An experimental test of exchange degeneracy in the hyperchange exchange reactions π+p→K+Σ+and K−→π−Σ+

For the first time, the line reversed reactions π⁺p→K⁺Σ⁺and K^?p→π^?Σ⁺ have been studied in the same apparatus. We present the differential cross sections and polarizations over a large t range and at two momenta, 7.0 and 10.1 GeV/c. The differential cross sections as a function of t are shown for the first time to cross over. Going from the lower to the higher momentum, the differences in cross section between the two reactions diminish at low |t| by about a factor 2. We find large polarizations of opposite sign for the two reactions. The momentum dependence, presented in the form of α_eff(t) for the t range 0 to ?2 (GeV/c)², is compared with the expectations from the K¹?K⁷ trajectory.     

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Performance bounds on single-particle tracking by fluorescence modulation

We consider fundamental bounds on the performance of single-particle tracking schemes based on non-imaging, fluorescence modulation methods. We calculate the noise density of a linearized position estimate arising from photon-counting statistics and find the optimal estimate of a freely diffusing particle’s position in the presence of this noise. For the experimentally relevant case of a Gaussian laser rapidly translated in a circular pattern, explicit expressions are derived for the noise density. Tracking performance limits are obtained by considering the variance in the estimated position of a Brownian particle with diffusion coefficient D in the presence of a noise density n_m, which we find scales generically as (Dn_m ²)^1/2. For reasonable experimental parameters, a particle with diffusion coefficient D=1 μm²/s cannot be tracked with accuracy better than approximately 100 nm in three dimensions or 80 nm in two dimensions. Using a combination of exact results and numerical simulation, we construct a ‘phase diagram’ for determining parameter regimes in which a particle can be tracked in the presence of measurement noise. PACS 87.64.Tt; 87.64.Ni; 87.15.Vv     

Geometric singular perturbation theory for stochastic differential equations

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers’ time). We obtain exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.