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Concept# Fokker–Planck equation

Summary

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.
It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. When applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case w

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This thesis focuses on the physics of suprathermal electrons generated by electron cyclotron (EC) waves in tokamak plasmas, which play an important role in the physics of current drive and energetic particle-driven instabilities. The suprathermal electron dynamics and their effect on the plasma stability have been experimentally studied utilizing high power EC waves, in the TCV tokamak of the Swiss Plasma Center at EPFL, Switzerland. A hard X-ray diagnostic, which measures the bremsstrahlung radiation of the suprathermal electrons in radial and energy spaces, has been mainly used for the analysis, and the measurement has been compared to an estimation made by Fokker-Planck modeling coupled with a hard X-ray synthetic diagnostic. In order to study the response of the suprathermal electrons to ECCD, ECCD modulation discharges have been developed. The time evolution of the hard X-ray profiles has been measured using coherent averaging techniques in order to observe the creation and relaxation of suprathermal electrons. Time-dependent Fokker-Planck modeling coupled with the hard X-ray synthetic diagnostic has been used to compare the experimental and simulation results, with various suprathermal electron transport models. A dependency of the radial transport of suprathermal electrons on the EC wave power has been demonstrated and a possibility of EC wave scattering has been addressed. The effect of the suprathermal electron population on the plasma stability has been studied, and in particular the destabilization and dynamics of the electron fishbone mode. The response of hard X-ray profiles to the internal kink mode has been observed directly by the hard X-ray diagnostic for the first time, at the frequency of the mode. The experimental evidence and a solution of a linear fishbone dispersion relation coupled to the Fokker-Planck modeling demonstrate the role of suprathermal electrons in destabilizing the fishbone mode and in particular the interaction of trapped electrons with the mode. This work provides the framework for a comprehensive understanding of the physics of suprathermal electrons related to ECCD, and explains how ECCD-generated suprathermal electrons behave in real and velocity spaces, and how they interact with and are redistributed by the MHD mode.

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This work is dedicated to the construction of numerical techniques for the models of viscoelastic fluids that result from polymer kinetic theory. Our main contributions are as follows: Inspired by the interpretation of the Oldroyd B model of dilute polymer solutions as a suspension of Hookean dumbbells in a Newtonian solvent, we have constructed new numerical methods for this model that respect some important properties of the underlying differential equations, namely the positive definiteness of the conformation tensor and an energy estimate. These methods have been implemented on the basis of a spectral discretization for simple Couette and Poiseuille planar flows as well as flow past a cylinder in a channel. Numerical experiments confirm the enhanced stability of our approach. Spectral methods have been designed and implemented for the simulation of mesoscopic models of polymeric liquids that do not possess closed-form constitutive equations. The methods are based on the Fokker-Planck equations rather than on the equivalent stochastic differential equations. We have considered the FENE dumbbell model of dilute polymer solutions and the Öttinger reptation model of concentrated polymer solutions. The comparison with stochastic simulation techniques has been performed in the cases of both homogeneous flows and the flow past a cylinder in a channel. Our method turned out to be more efficient in most cases.

The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n - d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n - d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n-d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(-beta H) with respect to the flat measure dqdp in these 2(n - d) dimensional curvilinear phase space coordinates. The existence of (n - d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n - d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.