Search results for "Fokker–Planck equation"
showing 10 items of 38 documents
Fokker–Planck equation with respect to heat measures on loop groups
2011
Abstract The Dirichlet form on the loop group L e ( G ) with respect to the heat measure defines a Laplacian Δ DM on L e ( G ) . In this note, we will use Wasserstein distance variational method to solve the associated heat equation for a given data of finite entropy.
Tunnel effect and symmetries for Kramers–Fokker–Planck type operators
2011
AbstractWe study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.
Approximate solution of the Fokker-Planck-Kolmogorov equation
2002
The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker-Planck-Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions i…
Complex fractional moments for the characterization of the probabilistic response of non-linear systems subjected to white noises
2019
In this chapter the solution of Fokker-Planck-Kolmogorov type equations is pursued with the aid of Complex Fractional Moments (CFMs). These quantities are the generalization of the well-known integer-order moments and are obtained as Mellin transform of the Probability Density Function (PDF). From this point of view, the PDF can be seen as inverse Mellin transform of the CFMs, and it can be obtained through a limited number of CFMs. These CFMs’ capability allows to solve the Fokker-Planck-Kolmogorov equation governing the evolutionary PDF of non-linear systems forced by white noise with an elegant and efficient strategy. The main difference between this new approach and the other one based …
Fokker Planck equation solved in terms of complex fractional moments
2014
Abstract In this paper the solution of the Fokker Planck (FPK) equation in terms of (complex) fractional moments is presented. It is shown that by using concepts coming from fractional calculus, complex Mellin transform and related ones, the solution of the FPK equation in terms of a finite number of complex moments may be easily found. It is shown that the probability density function (PDF) solution of the FPK equation is restored in the whole domain, including the trend at infinity with the exception of the value of the PDF in zero.
Ideal and physical barrier problems for non-linear systems driven by normal and Poissonian white noise via path integral method
2016
Abstract In this paper, the probability density evolution of Markov processes is analyzed for a class of barrier problems specified in terms of certain boundary conditions. The standard case of computing the probability density of the response is associated with natural boundary conditions, and the first passage problem is associated with absorbing boundaries. In contrast, herein we consider the more general case of partially reflecting boundaries and the effect of these boundaries on the probability density of the response. In fact, both standard cases can be considered special cases of the general problem. We provide solutions by means of the path integral method for half- and single-degr…
An approximate technique for determining in closed-form the response transition probability density function of diverse nonlinear/hysteretic oscillat…
2019
An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokke…
Electron channeling experiments with bent silicon single crystals—a reanalysis based on a modified Fokker-Planck equation
2020
A surprising small dechanneling length was observed at (111) channeling of ultrarelativistic electrons in a 60 $\mu$m thick silicon single crystal with a bending radius of 0.15 m. The experiments were conducted at beam energies between 3.35 and 14 GeV at the Facility for Advanced Accelerator Experimental Tests (FACET at SLAC, USA). It is shown in this paper that the small dechanneling lengths can well be reproduced with a modified Fokker-Planck equation for plane crystals in which a crystal bending has been heuristically introduced. Encouraged by this result experiments have been reconsidered which were performed at the Mainz Microtron MAMI with (110) silicon undulator crystals. The results…
Semiadiabatic High-Field Polarization Response in Ferroelectrics I: Hysteresis and Nonlinear Susceptibility
2004
Ginzburg-Landau theory for ferroelectric phase instability is combined with Langevin, Fokker-Planck and imaginary time Schrodinger equation techniques to model impact of thermal noise in the kinetics of ferroelectric polarization. The proposed real space/real time numerical method allows to efficiently simulating relaxation, dynamic hysteresis and general dielectric response.