11,016 research outputs found
Exact statistical properties of the Burgers equation
The one dimensional Burgers equation in the inviscid limit with white noise
initial condition is revisited. The one- and two-point distributions of the
Burgers field as well as the related distributions of shocks are obtained in
closed analytical forms. In particular, the large distance behavior of spatial
correlations of the field is determined. Since higher order distributions
factorize in terms of the one and two points functions, our analysis provides
an explicit and complete statistical description of this problem.Comment: 21 pages, 6 figures include
Symmetry-preserving Observers
This paper presents three non-linear observers on three examples of
engineering interest: a chemical reactor, a non-holonomic car, and an inertial
navigation system. For each example, the design is based on physical
symmetries. This motivates the theoretical development of invariant observers,
i.e, symmetry-preserving observers. We consider an observer to consist in a
copy of the system equation and a correction term, and we give a constructive
method (based on the Cartan moving-frame method) to find all the
symmetry-preserving correction terms. They rely on an invariant frame (a
classical notion) and on an invariant output-error, a less standard notion
precisely defined here. For each example, the convergence analysis relies also
on symmetries consideration with a key use of invariant state-errors. For the
non-holonomic car and the inertial navigation system, the invariant
state-errors are shown to obey an autonomous differential equation independent
of the system trajectory. This allows us to prove convergence, with almost
global stability for the non-holonomic car and with semi-global stability for
the inertial navigation system. Simulations including noise and bias show the
practical interest of such invariant asymptotic observers for the inertial
navigation system.Comment: To be published in IEEE Automatic Contro
The Casimir Effect
After a review of the standard calculation of the Casimir force between two
metallic plates at zero and non-zero temperatures, we present the study of
microscopic models to determine the large-distance asymptotic force in the
high-temperature regime. Casimir's conducting plates are modelized by plasmas
of interacting charges at temperature T. The charges are either classical, or
quantum-mechanical and coupled to a (classical) radiation field. In these
models, the force obtained is twice weaker than that arising from standard
treatments neglecting the microscopic charge fluctutations inside the bodies.
The enforcement of inert boundary conditions on the field in the usual
calculations turns out to be inadequate in this regime.
Other aspects of dispersion forces are also reviewed. The status of
(non-retarded) van der Waals-London forces in a dilute medium of non-zero
temperature and density is investigated. In a proper scaling regime called the
atomic limit (high dilution and low temperature), one is able to give the exact
large-distance atomic correlations up to exponentially small terms as T->0.
Retarded van der Waals forces and forces between dielectric bodies are also
reviewed.
Finally, the Casimir effect in critical phenomena is addressed by considering
the free Bose gas. It is shown that the grand-canonical potential of the gas in
a slab at the critical value of the chemical potential has finite size
corrections of the standard Casimir type. They can be attributed to the
existence of long-range order generated by gapless excitations in the phase
with broken continuous symmetry.Comment: Lecture notes prepared for the proceedings of the 1st Warsaw School
of Statistical Physics, Kazimierz, Poland, June 2005. To appear in Acta
Physica Polonica (2006). 52 pages, 0 figures. Available at
http://th-www.if.uj.edu.pl/acta/vol37/pdf/v37p2503.pd
Quintessence Model Building
A short review of some of the aspects of quintessence model building is
presented. We emphasize the role of tracking models and their possible
supersymmetric origin.Comment: 14 pages, to appear in the proceedings of the sixth workshop of the
American University of Pari
Bose-Einstein Condensation: A MathematicallyUnsolved Problem
The interacting Bose gas with repulsive potential is considered in the polymer representation and some of the yet unsolved mathematical questions for establishing the existence of Bose condensation in this setting are discusse
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