Mathematical Physics Seminar
February Schedule
Organizer- Joel L. Lebowitz
email: lebowitz@math.rutgers.edu
- Speaker- D. Friedan, Rutgers University
- Title-
A Tentative Theory of Large Distance Physics
- Time/place- 2/6/2003 11:30am in Hill 705
- Abstract-
A certain mathematically natural, scale invariant two dimensional
nonlinear model, the lambda-model, is proposed to produce spacetime
physics. The lambda-model constructs a generally covariant, gauge
invariant spacetime quantum field theory which governs spacetime
physics at large distance (in dimensionless units). The spacetime
quantum field theory is the 2d model's a priori measure. Spacetime
physics at small distance is described by string scattering
amplitudes in an effective background spacetime, also produced by
the lambda-model. Quantum field theory and the S-matrix become
complementary descriptions of nature, quantum field theory working
at large distance, the S-matrix at small distance.
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: R. Tumulka , University of Munich, Germany/Rutgers University
Title:Bohmian mechanics and quantum field theory
Time/place-1:30pm, 2/6/2003 in Hill 705
Abstract:
Bohmian mechanics is a version of (nonrelativistic) quantum
mechanics that ascribes world lines to the particles, depending on the
wave function. I will describe a recently developed extension of Bohmian
mechanics to quantum field theory. The main innovation is that the number
of particles is not conserved, which means that world lines can begin and
end. As a consequence, the configuration Q(t) evolves no longer
deterministically, but follows a Markov jump process, depending on the
state vector in Fock space.
Speaker- S. Nussinov, Tel Aviv University/IAS
Title-
A Simple Physicist Approach to Complex Problems
Time/place-11:30am, 2/13/2003 in Hill 705
Abstract-
[Based on work with V.Gudkov, J. Johnson, & Z. Nussinov]
"Complex Problems" involving a large number, n, of elements
and for which any general method of solution {Presumeabley!}
requires a large number (Growing faster than any finite power of n)
of elementary steps, and the P=(?)NP issue are briefly described.
This is done in the context of a " proverbial" n students in a dorm
example - a laymen description of the largest "clique" problem-
We suggest a simple physical analog model which is easy to simulate.
The n students or n vertices in a graph are represented by
n points in d=n-1 dimensions, initially residing at the n vertices of
symmetric n simplex and which move due to (constant) attractive/repulsive
forces introduced between compatible/incompatible students or between
connected/disconnected vertices in the graph.
The deterministic evolution of the n points is free from local minima
traps, easy to simulate, and appears(?!) to solve in polynomial time :
1) The Heuristic problem of finding" clusters" in a network
{i.e in graphs or in communication,commercial,biological,etc systems}
by physically (geometrically) clustering the representative points.
2) The Graph isomorphism problem by evolving independently via
identical dynamics the simplexes corresponding to the two graphs and
checking the (Geometric) congruence of the later.
3) the largest clique problem.
These problems are of increasing intrinsic difficulty and this
reflects in our "Solutions".
We briefly speculate on possible extensins to other "Complex Problems"
such as the Traveling- salesman or Hamiltonian circuit problem and on
possible implications in sociology,bilogy, neural-nets and other
areas.
No previouse background beyond the most elementary geometry and
physics {the latter even at the Aristotelean level} is required.
Speaker- D. Holcman, Weizmann/UCSF
Title- Modeling Calcium Dynamics in Dentritic Spines and Spine Motility
Time/place- 2/20/2003 11:30am in Hill 705
Abstract-
A dendritic spine is a cell-like structure located on a dendrite of a
neuron. It conducts calcium ions from the synapse to the
dendrite. A dendritic spine can contain anywhere between a few and up to
thousands of calcium ions at a time.
Internal calcium is known to bring about fast
contractions of dendritic spines (twitching) after a burst, an
action potential, or a back-propagating action potential. In this
work, we propose an explanation of the cause and effect of the twitching
and its role in the functioning of the spine as a conductor of
calcium.
We model the spine as a machine powered by the calcium it
conducts and we describe its moving parts. The latter are proteins
that are involved in the conduction process. These proteins are
found inside the dendritic spine and their spatial distribution
can be measured.
We propose a molecular model of calcium dynamics in a dendritic spine,
which shows that the rapid calcium motility in the spine is due to the
concerted contraction of certain proteins that bind calcium. The
contraction induces a stream of the cytoplasmic fluid in the direction of
the dendritic shaft, thus speeding up the time course of spinal calcium
dynamics, relative to pure diffusion. According to the proposed model, the
diffusive motion of the calcium ions is described by a system of
Langevin equations, coupled to the hydrodynamical fluid flow field
induced by contraction of proteins. These contractions occur when
enough calcium binds to specific protein molecules inside the
spine. By following the random ionic trajectories, we compute the
distribution of calcium exit time from the spine, the evolution
of concentration of calcium bound to specific proteins, the
relative number of ions pumped out, compared to the number of ions
that leave at the dendritic shaft, and so on.
A computer simulation of this model of calcium dynamics in a dendritic
spine was run with any the number of calcium ions varying from one or
two, up to the hundreds. The simulation indicates that spine
motility can be explained by the basic rules of chemical reaction
rate theory at the molecular level . Analysis of the simulation
data reveals two time periods in the calcium dynamics. In the
first period calcium motion is driven by a hydrodynamical push,
while there are no push effects in the second, when ionic motion
is mainly diffusion in a domain with obstacles.
A biological conclusion is that the role of rapid motility in
dendritic spines is to increase the efficiency of calcium conduction to
the dendrite and to speed up the emptying of the spine.
In this talk, I will present the mathematical models, the simulation
results and the biological implications.
If time permits I will also present a coarse-grained model
of chemical reactions in biological micro-structures, which allow
to understand the role of the geometry in the function of
neurobiological microstrustures.
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: R. Esposito , University of Rome/Rutgers University
Title- A Possible Approach to the Derivation of the Non-Linear Quantum Boltzmann Equation
Time/place-1:30pm, 2/20/2003 in Hill 705
Abstract:
We analyze a possible way to derive the Nonlinear Quantum
Boltzmann equation form a system of $N$ identical quantum
particles in a weak-coupling regime, evolving according to the
Schr\"odinger equation. This is done by looking at the time
evolution of the Wigner transform of the one-particle reduced
density matrix, which is represented by means of a perturbative
series obtained upon iterating the Duhamel formula. For short
times, we rigorously prove that a subseries of the latter,
converges to the solution of the Boltzmann equation which is
Speaker- J. Cardy, IAS and Oxford University
Title- Stochastic Loewner Evolution and Dyson's Circular Ensembles
Time/place- 2/27/2003 11:30am in Hill 705
Abstract-
SLE is a new approach to describing the statistics of cluster
boundaries in 2d critical systems. We show that the problem of $N$
radial SLEs in the unit disc is equivalent to Dyson's Brownian
motion on the boundary of the disc, with a parameter
$\beta=4/\kappa$. As a result various equilibrium critical models
give realisations of circular ensembles with $\beta$ different from the
classical values of 1, 2 and 4 corresponding to random matrices.
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. Carlen, Georgia Tech
Title-Fast and slow convergence to equilibrium for the Boltzmann equation
Time/place- 2/27/2003 1:30pm in Hill 705
Abstract-
The Boltzmann collision kernel for Maxwellian molecules has a contrction
property
first discovered by Tanaka. When combined with estimates on the probability of
unfavorable collision histories in the Wild summation, this
allows one to bypass entropy estimates, and directly obtain precise
information on the rate of relaxation, and how it depends on the initial data.
This program was carried out in joint work with Xuguang Lu, and we show that
the rate of convergence depends on the initial
data $F$ essentially only through on the behavior near $r=0$ of the function
$J_F(r) = \int_{|v|>1/r}|v|^2dF(v)$. Our estimates on the
terms in the Wild sum yield not only a quantitative
estimate, in the strongest physical norm, on the rate at which
the solution converges to equilibrium,
but also a global stability estimate.
We show that our upper bounds are qualitatively sharp by
producing examples of
solutions for which the convergence is as slow as
permitted by our bounds. These are the first examples of
solutions of the Boltzmann
equation that converge to equilibrium more slowly than exponentially.