arXiv subjects
Publications and source records attributed to R. Friedrich.
In recent years, several fractional generalizations of the usual Kramers-Fokker-Planck equation have been presented. Using an idea of Fogedby [H.C. Fogedby, Phys. Rev. E {\bf 50}, 041103 (1994), we show how these equations are related to Langevin equations via the procedure of subordination.
The maximum likelihood approach is adapted to the problem of estimation of drift and diffusion functions of stochastic processes from measured time series. We reconcile a previously devised iterative procedure [Kleinhans et al., Physics Letters A (346), 2005] and put the application of the method on a firm theoretical basis.
The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.
We obtain the exact solution for the Burgers equation with a time dependent forcing, which depends linearly on the spatial coordinate. For the case of a stochastic time dependence an exact expression for the joint probability distribution for the velocity fields at multiple spatial points is obtained. A connection with stretched vortices in hydrodynamic flows is discussed.
A general method is proposed which allows one to estimate drift and diffusion coefficients of a stochastic process governed by a Langevin equation. It extends a previously devised approach [R. Friedrich et al., Physics Letters A 271, 217 (2000)], which requires sufficiently high sampling rates. The analysis is based on an iterative procedure minimizing the Kullback-Leibler distance between measured and estimated two time joint probability distributions of the process.
This letter reports on a new method of analysing experimentally gained time series with respect to different types of noise involved, namely, we show that it is possible to differentiate between dynamical and measurement noise. This method does not depend on previous knowledge of model equations. For the complicated case of a chaotic dynamics spoiled at the same time by dynamical and measurement noise, we even show how to extract from data the magnitude of both types of noise. As a further result, we present a new criterion to verify the correct embedding for chaotic dynamics with dynamical noise.
We adress the problem of interactions between the longitudinal velocity increment and the energy dissipation rate in fully developed turbulence. The coupling between these two quantities is experimentally investigated by the theory of stochastic Markovian processes. The so-called Markov analysis allows for a precise characterization of the joint statistical properties of velocity increment and energy dissipation. In particular, it is possible to determine the differential equation that governs the evolution along scales of the joint probability density of these two quantities. The properties of this equation provide interesting new insights into the coupling between energy dissipation and velocity incrementas leading to small scale intermittency.
We outline a statistical theory of turbulence based on the Lagrangian formulation of fluid motion. We derive a hierarchy of evolution equations for Lagrangian N-point probability distributions as well as a functional equation for a suitably defined probability functional which is the analog of Hopf's functional equation. Furthermore, we adress the derivation of a generalized Fokker-Plank equation for the joint velocity - position probability density of N fluid particles.
Comment on "Indispensable Finite Time Correlations for Fokker-Planck Equations from Time Series Data"
Traffic flow data collected by an induction loop detector on the highway close to Koeln-Nord are investigated with respect to their dynamics including the stochastic content. In particular we present a new method, with which the flow dynamics can be extracted directly from the measured data. As a result a Langevin equation for the traffic flow is obtained. From the deterministic part of the flow dynamics, stable fixed points are extracted and set into relation with common features of the fundamental diagram.
The proposed universality of small scale turbulence is investigated for a set of measurements in a cryogenic free jet with a variation of the Reynolds number (Re) from 8500 to 10^6. The traditional analysis of the statistics of velocity increments by means of structure functions or probability density functions is replaced by a new method which is based on the theory of stochastic Markovian processes. It gives access to a more complete characterization by means of joint probabilities of finding velocity increments at several scales. Based on this more precise method our results call in question the concept of universality.
We present a stochastic analysis of a data set consisiting of 10^6 quotes of the US Doller - German Mark exchange rate. Evidence is given that the price changes x(tau) upon different delay times tau can be described as a Markov process evolving in tau. Thus, the tau-dependence of the probability density function (pdf) p(x) on the delay time tau can be described by a Fokker-Planck equation, a gerneralized diffusion equation for p(x,tau). This equation is completely determined by two coefficients D_{1}(x,tau) and D_{2}(x,tau) (drift- and diffusion coefficient, respectively). We demonstrate how these coefficients can be estimated directly from the data without using any assumptions or models for the underlying stochastic process. Furthermore, it is shown that the solutions of the resulting Fokker-Planck equation describe the empirical pdfs correctly, including the pronounced tails.
We present a method to determine fixed points in spatiotemporal signals. A 144-dimensioanl simulated signal, similar to a Kueppers-Lortz instability, is analyzed and its fixed points are reconstructed.
The transition to fully developed turbulence of a wake behind a circular cylinder is investigated with respect to its statistics. In particular, we evaluated the probability density functions of velocity increments on different length scales $r$. Evidence is presented that the $r$-dependence of the velocity increments can be taken as Markov processes in the far field, as well as, in the near field of the cylinder wake. With the estimation of the deterministic part of these Markov processes, as a function of the distance from the cylinder, we are able to set the transition to fully developed turbulence in analogy with a phase transition. We propose that the appearing order parameter corresponds to the presence of large scale coherent structures close to the cylinder.
We present a method how to estimate from experimental data of a turbulent velocity field the drift and the diffusion coefficient of a Fokker-Planck equation. It is shown that solutions of this Fokker-Planck equation reproduce with high accuracy the statistics of velocity increments in the inertial range. Using solutions with different initial conditions at large scales we show that they converge. This can be interpreted as a signature of the universality of small scale turbulence in the limit of large inertial ranges.
This paper deals with the analysis of stochastic systems which can be described by a Langevin equation. By the method presented in this paper drift and diffusion terms of the corresponding Fokker-Planck equation can be extracted from the noisy data sets, and deterministic laws and fluctuating forces of the dynamics can be identified. The method is validated by the application to simulated one- and two-dimensional noisy data sets.
The $ρω$--mixing generated by the isospin breaking of the current quark masses $m_{u} \neq m_{d}$ is studied within the bosonized NJL model in the gradient expansion. The resulting effective meson lagrangian naturally incorperates vector meson dominance. By including pion loops an excellent description of both the pion electromagnetic form factor and of the $π^+π^-$ phase shifts in the vector--isovector channel is obtained. The $ρω$--mixing can be treated in the static approximation but is absolutely necessary to reproduce the fine structure of the electromagnetic form factor, while the pion loops are necessary to obtain the correct energy dependence of the phase shifts.
In this letter we reconsider the previously given description of heavy mesons within a bosonized extended NJL model that combines heavy quark and chiral symmetry. In that work the naive gradient expansion of the quark determinant was used, which satisfactorily works in the light sector but does not adequately describe the heavy $(0^+,1^+)$ mesons. By investigating the exact momentum dependence of the quark loop we demonstrate that the naive gradient expansion in the heavy sector is not the right method to treat the unphysical $q\bar{q}$--thresholds which would be absent in confining theories. We propose a modified gradient expansion which adequately extrapolates from the low--momentum region beyond threshold. This expansion gives a satisfactory description even of the $(0^+,1^+)$ heavy mesons whose masses are significantly above threshold.