Statistical model for collisions and recollisions of inertial particles in mixing flows

Finding a quantitative description of the rate of collisions between small particles suspended in mixing flows is a long-standing problem. Here we investigate the validity of a parameterisation of the collision rate for identical particles subject to Stokes force, based on results for relative velocities of heavy particles that were recently obtained within a statistical model for the dynamics of turbulent aerosols. This model represents the turbulent velocity fluctuations by Gaussian random functions. We find that the parameterisation gives quantitatively good results in the limit where the \lq ghost-particle approximation' applies. The collision rate is a sum of two contributions due to \lq caustics' and to \lq clustering'. Within the statistical model we compare the relative importance of these two collision mechanisms. The caustic formation rate is high when the particle inertia becomes large, and we find that caustics dominate the collision rate as soon as they form frequently. We compare the magnitude of the caustic contribution to the collision rate to the formation rate of caustics.Comment: 9 pages, 4 figures, final version as publishe

Semiclassical trace formulae using coherent states

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate-space formulation. Using a coherent-state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum-mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.Comment: 19 pages, 1 figur

Planet Formation by Concurrent Collapse

After reviewing the difficulties faced by the conventional theory of planet formation (based upon the aggregation of microscopic dust particles), we describe an alternative hypothesis. We propose that planets form by gravitational collapse at the same time as the star about which they orbit. This 'concurrent collapse' hypothesis avoids theoretical difficulties associated with the conventional model and suggests satisfying explanations for various poorly understood phenomena. We introduce new explanations for FU Orionis outbursts seen in young stars, the discovery of exoplanets with eccentric orbits and the existence of small rocky objects such as chondrules in the solar system.Comment: 16 pages, no figures, extended and refined version of earlier submissio

Precise asymptotics for a variable-range hopping model

For a system of localised electron states the DC conductivity vanishes at zero temperature, but localised electrons can conduct at finite temperature. Mott gave a theory for the low-temperature conductivity in terms of a variable-range hopping model, which is hard to analyse. Here we give precise asymptotic results for a modified variable-range hopping model proposed by S. Alexander [Phys. Rev. B 26, 2956 (1982)].Comment: 7 pages, 2 figure