Noise-Induced Stabilization of Planar Flows I

We show that the complex-valued ODE \begin{equation*} \dot z_t = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0, \end{equation*} which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II of this paper extends the main results to the general setting.Comment: Part one of a two part pape

Boundary between the thermal and statistical polarization regimes in a nuclear spin ensemble

As the number of spins in an ensemble is reduced, the statistical uctuations in its polarization eventually exceed the mean thermal polarization. This transition has now been surpassed in a number of recent nuclear magnetic resonance experiments, which achieve nanometer-scale detection volumes. Here, we measure nanometer- scale ensembles of nuclear spins in a KPF6 sample using magnetic resonance force microscopy. In particular, we investigate the transition between regimes dominated by thermal and statistical nuclear polarization. The ratio between the two types of polarization provides a measure of the number of spins in the detected ensemble

Cosmogenic nuclides in cometary materials: Implications for rate of mass loss and exposure history

As planned, the Rosetta mission will return to earth with a 10-kg core and a 1-kg surface sample from a comet. The selection of a comet with low current activity will maximize the chance of obtaining material altered as little as possible. Current temperature and level of activity, however, may not reliably indicate previous values. Fortunately, from measurements of the cosmogenic nuclide contents of cometary material, one may estimate a rate of mass loss in the past and perhaps learn something about the exposure history of the comet. Perhaps the simplest way to estimate the rate of mass loss is to compare the total inventories of several long-lived cosmogenic radionuclides with the values expected on the basis of model calculations. Although model calculations have become steadily more reliable, application to bodies with the composition of comets will require some extension beyond the normal range of use. In particular, the influence of light elements on the secondary particle cascade will need study, in part through laboratory irradiations of volatile-rich materials. In the analysis of cometary data, it would be valuable to test calculations against measurements of short-lived isotopes

Localized excited charge carriers generate ultrafast inhomogeneous strain in the multiferroic BiFeO3_3

We apply ultrafast X-ray diffraction with femtosecond temporal resolution to monitor the lattice dynamics in a thin film of multiferroic BiFeO$_3$ after above-bandgap photoexcitation. The sound-velocity limited evolution of the observed lattice strains indicates a quasi-instantaneous photoinduced stress which decays on a nanosecond time scale. This stress exhibits an inhomogeneous spatial profile evidenced by the broadening of the Bragg peak. These new data require substantial modification of existing models of photogenerated stresses in BiFeO$_3$: the relevant excited charge carriers must remain localized to be consistent with the data