ISBN 978-3-86853-996-7

978-3-86853-996-7, Reihe Mathematik

Florian Conrad
Construction and analysis of Langevin dynamics in continuous particle systems

264 Seiten, Dissertation Technische Universität Kaiserslautern (2010), Softcover, A5

Zusammenfassung / Abstract

The Langevin equation is a stochastic differential equation, which describes the evolution of the positions and velocities of finitely of infinitely many stochastically perturbed interacting particles. Results are derived on the existence of martingale solutions to this equation for regular initial distributions, but for a huge class of continuous interaction potentials, allowing a nonintegrable singularity in 0 and forces which are merely weakly differentiable (away from 0). We also obtain results on the long-time behavior of the finite particle dynamics and some conditional results in the infinite dimensional case.

In Part I of the thesis we collect the necessary (essentially well-known) functional analytic framework for describing the correspondence of strongly continuous semigroups in L^p spaces, their generators and Markov processes.

In Part II two results are derived on finite volume (grand) canonical Gibbs measures with periodic boundary conditions: First, a Ruelle bound is shown for these measures for a large class of pair interactions. Second, (based on a work by Georgii) the accumulation points (in the thermodynamic limit) of finite volume canonical Gibbs measures with periodic boundary condition are identified as tempered translation invariant grand canonical Gibbs measures.

Part III starts with the construction of the finite particle Langevin dynamics, based on proving essential m-dissipativity of its non-sectorial, degenerate elliptic generator. Also, some results on the long-time behavior are derived, namely the weak mixing property and a convergence rate for the L^2-convergence of time avarages to equilibrium. We then construct the infinite particle dynamics (having a grand canonical Gibbs measure as initial distribution) as a weak limit of finite particle dynamics living on the d-dimensional torus. Here the Ruelle bound from Part II is important. Moreover, it is explained in how far essential m-dissipativity for the generator, if it could be shown, would allow us to prove the weak mixing property also in the infinite dimensional case. The thesis ends with a result on the tagged particle process corresponding to the infinite particle stochastic gradient dynamics. Using methods developed for the finite particle Langevin dynamics, we prove an essential m-dissipativity result allowing the identification of the environment component of a process consisting of the coupled motion of one single particle and its environment.