Mathematical Physics
Seminar
February Schedule
Please join us for cookies and coffee in Hill 705
kitchen at 11:40am
Speaker: M. Kiessling, Rutgers University
Date: Thursday, Feb. 1, 2007
Location & Time: Hill 705; 12:00pm
Title: "Restricted micro-canonical ensembles"
Abstract: I will talk about joint work with Carlo Lancellotti on
the mean-field thermodynamic limit of stationary ensemble measures
describing restricted micro-canonical ensembles; restricted in the
sense
that beside energy additional conservation laws hold. Concrete
examples
of physically interesting systems where such ensembles are relevant
are
Coulomb (jellium) systems. Distinct from the unrestricted
micro-canonical
ensembles, the individual members of a restricted ensemble are
generally
not themselves in a state of global thermodynamic equilibrium in the
strict sense "equilibrium." Instead, they are locally in thermodynamic
equilibrium, evolving in time according to the Vlasov dynamics along
an
orbit of maximum entropy configurations, each of which is the solution
of
the familiar nonlinear fixed point equation of Poisson-Boltzmann type.
Speaker: R. Costin, Ohio State University
Date: Thursday, Feb. 8, 2007
Location & Time: Hill 705; 12:00pm
Title: "Results on the classification of ODEs in regions containing
several regular singular points"
Abstract: In a region with one regular singularity (of the linear
part) generic
differential equations are analytically equivalent to their linear
part; if the
region contains several such singularities, this no longer
happens. The talk
will present the obstructions to linearization and the classification
of equations
in such regions.
There will be a brown bag lunch from 1:00-2:00pm.
Speaker: O. Costin, Ohio State University
Date: Thursday, Feb. 8, 7007
Location & Time: Hill 705; 2:00pm
Title: "Borel summability in PDEs and applications"
Abstract:The talk focuses on recent developments in Borel summability
of formal solutions of PDEs, with special emphasis on the
Schrödinger equation in
time periodic forcings and nonlinear equations such as Navier-Stokes.
We study the time-periodically forced nonrelativistic Hydrogen atom
with potential
$-b/r+\Omega(r)\cos\omega t$. $\Omega$ is real-valued, compactly
supported in $r$,
and with no zeros on its support. As $t\to\infty$ the particle
undergoes, with probability
1, transition to the continuum from an arbitrarily localized state
(ionizes).
In nonlinear equations, the methods are used to show existence and
uniqueness of
solutions in relatively general settings, and will be illustrated on a
constructive proof
of short term existence of solutions of Navier-Stokes.
Work in collaboration with J. Lebowitz and S. Tanveer.
Speaker: C. Villani, ENS
Date: Thursday, Feb. 15, 2007
Location & Time: Hill 705; 12:00pm
Title: "Hypocoercive diffusions"
Abstract:In many problems coming from mathematical physics, the
association of
a degenerate diffusion operator with a conservative operator may lead
to dissipation in all variables and convergence to equilibrium.
One can draw an analogy with the well-studied phenomenon
of hypoellipticity in regularity theory, and actually both phenomena
have been studied together. Now a distinctive theory of
"hypocoercivity" is starting to emerge, with already some striking
results, and several challenging open problems.
There will be a brown bag lunch from
1:00-2:00pm.
Speaker: R. Seiringer, Princeton University
Date: Thursday, Feb. 15, 2007
Location & Time: Hill 705; 2:00pm
Title: "Correlation Estimates for Quantum Many-Body Systems at
Positive Temperature"
Abstract: We present a method for obtaining bounds on the expectation
value of certain two-body interaction potentials in a general state on
Fock space in terms of the corresponding expectation value for thermal
equilibrium states of non-interacting systems. The difference can be
dominated by the difference in the free energies. This method can be
viewed as a rigorous form of first order perturbation theory for
many-body systems at positive temperature. One of the key ingredients
is the strong subadditivity of the von-Neumann entropy. Possible
applications include, for instance, dilute Bose gases, or jellium with
Coulomb interactions at high density.
Speaker: A. Giuliani, Princeton University
Date: Thursday, Feb. 22, 2007
Location & Time: Hill 705; 12:00pm
Title:"Spin Models with Long Range Competing
Interactions: Spontaneous Pattern Formation in the Ground States"
Abstract: Thin magnetic films at low temperatures show a
wide range of ordering effects and related pattern-formation
phenomena. The origins of
these phenomena can often be traced to competition between the long
ranged dipole-dipole interaction and short ranged exchange
interactions.
In this talk I will introduce a class of spin models (either of Ising
or Heisenberg type) describing different experimental realizations
of thin magnetic films. I will review the behavior of some of these
models, as expected from MonteCarlo simulations and variational
computations. Then I will discuss some recent rigorous results on the
nature of ground states for these and for closely related models in
one or more dimensions. The talk is based on joint work with Joel
Lebowitz and Elliott Lieb.
There will be a brown bag lunch from 1-2pm
Speaker: M.D. Jara, IMPA
Date: Thursday, Feb. 22, 2007
Location & Time: Hill 705; 2:00pm
Title: "Scaling Limits for Gradient Systems in Random
Environment"
Abstract: It is well known that the hydrodynamic limit
of an interacting particle system satisfying a gradient condition
(such as the zero-range process or the symmetric simple exclusion
process) is given by a non-linear parabolic heat equation, and that
the fluctuations from this limit are given by a generalized
Ornstein-Uhlenbeck process. We prove that in the presence of a random
environment in divergence form, these scaling limits hold for almost
every choice of the random environment, with an homogenized diffusion
coefficient that does not depend on the realization of the random
environment.
Joint work with Patricia Golcalves.