Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium which is an autonomous classical super-Brownian motion. We characterize both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K ^η and mass by K ^−η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.      

The role of energy density and acoustic cavitation in shock wave lithotripsy

Today a high percentage of urinary stones are successfully treated by extracorporeal shockwave lithotripsy (SWL); however, misconceptions regarding fragmentation mechanisms, as well as treatment parameters like dose, applied energy and focal area are still common. A main stone comminution mechanism during SWL is acoustic cavitation. The objective of this study was to analyze the influence of cavitation and energy density on stone fragmentation. A research lithotripter was used to expose a large set of artificial kidney stones to shock waves varying different parameters. Hundreds of pressure records were used to calculate the energy density of the lithotripter at different settings. Results indicate that energy density is a crucial parameter and that better SWL treatment outcomes could be obtained placing the calculus at a prefocal position.     

Information theory, spectral geometry, and quantum gravity

We show that there exists a deep link between the two disciplines of information theory and spectral geometry. This allows us to obtain new results on a well-known quantum gravity motivated natural ultraviolet cutoff which describes an upper bound on the spatial density of information. Concretely, we show that, together with an infrared cutoff, this natural ultraviolet cutoff beautifully reduces the path integral of quantum field theory on curved space to a finite number of ordinary integrations. We then show, in particular, that the subsequent removal of the infrared cutoff is safe.     

Microfluidic mixing via acoustically driven chaotic advection

Mixing presents a notoriously difficult problem in small amounts of fluids. Herein, surface acoustic waves provide a convenient technique to generate time-dependent flow patterns. These flow patterns can be optimized in such a way that advected particles are mixed most efficiently in the fluid within a short time compared to the time pure diffusion would take. Investigations are presented for the mixing efficiency of a flat cylinder that is driven by two surface acoustic waves. The experimental results favorably agree with model calculations of the flow patterns and the advective transport.