynamics and kinetics. i.theory.
The Journal of chemical physics , 94(12):8310–8328, 1991.[31] Atanasiu Stefan Demian and Stephen Wiggins. Detection of periodic orbitsin hamiltonian systems using lagrangian descriptors.
International Journalof Bifurcation and Chaos , 27(14):1750225, 2017.[32] Shibabrat Naik, V´ıctor J Garc´ıa-Garrido, and Stephen Wiggins. Findingnhim: Identifying high dimensional phase space structures in reaction dy-namics using lagrangian descriptors.
Communications in Nonlinear Scienceand Numerical Simulation , 79:104907, 2019.[33] Shibabrat Naik and Stephen Wiggins. Detecting reactive islands in asystem-bath model of isomerization.
Physical Chemistry Chemical Physics ,2020.[34] F Gonzalez and C Jung. Rainbow singularities in the doubly differentialcross section for scattering off a perturbed magnetic dipole.
Journal ofPhysics A: Mathematical and Theoretical , 45(26):265102, 2012.[35] G´abor Dr´otos, Francisco Gonz´alez Montoya, Christof Jung, and Tam´asT´el. Asymptotic observability of low-dimensional powder chaos in a three-degrees-of-freedom scattering system.
Phys. Rev. E , 90(2):22906, aug 2014.[36] Francisco Gonzalez Montoya, Florentino Borondo, and Christof Jung.Atom scattering off a vibrating surface: An example of chaotic scatter-ing with three degrees of freedom.
Communications in Nonlinear Scienceand Numerical Simulation , 90:105282, 2020.[37] T´amas T´el. The joy of transient chaos.
Chaos: An Interdisciplinary Journalof Nonlinear Science , 25(9):97619, 2015.[38] Ying-Cheng Lai and Tam´as T´el.
Transient Chaos . Springer-Verlag NewYork, 2011.[39] D´aniel J´anosi and Tam´as T´el. Chaos in hamiltonian systems subjected toparameter drift.
Chaos: An Interdisciplinary Journal of Nonlinear Science ,29(12):121105, 2019.[40] M Katsanikas, P A Patsis, and G Contopoulos. The structure and evolutionof confined tori near a Hamiltonian Hopf bifurcation.
International Journalof Bifurcation and Chaos , 21(08):2321–2330, aug 2011.2241] Jes´us M Seoane and Miguel A F Sanju´an. New developments in classicalchaotic scattering.
Reports on Progress in Physics , 76(1):16001, 2013.[42] George Contopoulos.
Order and Chaos in Dynamical Astronomy . Springer-Verlag Berlin Heidelberg, 2002.[43] Roberto Venegeroles. Universality of algebraic laws in hamiltonian systems.
Phys. Rev. Lett. , 102:064101, Feb 2009.[44] E. Wigner. The transition state method.