Transitions Through Avoided Crossings: A Complex, Multiscale Problem
Complex, multiscale systems are almost ubiquitous in science, ranging from electrons in atoms and molecules, through soft matter such as blood, inks, and foams, up to stellar clusters. They are often at the interface between applied science and mathematics, and as such are highly interdisciplinary and require a synergistic approach, combining modelling, analysis and numerics. In this talk, I shall briefly discuss some examples of recent progress in the study of such systems, before focussing on nonadiabatic transitions through avoided crossings.
A standard approximation in quantum mechanics is the Born-Oppenheimer (BO) approximation, in which the dynamics of the electrons are decoupled from those of the much heavier nuclei. However, in many cases, such as when the BO surfaces become close but do not cross (an avoided crossing), this approximation breaks down. A paradigmatic example is the photodissociation of diatomic molecules. The aim is to predict the size and shape of a wavepacket transmitted from one BO surface to another, through the avoided crossing.
The numerical complexity arises both from the rapidly oscillating nature of semiclassical wavepackets, and from the exponentially small transitions through avoided crossings. This multiscale nature is governed by a small parameter $\epsilon$ , the square root of the ratio of electron and nuclear masses. Standard numerical schemes utilise the adiabatic representation (which we will discuss), in which the transition probability is order $\epsilon$ near the transition point, but exponentially small (order exp(-1/$\epsilon$)) in the long-time limit. This strongly suggests that this representation is not the optimal one in which to study the problem.
Using the more general superadiabatic representations, and an approximation of the dynamics near the crossing region, we obtain an explicit formula for the transmitted wavepacket in systems with one effective degree of freedom, such as diatomic molecules. I will explain the physical meaning of the formula as well as demonstrating the excellent agreement with high precision abinitio calculations.
Joint work with Volker Betz (Warwick / TU Darmstadt) and Stefan Teufel (Tübingen).
Renormalization and Entropy Methods for a New Stochastic Mode Reduction
We are interested in a low-dimensional representation of dissipative systems, exemplified here by the generalized Kuramoto-Sivashinsky (gKS) equation. The gKS equation is a prototype modelling nonlinear media with energy supply, energy dissipation, and dispersion. Flame propagation, wave fronts in reaction-diffusion systems and a viscous film flowing down an inclined plane represent just a few of the many examples of the wide applicability of the gKS equation. By our new methodology, we approximate nonlinear evolution equations by renormalized, lower dimensional problems. The resulting equations are qualitatively characterized by rigorous error estimates which additionally provide information on how many resolved degrees of freedom are required. This forms the basis for a new stochastic mode reduction strategy guaranteeing optimality in the sense of maximum information entropy appropriately extended for dissipative systems. Herewith, noise is rigorously introduced and gives an analytical explanation for its origin. The proposed stochastic mode reduction strategy allows us to construct reliable, low-dimensional numerical schemes for a large class of nonlinear higher-order partial differential equations by systematically adding noise. It follows that noise is of increasing relevance under decreasing resolved degrees of freedom in computational methods.
(This is joint work with Marc Pradas, Gregorios A. Pavliotis, and Serafim Kalliadasis )