Mathematical Physics Seminar
October Schedule
Organizer- Joel L. Lebowitz
email: lebowitz@math.rutgers.edu
- Speaker- D. Ruelle, IHES, France
- Title- Dynamical zeta functions: an introduction
- Time/place- 10/3/2002 11:30am in Hill 705
- Abstract-Zeta functions are generating functions encoding the properties of certain classes of objects. For instance the Riemann zeta function
encodes properties of primes. Dynamical zeta functions encode
properties of periodic orbits for a dynamical system, with a weight.
Because the weights are fairly general functions, the topic has a
more function-theoretic flavor than the study of number-theoretic
zeta functions. The talk will follow a recent article in the
Notices of the AMS (September 2002) in relating DZF with various
other mathematical objects: operators ("transfer operators"),
determinants (Fredholm determinants, kneading determinants), etc.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- E. Van den Eijnden, Courant Institute
Title- Conformational Dynamics in Complex Systems: Theory
and Computational Aspects
Time/place- 10/3/2002 1:30pm in Hill 705
Abstract- Many problems in physics, material sciences, chemistry and biology can
be abstractly formulated as a system that navigates over a complex
energy landscape of high or infinite dimensions. Well-known examples
include phase transitions of condensed matter, conformational changes
of biopolymers, and chemical reactions. The state of these systems is
confined for long periods of time in metastable regions in
configuration space and only rarely switches from one region to
another. The separation of time scale is a consequence of the
disparity between the effective thermal energy and typical energy
barrier in these systems, and their dynamics effectively reduces to a
Markov chain on the metastables regions. The analysis and computation
the transition pathways and rates between the metastable states is a
major challenge, especially when the energy landscape exhibits
multiscale features. I will review recent work done by scientists from
several disciplines on probing such energy landscapes. I will then
present a new method, the string method, that has proven to be very
effective for some truly complex systems in material science and
chemistry.
Speaker- G. Gallavotti, Universita di Roma, Rome
Title- Equivalence between reversible and irreversible low
dimensional truncations of the Navier Stokes equation. Numerical results and
conjectures.
Time/place-11:30am 10/10/2002 in Hill 705
Abstract- We perform numerical experiments to study the Lyapunov spectra of
dynamical systems associated with the Navier--Stokes (NS) equation in
two spatial dimensions truncated over the Fourier basis. Recently new
equations, called GNS equations, have been introduced and conjectured
to be equivalent to the NS equations at large Reynolds numbers. The
Lyapunov spectra of the NS and of the corresponding GNS systems
overlap, adding evidence in favor of the conjectured equivalence
already studied and partially extended in previous papers. We make
use of the Lyapunov spectra to study a fluctuation relation which had been
proposed to extend the ``fluctuation theorem'' to strongly dissipative
systems. Preliminary results towards the formulation of a local
version of the fluctuation formula are also presented.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- H. Brezis*, University of Paris/Rutgers University
Title- New Results from the Ginzburg-Landau Model in 3-D(Continuation)
Time/place- 1:30pm 10/10/2002
*NOTE: THIS IS THE SECOND PART OF A TALK
GIVEN ON 9/19/02 BY HAIM BREZIS. THE FIRST PART DEALT MOSTLY WITH
3-DIMENSIONAL CASES WHICH BREZIS WILL SUMMARIZE
Abstract- I will report on a recent joint work with
J. Bourgain and P. Mironescu.
Consider a domain $G$ in $\Bbb R^3$ and a boundary data $g$ in the
Sobolev space $H^{1/2}(\partial G; S^1)$. To every such $g$ we attach
a distribution $T(g)$ of the form $\sum_i(\delta_{P_i} -
\delta_{N_i}), P_i, N_i \in \partial G$ and a number $L(g); T(g)$
describes the location of the singularities of $g$ and $L(g)$ is the
length of a minimal connection connecting the singularities. Such
objects play a fundamental role in the study of the Ginzburg-Landau
energy
E_\varepsilon (u) = \frac{1}{2} \int\limits_G |\nabla u|^2 +
\frac{1}{4\varepsilon^2} \int\limits_G (|u|^2 - 1)^2
as $\varepsilon \to 0$, where $u$: $G \to \Bbb C$ and $u=g$ on
$\partial G$. For example, the minimum energy is of the order of $\pi
L(g)\log (1/\varepsilon)$ and minimizers concentrate along vortex
lines in $G$ emanating from $P_i$'s and terminating at the $N_i$'s.
Speaker- F. Rezakhanlou, UC Berkeley
Title- Boltzmann-Grad Limit for the Stochstic Hard Sphere Model
Time/place-11:30am 10/17/2002
Abstract-A long-standing open problem in statistical
mechanics is the
derivation of the Boltzmann equation from the hard sphere model.
In the hard sphere model, one starts with $N$ spheres of diameter
$\epsilon$ that travel according to their velocities and collide
elastically. In a Boltzmann-Grad limit, we send $N\to\infty$, $\epsilon\to 0$
in such a way that $N\epsilon^{d -1}\to Z$ where $Z$ is a positive finite
number.
If $f(x,v,t)$ denotes the density of the particles of velocity $v$, then $f$
satisfies the Boltzmann equation. I consider a variant
of the hard sphere model in which the collisions are still
elastic but now occur with probability
$\epsilon^a$.
When the number of particles $N$ goes to infinity and
$N\epsilon^{d+a -1}$ goes to a nonzero constant, I show that
the probability density converges to a solution of the Boltzmann equation
provided that $a\geq d+1$. Here a solution is understood in the
renormalized sense of DiPerna-Lions.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- M. Feigenbaum, Rockefeller University
Title- Patterns: Determined by Arbitrarily Distant Boundaries
Time/place- 1:30pm 10/17/2002
Speaker- J. Kurchan, Harvard University
Title- Strategies for Saddles In Phase-Space
Time/place- 11:30am 10/24/2002
Abstract-A well-known strategy for studying the states of a system consists of
coupling it to a thermal bath with friction and random noise. The
Langevin process thus obtained reproduces the equilibrium
Boltzmann-Gibbs
distribution, and also gives information on all states via the low-lying
eigenvalues of the Fokker-Planck operator.
It turns out that there is a natural generalization of this approach
that allows to define unambiguously (and detect) saddle-points in
phase-space, even in the absence of a known order parameter.
The technique used (supersymmetry) has been applied years ago to
construct
Morse theory in an extremely elegant way, but has not been exploited to
study systems at finite temperatures in the thermodynamical limit.
After reviewing past results, I will describe how this can be done,
stressing the elementary nature of all calculations.
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker- L. Rey-Bellet, University of Virginia
Title- The fluctuation theorem for classical and quantum open systems.
Time/place- 1:30pm 10/24/2002
Abstract-The (Cohen-Galavotti) fluctuation theorem refers to a symmetry
of the fluctuation of the entropy production in a stationary
nonequilibrium state. We discuss two examples of Hamiltonian open
systems, one classical and one quantum, where the fluctuation theorem
for the entropy production has been proved.
Speaker-M. Vogelius, Rutgers University
Title-A very general representation formula for
voltage potentials in the presence of low volume
fraction inhomogeneities
Time/place- 11:30am 10/31/2002
Abstract-TBA
Please note there will be a brown bag lunch between the 2
seminars this morning. Bring your sandwich.
Coffee and homemade cookies will be
available.
Speaker-I. Rodnianski, Princeton University
Title-Asymptotic stability of N-soliton states of NLS
Time/place- 1:30pm 10/31/2002
Abstract-
We establish existence and stability of special profile solutions of the
time-dependent nonlinear Schrodinger equation. The asymptotic profile
of these solutions as $t\to +\infty$ is given by a sum of non-colliding
solitons constructed from the ground state of the corresponding
time-independent elliptic problem. This is joint work with W. Schlag and A. Soffer.