TITLES
AND ABSTRACTS OF THE TALKS
Stefan
ADAMS
Probabilistic approaches to Bose-Einstein
condensation
Interacting many-particle systems of Bosons have been studied widely in
the physics and mathematics literature. They pose hard mathematical
problems, which
have been attacked by a variety of methods. We outline how a stochastic
analysis of interacting Brownian motions via large deviations
techniques leads to effective descriptions of large systems of Bosons
and how it raises new mathematical questions. The starting point are
systems of Brownian bridges under certain initial distributions, which
are linked via Fenyman-Kac formulae to traces of quantum mechanical
density matrices. We study the large particle limits of random measures
like the mean of occupation or the mean of path measures. We will
outline how the specific symmetry for systems of Bosons can be
implemented in the stochastic model. With this approach we can prove
the Gross-Pitaevskii formula, a variational formula for the ground
state of Bosons. Furthermore, we will discuss an empirical path measure
phase transition and its interpretation as Bose-Einstein condensation.
Walter ASCHBACHER
Landauer-Büttiker formulas in systems
of independent fermions
Using the scattering approach to the construction of non-equilibrium
steady states in the sense of Ruelle, we establish
Landauer-Büttiker type formulas in systems of independent fermions
under general conditions. This is a joint work with V. Jaksic, Y.
Pautrat, and C.-A. Pillet.
Laurent BRUNEAU
Repeated Interaction Systems II: Random
(Part I given by Marco Merkli)
In this second talk on repeated interaction systems, we consider the
case where the various interactions are not identical but taken at
random. This randomness may have various origins: the interaction time,
the elements of the chain, the coupling between the system S and the
various elements. In a first part we show how to extend the approach of
identical interactions to this situation, and reduce the study of the
large time limits to an infinite product of iid random matrices
satisfying two basic properties: for some suitable norm they are
contractions and they have a common invariant vector. We then study the
convergence of such infinite products of random matrices, and show how
they translate into the large time behaviour of repeated interaction
systems (existence of an asymptotic state and its thermodynamical
properties).
Horia CORNEAN
Adiabatically switched-on electrical bias
in continuous systems, and the Landauer-Buttiker formula
This is joint work with P. Duclos, G. Nenciu and R. Purice. Consider a
three dimensional system which looks like a cross-connected pipe
system, i.e. a small sample coupled to a finite number of leads. We
investigate the current which runs through this system, in the linear
response regime, when we adiabatically turn on an electrical bias
between leads. The main technical tool is the use of the finite volume
regularization, which allows us to define the current coming out of a
lead as the time derivative of its charge. We then prove that in
virtually all physically interesting situations, the conductivity
tensor is given by a Landauer-Buttiker type formula. Finally we discuss
the connection with the Jaksic-Pillet-Ruelle approach.
Tony DORLAS
The HSW theorem for a chennal
with long-term memory
Patrik FERRARI
A stochastic growth of an interface on a
plane: fluctuations for the KPZ universality class
For some classes of growth models universal behaviors appear for large
growth time t. We consider a model in the Kardar-Parisi-Zhang
universality class in 1+1 dimensions. For large t, the surface is
described by the Airy_2 process or Airy_1 process depending on classes
of initial conditions. The transition between the two processes has
been recently analyzed too.
Ostap HRYNIV
Birth of the critical droplet and
related topics
Phase transition in various models of statistical mechanics often leads
to phase coexistence. We shall describe typical configurations in the
2D Ising ferromagnet at the edge of the phase coexistence region, and
compare our results with the "birth of the giant component" phenomenon
in the random graphs theory.
(Partially based on a joint work with D.Ioffe and R.Kotecky.)
Marco MERKLI
Repeated
Interaction Systems I: Deterministic
(Part II given by Laurent Bruneau)
The goal of this talk is to explain what repeated interaction systems
are and why they are important, to set up a suitable mathematical
framework, and to present results on the dynamics of such systems. In
this talk we focus on deterministic systems, while a second talk is
devoted to random repeated interaction systems.
A quantum system S interacts successively with elements E of a
chain of identical independent quantum subsystems. Each interaction
lasts for a duration T and is governed by a fixed coupling
between S and E. We show that the system approaches a "repeated
interaction asymptotic state" in the limit of large times. Our approach
is constructive, it allows us to examine the thermodynamic properties
of the asymptotic state, and to analyze it by rigorous perturbation
theory (in the interaction strength). Our method takes advantage of the
structural properties of the systems in question and involves a
generalization of some recent ideas from the analysis of open quantum
systems far from equilibrium.
Bruno NACHTERGAELE
A
multi-dimensional Lieb-Schultz-Mattis theorem
For a large class of finite-range quantum spin models with half-integer
spins, we prove that uniqueness of the ground state implies the
existence of a low-lying excited state. For systems of linear size L,
of arbitrary finite dimension, we obtain an upper bound on the
excitation energy (i.e., the gap above the ground state) of the form C
log(L)/L. This result is a multi-dimensional version of the
Lieb-Schultz-Mattis theorem and provides a rigorous proof of a recent
result by Hastings. We will discuss the main ideas in our proof.
Robert SEIRINGER
The Free Energy of Dilute
Quantum Gases
We discuss bounds on the free energy of homogeneous Bose and Fermi
gases at non-zero temperature. In the dilute regime, the leading order
correction compared to an ideal quantum gas depends on the particle
interaction only through the scattering length. Moreover, in the Bose
case it depends non-trivially on temperature through the critical
density for Bose-Einstein condensation. Some of the key ingredients in
the proof are the use of coherent states to deal with the condensation,
as well as new correlation estimates relying on the monotonicity of the
relative entropy under partial traces.
Valeriy SLASTIKOV
Fast rotating Bose-Einstein condensates:
variational approach.
András SUTO
Remarks on the variational quasiparticle
theory for bosons
Forian THEIL
Towards a Mathematical Justification of
Kinetic Theory
Jakob YNGVASON
Rotating Bose Gases
The talk will give a brief overview of the mathematical physics of
rotating Bose gases and present some recent results on gases in rapid
rotation.
Valentin ZAGREBNOV
About
Variational Principle for a Pair Hamiltonian Boson Model
We give a two parameter variational formula for the grand-canonical
pressure of the Pair Boson Hamiltonian model. By using the
Approximating Hamiltonian Method we provide a rigorous proof of this
variational principle. The Euler-Lagrange equations allows a detailed
study of Bose-Einstein condensation. In particular, this concerns the
sequence of phase transitions in the Pair-Boson (BCS) model and the
conditions for the coexistence of the generalized Bose-condensation and
the condensation of boson pairs.
Miloš ZAHRADNIK
Combinatorial approach to
cluster (Mayer) expansions
I will present a selfcontained introduction to this method, which is
really a "theory of general exponential function of many
variables". Applications range from the elementary expansion of
log(1+x), or more generally of det(J-A), to complicated, possibly
also non absolutely convergent series for logarithms of polynomials of
many variables. These describe partition functions of some spin models
with continuous symmetry (perturbed zero mass gaussians).
Most of the presented material is well known to specialists
and many crucial features of these expansions, like the
marvellous simplifications obtained after the cancellations of
terms indexed by different graphs (especially for the hard repulsion
case), positivity of the final coefficients (if they are written
properly) or K-P type bounds (which I will present as equations) were
stressed recently by other authors. I will mention also
interesting relations to exactly computable models (determinants of
lattice Laplacians, Kramers - Wannier- Onsager solution)
through a suitable, (possibly) new method of resummation of the
cluster expansion terms. The method is especially useful for models
where incompatibility of polymers is determined by the nonempty
intersection of their "supports", and polymers itself are
given as closed paths (possibly with "decorations" influencing the
compatibility) in d-dimensional lattice, d>2.