Random Walk with Shrinking Steps: First Passage Characteristics

We study the mean first passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like $\lambda^{n}$ with $\lambda\leq1$ . We also present, for pedagogical purposes, a continuum system with a diffusion constant decaying exponentially in continuous time. Qualitatively both systems are alike in their global properties. However, the discrete case shows very rich mathematical structure, depending on the value of the shrinking parameter, such as self-repetitive and fractal-like structure for the first passage characteristics. The results we present show that the most important quantitative behavior of the discrete case is that the support of the distribution function evolves in time in a rather complicated way in contrast to the time independent lattice structure of the ordinary random walker. We also show that there are critical values of $\lambda$ defined by the equation $\lambda^{K}+2\lambda^{P}-2=0$ with $\{K,N\}\in{\mathcal N}$ where the mean first passage time undergo transitions.Comment: Major Re-Editing of the article. Conclusions unaltere

Hamiltonian model of capture into mean motion resonance

Mean motion resonances are a common feature of both our own Solar System and of extrasolar planetary systems. Bodies can be trapped in resonance when their orbital semi-major axes change, for instance when they migrate through a protoplanetary disc. We use a Hamiltonian model to thoroughly investigate the capture behaviour for first and second order resonances. Using this method, all resonances of the same order can be described by one equation, with applications to specific resonances by appropriate scaling. We focus on the limit where one body is a massless test particle and the other a massive planet. We quantify how the the probability of capture into a resonance depends on the relative migration rate of the planet and particle, and the particle's eccentricity. Resonant capture fails for high migration rates, and has decreasing probability for higher eccentricities, although for certain migration rates, capture probability peaks at a finite eccentricity. We also calculate libration amplitudes and the offset of the libration centres for captured particles, and the change in eccentricity if capture does not occur. Libration amplitudes are higher for larger initial eccentricity. The model allows for a complete description of a particle's behaviour as it successively encounters several resonances. The model is applicable to many scenarios, including (i) Planet migration through gas discs trapping other planets or planetesimals in resonances; (ii) Planet migration through a debris disc; (iii) Dust migration through PR drag. Full details can be found in \cite{2010submitted}. (Abridged)Comment: 4 pages, Proceedings of IAUS276 "The Astrophysics of Planetary Systems: Formation, Structure, and Dynamical Evolution

Clinical specialty training in UK undergraduate medical schools: A retrospective observational study

Objectives: To determine if increased exposure to clinical specialties at medical school is associated with increased interest in pursuing that specialty as a career after foundation training. Design: A retrospective observational study. Setting: 31 UK medical schools were asked how much time students spend in each of the clinical specialties. We excluded two schools that were solely Graduate Entry, and two schools were excluded for insufficient information. Main outcome measures: Time spent on clinical placement from UK undergraduate medical schools, and the training destinations of graduates from each school. A general linear model was used to analyse the relationship between the number of weeks spent in a specialty at medical school and the percentage of graduates from that medical school entering each of the Core Training (CT1)/Specialty Training (ST1) specialties directly after Foundation Year 2 (FY2). Results: Students spend a median of 85 weeks in clinical training. This includes a median of 28 weeks on medical firms, 15 weeks in surgical firms, and 8 weeks in general practice (GP). In general, the number of training posts available in a specialty was proportionate to the number of weeks spent in medical school, with some notable exceptions including GP. Importantly, we found that the number of weeks spent in a specialty at medical school did not predict the percentage of graduates of that school training in that specialty at CT1/ST1 level (ß coefficient=0.061, p=0.228). Conclusions: This study found that there was no correlation between the percentage of FY2 doctors appointed directly to a CT1/ST1 specialty and the length of time that they would have spent in those specialties at medical school. This suggests that curriculum adjustments focusing solely on length of time spent in a specialty in medical school would be unlikely to solve recruitment gaps in individual specialties