1,470 research outputs found
Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005
A Smoothed Dual Approach for Variational Wasserstein Problems
Variational problems that involve Wasserstein distances have been recently
proposed to summarize and learn from probability measures. Despite being
conceptually simple, such problems are computationally challenging because they
involve minimizing over quantities (Wasserstein distances) that are themselves
hard to compute. We show that the dual formulation of Wasserstein variational
problems introduced recently by Carlier et al. (2014) can be regularized using
an entropic smoothing, which leads to smooth, differentiable, convex
optimization problems that are simpler to implement and numerically more
stable. We illustrate the versatility of this approach by applying it to the
computation of Wasserstein barycenters and gradient flows of spacial
regularization functionals
Colloquium: Trapped ions as quantum bits -- essential numerical tools
Trapped, laser-cooled atoms and ions are quantum systems which can be
experimentally controlled with an as yet unmatched degree of precision. Due to
the control of the motion and the internal degrees of freedom, these quantum
systems can be adequately described by a well known Hamiltonian. In this
colloquium, we present powerful numerical tools for the optimization of the
external control of the motional and internal states of trapped neutral atoms,
explicitly applied to the case of trapped laser-cooled ions in a segmented
ion-trap. We then delve into solving inverse problems, when optimizing trapping
potentials for ions. Our presentation is complemented by a quantum mechanical
treatment of the wavepacket dynamics of a trapped ion. Efficient numerical
solvers for both time-independent and time-dependent problems are provided.
Shaping the motional wavefunctions and optimizing a quantum gate is realized by
the application of quantum optimal control techniques. The numerical methods
presented can also be used to gain an intuitive understanding of quantum
experiments with trapped ions by performing virtual simulated experiments on a
personal computer. Code and executables are supplied as supplementary online
material (http://kilian-singer.de/ent).Comment: accepted for publication in Review of Modern Physics 201
Stochastic Particle Flow for Nonlinear High-Dimensional Filtering Problems
A series of novel filters for probabilistic inference that propose an alternative way of performing Bayesian updates, called particle flow filters, have been attracting recent interest. These filters provide approximate solutions to nonlinear filtering problems. They do so by defining a continuum of densities between the prior probability density and the posterior, i.e. the filtering density. Building on these methods' successes, we propose a novel filter. The new filter aims to address the shortcomings of sequential Monte Carlo methods when applied to important nonlinear high-dimensional filtering problems. The novel filter uses equally weighted samples, each of which is associated with a local solution of the Fokker-Planck equation. This hybrid of Monte Carlo and local parametric approximation gives rise to a global approximation of the filtering density of interest. We show that, when compared with state-of-the-art methods, the Gaussian-mixture implementation of the new filtering technique, which we call Stochastic Particle Flow, has utility in the context of benchmark nonlinear high-dimensional filtering problems. In addition, we extend the original particle flow filters for tackling multi-target multi-sensor tracking problems to enable a comparison with the new filter
Phase-space structure of resonance eigenfunctions for chaotic systems with escape
Physical systems are usually not closed and insight about their internal structure is experimentally derived by scattering. This is efficiently described by resonance eigenfunctions of non-Hermitian quantum systems with a corresponding classical dynamics that allows for the escape of particles. For the phase-space distribution of resonance eigenfunctions in chaotic systems with partial and full escape we obtain a universal description of their semiclassical limit in terms of classical conditional invariant measures with the same decay rate. For partial escape, we introduce a family of conditionally invariant measures with arbitrary decay rates based on the hyperbolic dynamics and the natural measures of forward and backward dynamics. These measures explain the multifractal phase-space structure of resonance eigenfunctions and their dependence on the decay rate. Additionally, for the nontrivial limit of full escape we motivate the hypothesis that resonance eigenfunctions are described by conditionally invariant measures that are uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. Overall we confirm quantum-to-classical correspondence for the phase-space densities, for their fractal dimensions, and by evaluating their Jensen–Shannon distance in a generic chaotic map with partial and full escape, respectively.Typische physikalische Systeme sind nicht geschlossen, sodass ihre innere Struktur mit Hilfe von Streuexperimenten untersucht werden kann. Diese werden mit Hilfe einer nicht-Hermiteschen Quantendynamik und deren Resonanzeigenzuständen beschrieben. Die dabei zugrunde liegende klassische Dynamik berücksichtigt den Verlust von Teilchen. Für die semiklassische Phasenraumverteilung solcher Resonanzeigenzustände in chaotischen Systemen mit partieller und voller Öffnung entwickeln wir eine universelle Beschreibung mittels bedingt invarianter Maße gleicher Zerfallsrate. Für partiellen Zerfall stellen wir eine Familie bedingt invarianter Maße mit beliebiger Zerfallsrate vor, welche auf der hyperbolischen Dynamik und den natürlichen Maßen der vorwärts gerichteten und der invertierten Dynamik aufbauen. Diese Maße erklären die multifraktale Phasenraumstruktur der Resonanzzustände und deren Abhängigkeit von der Zerfallsrate. Darüber hinaus motivieren wir für den nicht trivialen Grenzfall voll geöffneter Systeme die Hypothese, dass Resonanzeigenzustände durch ein bedingt invariantes Maß beschrieben werden, welches gleichverteilt auf solchen Mengen ist, die den gleichen zeitlichen Abstand zum quantenunscharfen chaotischen Sattel haben. Insgesamt bestätigen wir die quantenklassische Korrespondenz für die Phasenraumdichten, deren fraktale Dimensionen und durch Auswertung ihres Jensen–Shannon Abstandes in einer generischen chaotischen Abbildung sowohl für partielle als auch für volle Öffnung
New Directions for Contact Integrators
Contact integrators are a family of geometric numerical schemes which
guarantee the conservation of the contact structure. In this work we review the
construction of both the variational and Hamiltonian versions of these methods.
We illustrate some of the advantages of geometric integration in the
dissipative setting by focusing on models inspired by recent studies in
celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282
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