652 research outputs found
Hydrodynamics of the interacting Bose gas in the Quantum Newton Cradle setup
Describing and understanding the motion of quantum gases out of equilibrium
is one of the most important modern challenges for theorists. In the
groundbreaking Quantum Newton Cradle experiment [Kinoshita, Wenger and Weiss,
Nature 440, 900, 2006], quasi-one-dimensional cold atom gases were observed
with unprecedented accuracy, providing impetus for many developments on the
effects of low dimensionality in out-of-equilibrium physics. But it is only
recently that the theory of generalized hydrodynamics has provided the adequate
tools for a numerically efficient description. Using it, we give a complete
numerical study of the time evolution of an ultracold atomic gas in this setup,
in an interacting parameter regime close to that of the original experiment. We
evaluate the full evolving phase-space distribution of particles. We simulate
oscillations due to the harmonic trap, the collision of clouds without
thermalization, and observe a small elongation of the actual oscillation period
and cloud deformations due to many-body dephasing. We also analyze the effects
of weak anharmonicity. In the experiment, measurements are made after release
from the one-dimensional trap. We evaluate the gas density curves after such a
release, characterizing the actual time necessary for reaching the asymptotic
state where the integrable quasi-particle momentum distribution function
emerges.Comment: v1: 7+10 pages, 3+7 figures. v2: references added, pictures with
refined discretization. v3: addition of discussion of integrability breaking
by trap + small improvement
Momentum and energy preserving integrators for nonholonomic dynamics
In this paper, we propose a geometric integrator for nonholonomic mechanical
systems. It can be applied to discrete Lagrangian systems specified through a
discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and
a (generally nonintegrable) distribution in TQ. In the proposed method, a
discretization of the constraints is not required. We show that the method
preserves the discrete nonholonomic momentum map, and also that the
nonholonomic constraints are preserved in average. We study in particular the
case where Q has a Lie group structure and the discrete Lagrangian and/or
nonholonomic constraints have various invariance properties, and show that the
method is also energy-preserving in some important cases.Comment: 18 pages, 6 figures; v2: example and figures added, minor correction
to example 2; v3: added section on nonholonomic Stoermer-Verlet metho
Mechanical Systems: Symmetry and Reduction
Reduction theory is concerned with mechanical systems with symmetries. It constructs a
lower dimensional reduced space in which associated conservation laws are taken out and
symmetries are \factored out" and studies the relation between the dynamics of the given
system with the dynamics on the reduced space. This subject is important in many areas,
such as stability of relative equilibria, geometric phases and integrable systems
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Invariant varieties of periodic points for the discrete Euler top
The behaviour of periodic points of discrete Euler top is studied. We derive invariant varieties of periodic points explicitly. When the top is axially symmetric they are specified by some particular values of the angular velocity along the axis of symmetry, different for each period
Growth and integrability of some birational maps in dimension three
Motivated by the study of the Kahan--Hirota--Kimura discretisation of the
Euler top, we characterise the growth and integrability properties of a
collection of elements in the Cremona group of a complex projective 3-space
using techniques from algebraic geometry. This collection consists of maps
obtained by composing the standard Cremona transformation
with projectivities that permute
the fixed points of and the points over which
performs a divisorial contraction. More specifically, we show that three
behaviour are possible: (A) integrable with quadratic degree growth and two
invariants, (B) periodic with two-periodic degree sequences and more than two
invariants, and (C) non-integrable with submaximal degree growth and one
invariant.Comment: 46 pages, 6 figures, 7 tables, comments are welcom
SDE based regression for random PDEs
A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour
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