291 research outputs found
Exponential rate of convergence to equilibrium for a model describing fiber lay-down processes
This paper is devoted to the adaptation of the method developed in [4,3] to a
Fokker-Planck equation for fiber lay-down which has been studied in [1,5].
Exponential convergence towards a unique stationary state is proved in a norm
which is equivalent to a weighted norm. The method is based on a micro /
macro decomposition which is well adapted to the diffusion limit regime.Comment: 8 page
Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt
We show existence and uniqueness of a stationary state for a kinetic
Fokker-Planck equation modelling the fibre lay-down process in the production
of non-woven textiles. Following a micro-macro decomposition, we use
hypocoercivity techniques to show exponential convergence to equilibrium with
an explicit rate assuming the conveyor belt moves slow enough. This work is an
extension of (Dolbeault et al., 2013), where the authors consider the case of a
stationary conveyor belt. Adding the movement of the belt, the global Gibbs
state is not known explicitly. We thus derive a more general hypocoercivity
estimate from which existence, uniqueness and exponential convergence can be
derived. To treat the same class of potentials as in (Dolbeault et al., 2013),
we make use of an additional weight function following the Lyapunov functional
approach in (Kolb et al., 2013)
Hypocoercivity for Kolmogorov backward evolution equations and applications
In this article we extend the modern, powerful and simple abstract Hilbert
space strategy for proving hypocoercivity that has been developed originally by
Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an
exponential decay to equilibrium with explicit computable rate of convergence.
Our extension is now made for studying the long-time behavior of some strongly
continuous semigroup generated by a (degenerate) Kolmogorov backward operator
L. Additionally, we introduce several domain issues into the framework.
Necessary conditions for proving hypocoercivity need then only to be verified
on some fixed operator core of L. Furthermore, the setting is also suitable for
covering existence and construction problems as required in many applications.
The methods are applicable to various, different, Kolmogorov backward evolution
problems. As a main part, we apply the extended framework to the (degenerate)
spherical velocity Langevin equation. The latter can be seen as some kind of an
analogue to the classical Langevin equation in case spherical velocities are
required. This model is of important industrial relevance and describes the
fiber lay-down in the production process of nonwovens. For the construction of
the strongly continuous contraction semigroup we make use of modern
hypoellipticity tools and pertubation theory
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Keller-Segel-Type Models and Kinetic Equations for Interacting Particles: Long-Time Asymptotic Analysis
This thesis consists of three parts: The first and second parts focus on long-time asymptotics of macroscopic and kinetic models respectively, while in the third part we connect these regimes using different scaling approaches.
(1) Keller–Segel-type aggregation-diffusion equations:
We study a Keller–Segel-type model with non-linear power-law diffusion and non-local particle interaction: Does the system admit equilibria? If yes, are they unique? Which solutions converge
to them? Can we determine an explicit rate of convergence? To answer these questions, we make use of the special gradient flow structure of the equation and its associated free energy functional
for which the overall convexity properties are not known. Special cases of this family of models have been investigated in previous works, and this part of the thesis represents a contribution towards
a complete characterisation of the asymptotic behaviour of solutions.
(2) Hypocoercivity techniques for a fibre lay-down model:
We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in non-woven textile production. Further, we prove convergence to equilibrium with an explicit rate. This part of the thesis is an extension of previous work which considered the case of a stationary conveyor belt. Adding the movement of the belt, the global equilibrium state is not known explicitly and a more general hypocoercivity estimate is needed. Although we focus here on a particular application, this approach can be used for any equation with a similar structure as long as it can be understood as a certain perturbation of a system for
which the global Gibbs state is known.
(3) Scaling approaches for collective animal behaviour models:
We study the multi-scale aspects of self-organised biological aggregations using various scaling techniques. Not many previous studies investigate how the dynamics of the initial models are preserved via these scalings. Firstly, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local kinetic 1D and 2D models to simpler models existing in the literature. Secondly, we investigate how some of the kinetic spatio-temporal patterns are preserved via these scalings using asymptotic preserving numerical methods
Hypocoercivity without confinement
In this paper, hypocoercivity methods are applied to linear kinetic equations
with mass conservation and without confinement, in order to prove that the
solutions have an algebraic decay rate in the long-time range, which the same
as the rate of the heat equation. Two alternative approaches are developed: an
analysis based on decoupled Fourier modes and a direct approach where, instead
of the Poincar\'e inequality for the Dirichlet form, Nash's inequality is
employed. The first approach is also used to provide a simple proof of
exponential decay to equilibrium on the flat torus. The results are obtained on
a space with exponential weights and then extended to larger function spaces by
a factorization method. The optimality of the rates is discussed. Algebraic
rates of decay on the whole space are improved when the initial datum has
moment cancellations
Optimal quantum control of atomic wave packets in optical lattices
In this work, I investigate the motional control and the transport of single neutral atoms trapped in an optical conveyor belt. The main goal is to prepare the atoms in the vibrational ground state of the trapping potential with high efficiency and keep the atoms in this state after fast non-adiabatic transport. In this group, the conveyor belt is used in two systems: (i) In an atom-cavity system, the three-dimensional ground state is prepared by means of carrier-free Raman sideband cooling for the first time. (ii) I use one-dimensional microwave sideband cooling in a state-dependent optical lattice and analyze with a new temperature model the influence of the anharmonic shape of the trapping potential. In the next step, I present a numerical simulation of atom transport. Optimal quantum control theory is used to find transport sequences for different durations without heating atoms out of the ground state. The measurements with these new sequences demonstrate that atoms can be transported by a factor two faster, with higher fidelity and robustness against experimental imperfections. Additionally, I analyze the dynamics of atom transport for sequences of multiple transport steps, which are required for quantum walk experiments. A proof-of-principle measurement demonstrates open-loop live feedback optimization of transport sequences with the experiment. This technique can further compensate experimental imperfections that are not taken into account in the numerical calculation. In the last part, I examine the fundamental limit of fast atom transport, the so-called quantum speed limit. It is defined as the minimum time that a quantum state requires to evolve into an orthogonal one. I investigate the dependencies of this boundary on different trap depths and the finite radial temperature
Steady states of an Elo-type rating model for players of varying strength
In this paper we study the long-time behaviour of a kinetic formulation of an
Elo-type rating model for a large number of interacting players with variable
strength. The model results in a non-linear mean-field Fokker-Planck equation
and we show the existence of steady states via a Schauder fixed point argument.
Our proof relies on the study of a related linear equation using hypocoercivity
techniques.Comment: 2 figures, 20 page
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