Mathematical Physics
Seminar

February Schedule

Organizer: Joel L. Lebowitz
lebowitz@math.rutgers.edu

  • Speaker- R. Zia, University of Virgina Tech.
  • Time/Place- Thursday 2/5/04, 11:30am in CoRE Bldg. 431
    (PLEASE NOTE BUILDING & RM. CHANGE)
  • Title- Biased Diffusion of Two Species: Novel Behavior in Two Lanes
  • Abstract- When two species of particles, with no interactions other than excluded volume, are driven in opposite directions on a periodic lattice many interesting phenomena emerge. Though it is proved that the system in d=1 is always disordered, MC simulations signals transitions to ordered states if there are just two lanes (2xL lattices)! A brief review of the background, including the relationship to zero-range processes and the conjecture that this system may revert to disorder at gargantuan lengths (e.g., 1070), will be provided. Recent developments, such as fast coarsening for realistic L's and the effects of "lane preference," will be presented. Many features are reminiscent of fast cars and slow trucks on two-lane highways.




  • Speaker- E. Carlen, Georgia Institute of Technology
  • Time/Place- Thursday 2/12/04, 11:30am in Hill 705
  • Title- Sharp Form of the Central Limit Theorem for Maxwellian Molecules
  • Abstract- Abstract: In the case of Maxwellian molecules, the Wild summation formula gives an expression for the solution of the spatially homogeneous Boltzmann equation in terms of its initial data F as a sum {\displaystyle f(v,t) = \sum_{n=0}^\infty e^{-t}(1 - e^{-t})^n Q_n^+(F)(v)}. Here, $Q_n^+(F)$ is an average over n--fold iterated Wild convolutions of F. If M denotes the Maxwellian equilibrium corresponding to F, then it is of interest to determine bounds on the rate at which $\|Q_n^+(F) - M\|_{L^1(\R)}$ tends to zero. We prove that for every $\epsilon>0$, if F has moments of every order and finite Fisher information, there is a constant C so that for all n, $\|Q_n^+(F) - M\|_{L^1(\R)} \le Cn^{\Lambda+\epsilon}$ where $\Lambda$ is the least negative eigenvalue for the linearized collision operator. We show that $\Lambda$ is the best possible exponent by relating this estimate to a sharp estimate for the rate of relaxation of $f(\cdot,t)$ to M. A key role in the analysis is played by a decomposition of $Q_n^+(F)$ into a smooth part and a small part. This depends in an essential way on a probabilistic construction of McKean. It allows us to circumvent difficulties stemming from the fact that the evolution does not improve the qualitative regualrity of the initial data. This is joint work with C. Carvalho and E. Gabetta.

    There will be a brown bag lunch in between the seminars
    Please bring your lunch
    Cookies and coffee will be available



  • Speaker- A. Soffer, Rutgers University
  • Time/Place- Thursday 2/12/04, 1:30pm in Hill 705
  • Title- Multidimensional Nonlinear Dispersive Waves:From Exact Results to Applications
  • Abstract- The Nonlinear Schrodinger equation, which appear in the Hartree Fock Approximation, and in nonlinear optics,is an example of dispersive wave equation which have many different asymptotic states depending on the initial data. Such time dependent equations play a central role in many of the latest scientific advances,such as Bose-Einstein condensates and optical devices , and general bistable systems. I will discuss the solutions of such equations,including the large time behavior. Rigorous results have shown for the first time the phenomena of ground state selection,asymptotic instability of the excited states and more.These results are obtained by deriving a novel Nonlinear Master equation and multitime scale analysis of its properties.


  • Speaker- J. Beck, Rutgers University
  • Time/Place- Thursday 2/19/04, 11:30am in Hill 705
  • Title- The Mathematics of Tic-Tac-Toe Like Games
  • Abstract- Tic-Tac-Toe is a simple and boring draw game, but playing it on a 4x4x4 cube instead of the usual 3x3 square, one gets a very entertaining and difficult game. It is called "Qubic", and it was analyzed by Patashnik in 1979 by extensive use of computer (first player wins). Other cases like the 5x5x5 cube are wide open (conjectured to be a draw game) because of the "combinatorial chaos": the "exhaustice search" is far too complex. If multi-dimensional Tic-Tac-Toe is far too complex, then what else can we do? How can we apply Combinatorics to analyze hopelessly complex games? What is the role of Probability Theory to understand games of complete information (which have nothing to do with Neumann's "Game Theory")? The object of my talk is to answer these questions.

    There will be a brown bag lunch in between the seminars
    Please bring your lunch
    Cookies and coffee will be available



  • Speaker: S. Torquato, Princeton University
  • Title: Particle Packings, Jamming, and Order Metrics
  • Time/Place: 2/19/04, 1:30pm, Room 705
  • Abstract: Dense particle packings have fascinated people since the dawn of civilization and the fascination persists. Resurgent interest comes from the recent proof of the Kepler conjecture that the face-centered-cubic lattice provides the densest packing of congruent spheres in three dimensions and from the problem of "random" sphere packings, which holds the clue to the formation of glasses and the structure of liquids. In this talk, I will discuss three topics.

    I demonstrate why the venerable 50-year old notion of "random close packing" (RCP) of spheres (the putative "random" analog of Kepler's conjecture) is mathematically ill-defined. To replace this traditional notion, we introduce a new concept of a maximally random jammed (MRJ) state, which can be made precise. The latter idea rests on devising precise meanings for "jamming" and "randomness," which I describe. We show that there are random packings of ellipsoids in three-dimensional Euclidean space that closely approach the densest lattice packing. Moreover, we have discovered a periodic (non-lattice) ellipsoid packing with the highest known density and one in which the ellipsoids are not highly degenerate in shape (i.e., highly eccentric). Present results do not exclude the possibility that even denser periodic packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids.

    Finally, I discuss a new approach to derive lower bounds on the density of random sphere packings in d-dimensional Euclidean space. One of these bounds is sharper than the well-known (nonconstructive) Minkowski lower bound that applies to the densest lattice packings in d-dimensional Euclidean space. Another improved bound is the sharpest to date.



    No seminar Thursday February 26th, 2004