Mathematical Physics
Seminar
September Schedule
Organizer: Joel L. Lebowitz
lebowitz@math.rutgers.edu
- Speaker- L. Shepp, Rutgers University
- Title-Olber's Paradox, Chandrasekhar's Model for the Stars,
Wireless Telephony, and Poisson Random Sets
- Time/place-Thursday 9/11/03, 11:30am in Hill 705
- Abstract-
Olber's famous 1823 paradox says that if the universe is either infinite in
age or extent the night sky should be infinitely bright. It is sometimes
put forward as a justification for assuming a finite universe, but I
will show that there is a simple quantum mechanical viewpoint for
the alleged paradox based on the Chandrasekhar Poisson model (for
gravitational forces due to) the stars under which destructive interference
of {\em waves} rather than particles and Kolmogorov's theorem keeps the
series convergent and removes this specious argument for a finite universe.
My attention was initially drawn by a new model for wireless under which
the interference noise, N, at the base station due to mobile phones at a
given frequency, f, is the sum over i of X_i/(R_i)^b, where R_i is the
distance from the base station of the ith mobile phone and X_i is the
signal at a fixed time at frequency f. The power, b, is usually taken as
4 in telephony and 2 in cosmology, but I will show that for any b, N is
a stable random variable if $X_i are iid with {\em any} distribution and the
phones are at the points of a Poisson random set. Chandrasekhar proved a
special case of this result and got a Nobel prize for it.
This is joint work with Susan Heath.
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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.
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Speaker - S. Adams, Technical University Berlin
Title - Large Deviations for the Random Fields of Gradients
Time/place-Thursday 9/11/03, 1:30pm in Hill 705
Abstract -
We consider continuous spin models as effective models for
interfaces. The random field gives the height of the interfacces. A
Gibbs measure exists only for dimensions $ d\ge 3 $. Instead of the
random field of heights we consider the random field of gradients, for
which Gibbs measures (Funaki-Spohn states) exist for any dimension $
d\ge 1 $. We discuss the problems of large deviations for these
models: the reference measure has no product structure and the
so-called plaquette condition gives additional long-range
constraints. We show that the random field of gradients can be studied
as a random field of heights with an appropriate gradient
$\sigma$-algebra. We obtain large deviation results for the empirical
field with free boundary conditions in the Gaussian case and discuss
related results. The talk is based on a common new work with
J.D. Deuschel and S. Sheffield.
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Speaker - Y. Brenier, Univ. Nice, France
Title - The Augmented Born--Infeld Field Equations
Time/place-Thursday 9/25/03, 1:30pm in Hill 423
Abstract -
The Born-Infeld system is a nonlinar version of Maxwell's equations. We
first show that, by using the energy density and the Poynting vector as
additional unknown variables, the BI system can be augmented as a 10x10
system of hyperbolic conservation laws. The resulting augmented system
has some similarity with Magnetohydrodynamics (MHD) equations and enjoys
remarkable properties (existence of a convex entropy, galilean invariance,
full linear degeneracy). In addition, the propagation speeds and the
characteristic fields can be computed in a very easy way, in contrast
with the original BI equations. We investigate several limit regimes of
the augmented BI equations by using a relative entropy method going back
to Dafermos, and recover, the Maxwell equations for weak fields, some
pressureless MHD equations for strong fields, and pressureless gas
equations for very strong fields.