"Nonequilibrium growth dynamics,
matrix models, and integrability "
Prof. Oded Agam
The Racah Institute of Physics,
The Hebrew University of Jerusalem
We consider the relation between nonequilibrium growth processes associated
with viscous fingering (e.g. diffusion limited aggregation, (DLA) and Saffman-Taylor
dynamics in Hele-Shaw cells) and the eigenvalue distribution of normal matrices
model (Complex matrices, M and M' which commute). Viscous fingering is a
classical paradigm model for pattern formation, and a long standing problem
is to understand its fractal properties. On the other hand, the normal matrix
model describes, apparently, very remote fields, for instance, Quantum Hall
Effect in nonuniform magnetic field [1], The Toda lattice hierarchy, and
two-dimensional quantum gravity. Yet, it is possible to show that in the
classical (dispersionless) limit (h→0), the normal matrix model describes
precisely viscous fingering with zero surface tension. Using this relation
we construct an algebro-geometric, genus one, solution for the two dimensional
Toda lattice equation [2]. This solution, analogous to Gurevich-Pitaevskii
solution of the KdV equation, enables us to give a detailed account of the
tip-splitting process. Tip splitting is the basic ingredient characterizing
viscous, and its properties allows one to deduce, e.g., the fractal dimension
of DLA and Saffman-Taylor droplets [3].
[1] O. Agam, E. Bettelheim, P. Wiegmann, and A. Zabrodin, Phys. Rev. Lett. 88, 236801 (2002).
[2] E. Bettelheim and O. Agam, in preparation.
[3] T.C. Halsey , Phys. Rev. Let. 72, 1228 (1994)
Host:
Dr. Ron Lifshitz, x5145
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