Convective Ripening and Initiation of Rainfall

This paper discusses the evolution of the droplet size distribution for a liquid-in-gas aerosol contained in a Rayleigh-B\'enard cell. It introduces a non-collisional model for broadening the droplet size distribution, termed 'convective ripening'. The paper also considers the initiation of rainfall from ice-free cumulus clouds. It is argued that while collisional mechanisms cannot explain the production of rain from clouds with water droplet diameters of $20\ \mu {\rm m}$, the non-collisional convective ripening mechanism gives a much faster route to increasing the size of the small fraction of droplets that grow into raindrops.Comment: 6 pages, no figure

An exact effective Hamiltonian for a perturbed Landau level

Considers the effect of a scalar potential V (x, y) on a Landau level in two dimensions. An exact effective Hamiltonian is derived which describes the effect of the potential on a single Landau level, expressed as a power series in V/Ec, where Ec is the cyclotron energy. The effective Hamiltonian can be represented as a function H (x, p) in a one-dimensional phase space. The function H (x, p) resembles the potential V (x, y): when the area of a flux quantum is much smaller than the square of the characteristic length scale of V, then H approximately=V. Also H (x, p) retains the translational and rotational symmetries of V(x, y) exactly, but reflection symmetries are not retained beyond the lowest order of the perturbation expansion

Non-adiabatic transitions in multi-level systems

In a quantum system with a smoothly and slowly varying Hamiltonian, which approaches a constant operator at times $t\to \pm \infty$, the transition probabilities between adiabatic states are exponentially small. They are characterized by an exponent that depends on a phase integral along a path around a set of branch points connecting the energy level surfaces in complex time. Only certain sequences of branch points contribute. We propose that these sequences are determined by a topological rule involving the Stokes lines attached to the branch points. Our hypothesis is supported by theoretical arguments and results of numerical experiments.Comment: 25 pages RevTeX, 9 figures and 4 tables as Postscipt file

Semilinear Response

We discuss the response of a quantum system to a time-dependent perturbation with spectrum \Phi(\omega). This is characterised by a rate constant D describing the diffusion of occupation probability between levels. We calculate the transition rates by first-order perturbation theory, so that multiplying \Phi(\omega) by a constant \lambda changes the diffusion constant to \lambda D. However, we discuss circumstances where this linearity does notextend to the function space of intensities, so that if intensities \Phi_i(\omega) yield diffusion constants D_i, then the intensity \sum_i \Phi_i(\omega) does not result in a diffusion constant \sum_i D_i. This `semilinear' response can occur in the absorption of radiation by small metal particles.Comment: 7 pages, 1 figur

Narrowly avoided crossings

In order to create a degeneracy in a quantum mechanical system without symmetries one must vary two parameters in the Hamiltonian. When only one parameter, lambda say, is varied, there is only a finite closest approach Delta E of two eigenvalues, and never a crossing. Often the gaps Delta E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as h(cross) to 0: Delta E=Ae-S/h(cross). The author reports the results of numerical calculations on a system with two parameters, epsilon , lambda , which is completely integrable when epsilon =0. It if found that the gaps Delta E obtained by varying lambda decrease exponentially as h(cross) to 0, consistent with the tunnelling conjecture. When epsilon =0, Delta E=0 because the system is completely integrable. As epsilon to 0, the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping h(cross) fixed, the gaps are of size Delta E=O( epsilon nu ), where nu is usually very close to an integer. Theoretical arguments are presented which explain this result