Mathematical Physics Seminar
Rutgers University
Hill Center, Room 705
October Schedule

Organized by: Joel L. Lebowitz
lebowitz@math.rutgers.edu

PLEASE JOIN US AT 11:40AM IN ROOM 705 KITCHEN FOR COFFEE AND COOKIES

Speaker: D. Ruelle, IHES
Time/Place: Thursday, 10/6, 12:00-1:00pm Hill 705
Title: "Is entropy production local in an infinite classical system?"
Abstract: We shall first show by an example how careful one has to be in using thermostats in nonequilibrium statistical mechanics. Then we shall discuss an infinite "little wheels" model, and argue whether the entropy production in a nonequilibrium steady state can be expresses as an average rate of volume contraction in phase space.

THERE WILL BE A BROWN BAG LUNCH BETWEEN THE TWO SEMINARS (1-2PM) PLEASE JOIN US.

Speaker: G. Gallavotti, University of Rome/Rutgers University
Time/Place:Thursday, 10/6, 2:00-3:00pm; Hill 705
Title: TBA
Abstract: TBA

THERE WILL BE NO SEMINAR THURSDAY OCTOBER 13. SEMINAR WILL RESUME ITS REGULAR SCHEDULE NEXT THURSDAY

leaf1 Speaker: R. Nussbaum, Rutgers University
Time/Place: Thursday, 10/20, 12:00-1:00pm; Room 705
Title: "Global Stability of fixed Points, Two Conjectures and Maple"
Abstract: There is considerable literature concerning second order, nonlinear difference equations of the form

x_(n+1) =g(x_(n-1), x_n)
(1) where g(u,v)=(a+bu+cv)/(A+Bu+Cv) and we assume for simplicity that a,b,c,A.B and C are nonnegative.Equivalently, one can study iterates of the planar map F(u,v )=(v.g(u,v)). There are several long-standing conjectures which ask the following sort of question: If L>0, g(L,L)=L and all eigenvalues z of dF(L,L), the Jacobian matrix of F at (L,L), satisfy |z|<1, is it true that for every positive x_0 and x_1, and for x_n as in equation (1), x_n---->L as n approaches infinity? In other words, does local stability imply global stability? We shall present some recent positive results concerning two special cases of this generic conjecture, discuss some general methods, indicate how Maple can be helpful and mention some open problems.

THERE WILL BE A BROWN BAG LUNCH BETWEEN THE TWO SEMINARS (1-2PM) PLEASE JOIN US.

Speaker: N. Pavlovic, Princeton University
Time/Place: Thursday, 10/20, 2:00-3:00pm; Room 705
Title: "Dyadic models for the equations of fluid motion"
Abstract: In this talk we shall introduce a scalar dyadic model for the Euler and the Navier-Stokes equations in three dimensions and will discuss some of the results that were obtained for these models. For the dyadic Euler equations we prove finite time blow-up, while in the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in case when the degree of dissipation is sufficiently small (joint work with Nets Katz). These results can be generalized to analogous results for a vector dyadic model (joint work with Susan Friedlander).

Also time permitting, we shall present some results for the actual Navier-Stokes equations that are inspired by observing similar phenomena present in dyadic models. In particular, we shall discuss long time behavior of solutions to the 3D Navier-Stokes equations that evolve from initial data in the space BMO^{-1} intersected with certain weighted space (joint work with James Colliander, Carlos Kenig and Gigliola Staffilani).

PLEASE JOIN US AT 11:40AM FOR COOKIES AND COFFEE IN ROOM 705 KICTHEN

Speaker: Y. Li, Rutgers University
Time/Place: Thursday, 10/27, 12:00-1:00pm; Room 705
Title: "A fully nonlinear version of the Yamabe problem"
Abstract: We will present some results concerning a fully nonlinear version of the Yamabe problem.

THERE WILL BE A BROWN BAG LUNCH BETWEEN THE TWO SEMINARS (1-2PM) PLEASE JOIN US.

Speaker: M. Loss, Georgia Tech./IAS
Time/Place: Thursday, 10/27, 2:00-3:00pm; Room 705
Title:"A lower bound on the free energy of matter interacting with radiation"
Abstract: The proof of the existence of the thermodynamic limit for matter was accomplished decades ago in the framework of non-relativistic quantum mechanics by Lieb and Leowitz. This result did not take account of interactions caused by magnetic fields, however, (the spin-spin interaction, in particular) or of the quantized nature of the electromagnetic field. Recent progress has made it possible to undertake such a proof in the context of non-relativistic QED. In this talk I will present one part of such a proof by giving a lower bound to the free energy that is proportional to the number of particles and which takes account of the fact that the field, unlike the particles, is never confined to a finite volume. This is joint work with Elliott Lieb.