Rate Theory, Return Jump Catastrophes, and the Center Manifold

Jacucci, Gianni; Toller, M.; DeLorenzi, G; Flynn, C. P.

doi:10.1103/physrevlett.52.295

Cited by 35 publications

(43 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, in condensed matter physics, this mechanism was described as much as 20 years ago, in the context of the development of a rate theory for the migration of atoms in solids. 21 More recently, such saddles have been shown to play a key role in the study of the ''landscape paradigm. '' 22 The landscape paradigm is central to the study of many complex systems, including glasses and biomolecules, but no tools have been introduced to study the complex deterministic dynamics of the resulting Hamiltonian systems.…”

Section: A Saddle Pointsmentioning

confidence: 99%

Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

Waalkens

Burbanks

Wiggins

2004

The Journal of Chemical Physics

112

173

View full text Add to dashboard Cite

The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary ͑finite͒ number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ''surface of no return.'' The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold ͑NHIM͒. A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.

show abstract

Section: A Saddle Pointsmentioning

confidence: 99%

Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

Waalkens

Burbanks

Wiggins

2004

The Journal of Chemical Physics

112

173

View full text Add to dashboard Cite

show abstract

“…This is is not only the fundamental mechanism for the evolution from reactants to products in chemical reaction, but also for "transformations" in general in a large, and diverse, number of applications as e.g. ionisation problems in atomic physics [5], rearrangements of clusters [6], cosmology [7], and solid state and semi-conductor physics [8,9]. Though it had been recognised that it is important to understand the dynamics near saddle-centre-...-centre equilibria it was only recently that new developments in dynamical systems theory offered the theoretical framework and computing power offered the means to study the phase space structure near saddle-centre-...-centres for systems with 3 or more DOF [10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill's problem

Waalkens¹,

Burbanks²,

Wiggins³

2004

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

A computational procedure that allows the detection of a new type of highdimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence a chaotic saddle. NHIMs control the phase space transport across an equilibrium point of saddle-centre-...-centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, "transformation" in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well known model in celestial mechanics and which gained much interest e.g. in the study of the formation of binaries in the Kuiper belt.

show abstract

“…Similarly, the transmission problem (again without resonances) has been studied for an axially symmetric hyperboloidal constriction in 3D by Torres, Pascual and Sáenz [19], and for the asymmetric case by Waalkens [20]. The main purpose of the present paper is to study the quantum transmission and the associated resonances through the 2D and 3D constrictions (2) and (4) in a coherent way using the perspective of transition state theory.…”

Section: The Transmission Problemmentioning

confidence: 96%

“…The region has a ''bottleneck" in the y-z plane with the shape of an ellipse with semimajor axis 1 and semiminor axisc ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 À c 2 p (in scaled coordinates). For the 2D case (c ¼ 0, or equivalently c ¼ 1), the accessible region is the area between the two branches of the hyperbola (2) in the x-y plane.…”

Section: The Transmission Problemmentioning

confidence: 99%

Classical and quantum transport through entropic barriers modeled by hardwall hyperboloidal constrictions

Hales¹,

Waalkens²

2009

Annals of Physics

View full text Add to dashboard Cite

a b s t r a c tWe study the quantum transport through entropic barriers induced by hardwall constrictions of hyperboloidal shape in two and three spatial dimensions. Using the separability of the Schrö-dinger equation and the classical equations of motion for these geometries, we study in detail the quantum transmission probabilities and the associated quantum resonances, and relate them to the classical phase structures which govern the transport through the constrictions. These classical phase structures are compared to the analogous structures which, as has been shown only recently, govern reaction type dynamics in smooth systems. Although the systems studied in this paper are special due their separability they can be taken as a guide to study entropic barriers resulting from constriction geometries that lead to non-separable dynamics.

show abstract

Rate Theory, Return Jump Catastrophes, and the Center Manifold

Cited by 35 publications

References 6 publications

Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

Phase space conduits for reaction in multidimensional systems: HCN isomerization in three dimensions

A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill's problem

Classical and quantum transport through entropic barriers modeled by hardwall hyperboloidal constrictions

Contact Info

Product

Resources

About