Fabio Armando Tal

Transition state theory: Variational formulation, dynamical corrections, and error estimates

Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate constant. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-separated metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate constants between these states). Exact and approximate criterions for the optimal TST dividing surface with minimum recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error estimates for the numerical procedure to evaluate the transmission coefficient kappaS of the TST dividing surface are given, and it is shown that the relative error on kappaS scales as 1/square root(kappaS) when kappaS is small. This implies that dynamical corrections to the TST rate constant can be computed efficiently if and only if the TST dividing surface has a transmission coefficient kappaS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise kappaS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has maximum kappaS, but this coefficient may still be very small).

Strictly toral dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus

Forcing theory for transverse trajectories of surface homeomorphisms

\mathbb {T}^{2}

Area-preserving irrotational diffeomorphisms of the torus with sublinear diffusion

which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points

Maximizing measures for endomorphisms of the circle

\operatorname {Ine}(f)

Transition state theory and dynamical corrections in ergodic systems

is shown to be a disjoint union of periodic topological disks (“elliptic islands”), while the set of essential points

On non-contractible periodic orbits for surface homeomorphisms

\operatorname {Ess}(f)

Transitivity and rotation sets with nonempty interior for homeomorphisms of the 2-torus

is an essential continuum, with typically rich dynamics (the “chaotic region”). This generalizes and improves a similar description by Jäger. The key result is boundedness of these “elliptic islands”, which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in

On generic rotationless diffeomorphisms of the annulus

\operatorname {Ess}(f)

Ergodicity and Annular Homeomorphisms of the Torus

is as rich as in