708 research outputs found
Noise and dissipation in rigid body motion
Using the rigid body as an example, we illustrate some features of stochastic geometric mechanics. These features include: i) a geometric variational motivation for the noise structure involving Lie-Poisson brackets and momentum maps, ii) stochastic coadjoint motion with double bracket dissipation, iii) the Lie-Poisson Fokker-Planck description and its stationary solutions, iv) random dynamical systems, random attractors and SRB measures connected to statistical physics
A stochastic Monte Carlo approach to model real star cluster evolution, II. Self-consistent models and primordial binaries
The new approach outlined in Paper I (Spurzem \& Giersz 1996) to follow the
individual formation and evolution of binaries in an evolving, equal point-mass
star cluster is extended for the self-consistent treatment of relaxation and
close three- and four-body encounters for many binaries (typically a few
percent of the initial number of stars in the cluster). The distribution of
single stars is treated as a conducting gas sphere with a standard anisotropic
gaseous model. A Monte Carlo technique is used to model the motion of binaries,
their formation and subsequent hardening by close encounters, and their
relaxation (dynamical friction) with single stars and other binaries. The
results are a further approach towards a realistic model of globular clusters
with primordial binaries without using special hardware. We present, as our
main result, the self-consistent evolution of a cluster consisting of 300.000
equal point-mass stars, plus 30.000 equal mass binaries over several hundred
half-mass relaxation times, well into the phase where most of the binaries have
been dissolved and evacuated from the core. In a self-consistent model it is
the first time that such a realistically large number of binaries is evolving
in a cluster with an even ten times larger number of single stars. Due to the
Monte Carlo treatment of the binaries we can at every moment analyze their
external and internal parameters in the cluster as in an N-body simulation.Comment: LaTeX MNRAS Style 21 pages, 34 figures, submitted to MNRAS Nov. 1999,
for preprint, see
ftp://ftp.ari.uni-heidelberg.de/pub/spurzem/warspaper-98.ps.gz for associated
mpeg-files (20 MB and 13 MB, respectively), see
ftp://ftp.ari.uni-heidelberg.de/pub/spurzem/movie1.mpg and
ftp://ftp.ari.uni-heidelberg.de/pub/spurzem/movie2.mp
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
We present approaches for the study of fluid-structure interactions subject
to thermal fluctuations. A mixed mechanical description is utilized combining
Eulerian and Lagrangian reference frames. We establish general conditions for
operators coupling these descriptions. Stochastic driving fields for the
formalism are derived using principles from statistical mechanics. The
stochastic differential equations of the formalism are found to exhibit
significant stiffness in some physical regimes. To cope with this issue, we
derive reduced stochastic differential equations for several physical regimes.
We also present stochastic numerical methods for each regime to approximate the
fluid-structure dynamics and to generate efficiently the required stochastic
driving fields. To validate the methodology in each regime, we perform analysis
of the invariant probability distribution of the stochastic dynamics of the
fluid-structure formalism. We compare this analysis with results from
statistical mechanics. To further demonstrate the applicability of the
methodology, we perform computational studies for spherical particles having
translational and rotational degrees of freedom. We compare these studies with
results from fluid mechanics. The presented approach provides for
fluid-structure systems a set of rather general computational methods for
treating consistently structure mechanics, hydrodynamic coupling, and thermal
fluctuations.Comment: 24 pages, 3 figure
Stochastic Gravity
Gravity is treated as a stochastic phenomenon based on fluctuations of the
metric tensor of general relativity. By using a (3+1) slicing of spacetime, a
Langevin equation for the dynamical conjugate momentum and a Fokker-Planck
equation for its probability distribution are derived. The Raychaudhuri
equation for a congruence of timelike or null geodesics leads to a stochastic
differential equation for the expansion parameter in terms of the
proper time . For sufficiently strong metric fluctuations, it is shown that
caustic singularities in spacetime can be avoided for converging geodesics. The
formalism is applied to the gravitational collapse of a star and the
Friedmann-Robertson-Walker cosmological model. It is found that owing to the
stochastic behavior of the geometry, the singularity in gravitational collapse
and the big-bang have a zero probability of occurring. Moreover, as a star
collapses the probability of a distant observer seeing an infinite red shift at
the Schwarzschild radius of the star is zero. Therefore, there is a vanishing
probability of a Schwarzschild black hole event horizon forming during
gravitational collapse.Comment: Revised version. Eq. (108) has been modified. Additional comments
have been added to text. Revtex 39 page
Single to double mill small noise transition via semi-Lagrangian finite volume methods
We show that double mills are more stable than single mills under stochastic perturbations in swarming dynamic models with basic attraction-repulsion mechanisms. In order to analyse this fact accurately, we will present a numerical technique for solving kinetic mean field equations for swarming dynamics. Numerical solutions of these equations for different sets of parameters will be presented and compared to microscopic and macroscopic results. As a consequence, we numerically observe a phase transition diagram in terms of the stochastic noise going from single to double mill for small stochasticity fading gradually to disordered states when the noise strength gets larger. This bifurcation diagram at the inhomogeneous kinetic level is shown by carefully computing the distribution function in velocity space
Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
- …