Feynman formulae and phase space Feynman path integrals for tau-quantization of some L\'evy-Khintchine type Hamilton functions

This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $\tau$) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of $n$-fold iterated integrals when $n$ tends to infinity (such representations are called Feynman formulae). Some of these representations are constructed with the help of another pseudo-differential operators, obtained by the same procedure of quantization (such representations are called Hamiltonian Feynman formulae). Some representations are based on integral operators with elementary kernels (these ones are called Lagrangian Feynman formulae and are suitable for computations). A family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented also as phase space Feynman path integrals with respect to these Feynman pseudomeasures. The obtained Lagrangian Feynman formulae allow to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures

Scaling limit of stochastic dynamics in classical continuous systems

We investigate a scaling limit of gradient stochastic dynamics associated to Gibbs states in classical continuous systems on ${\mathbb R}^d, d \ge 1$. The aim is to derive macroscopic quantities from a given micro- or mesoscopic system. The scaling we consider has been investigated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}, and \cite{GP86}, under the assumption that the underlying potential is in $C^3_0$ and positive. We prove that the Dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the Dirichlet form of a generalized Ornstein--Uhlenbeck process. The proof is based on the analysis and geometry on the configuration space which was developed in \cite{AKR98a}, \cite{AKR98b}, and works for general Gibbs measures of Ruelle type. Hence, the underlying potential may have a singularity at the origin, only has to be bounded from below, and may not be compactly supported. Therefore, singular interactions of physical interest are covered, as e.g. the one given by the Lennard--Jones potential, which is studied in the theory of fluids. Furthermore, using the Lyons--Zheng decomposition we give a simple proof for the tightness of the scaled processes. We also prove that the corresponding generators, however, do not converge in the $L^2$-sense. This settles a conjecture formulated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}

Geometric Langevin equations on submanifolds and applications to the stochastic melt-spinning process of nonwovens and biology

In this article we develop geometric versions of the classical Langevin equation on regular submanifolds in euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Leli\`evre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant absolute value. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jorgensen and Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occuring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of self-propelled interacting particle systems with roosting force are presented and further applications of the geometric Langevin equations are given

Breakdown characteristics of an isolated conducting object in a uniform electric field

A laboratory experiment was conducted to determine the physical processes involved in the electrical breakdown of a particular spark gap arrangement. The gap consists of an isolated conducting ellipsoid located midway between two large flat electrodes. Gradual increase of the applied electric field, E, in the gap produces corona on the ellipsoid tips followed by flashover in a leader-arc sequence. The leader phase consists of the abrupt formation of ionized channels which partially bridge the gap and then decay prior to the arc. Measurements of dE/dt and of current were made, and photographs were taken with an image converter. Experimental parameters are listed

Tagged particle process in continuum with singular interactions

By using Dirichlet form techniques we construct the dynamics of a tagged particle in an infinite particle environment of interacting particles for a large class of interaction potentials. In particular, we can treat interaction potentials having a singularity at the origin, non-trivial negative part and infinite range, as e.g., the Lennard-Jones potential.Comment: 27 pages, proof for conservativity added, tightened presentatio

Interpretation of F-106B in-flight lightning signatures

Various characteristics of the electromagnetic data obtained on a NASA F-106B aircraft during direct lightning strikes are presented. Time scales of interest range from 10 ns to 400 microsecond. The following topics are discussed: (1) Lightning current, I, measured directly versus I obtained from computer integration of measured I-dot; (2) A method of compensation for the low frequency cutoff of the current transformer used to measure I; (3) Properties of fast pulses observed in the lightning time-derivative waveforms; (4) The characteristic D-dot signature of the F-106B aircraft; (5) An RC-discharge interpretation for some lightning waveforms; (6) A method for inferring the locations of lightning channel attachment points on the aircraft by using B-dot data; (7) Simple, approximate relationships between D-dot and I-dot and between B and I; and (8) Estimates of energy, charge, voltage, and resistance for a particular lightning event

Is Cooled Radiofrequency Genicular Nerve Block and Ablation a Viable Option for the Treatment of Knee Osteoarthritis?

Background The purpose of this study was to determine demographic and psychosocial factors that influence the effectiveness of cooled radiofrequency genicular nerve ablation (C-RFA) and block in patients with chronic knee pain secondary to osteoarthritis (OA). Methods A retrospective review was completed including patients with knee OA who underwent genicular nerve ablation or block or both. Patient information collected included opioid use, psychological comorbidities, smoking history, body mass index, and medical comorbidities. Success was defined using the Osteoarthritis Research Society International criterion of greater than or equal to 50% reported pain relief from the procedure. Patients without a diagnosis of knee OA and patients with ipsilateral total knee arthroplasty were excluded. Patient factors were compared between (1) those that did or did not respond to the initial block and (2) those that did or did not respond to C-RFA. Results Of the 176 subjects that underwent genicular nerve block, 31.8% failed to respond to the procedure. Subjects that failed the initial block were significantly more likely to have psychological comorbidities, smoking history, and diabetes. Of the subjects that proceeded to genicular nerve ablation, 53.7% reported less than 50% pain relief, and 46.3% reported pain relief greater than or equal to 50% at the first follow-up visit. While the presence of psychological comorbidities, smoking, and diabetes were associated with first-stage block failures, these patient factors were not associated with second-stage ablation failures. Conclusions C-RFA may be an effective adjunct therapy as part of a multimodal pain regimen; however, individual patient characteristics must be considered

Diffusion approximation for equilibrium Kawasaki dynamics in continuum

A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $\mu$ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, $\phi$, (in particular, admitting a singularity of $\phi$ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of the gradient stochastic dynamics. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential $\phi$ is from $C_{\mathrm b}^3(\mathbb R^d)$ and sufficiently quickly converges to zero at infinity, we conclude the convergence of the processes from a result in [Choi {\it et al.}, J. Math. Phys. 39 (1998) 6509--6536]