Chjan C. Lim

International Conference on Dynamical Systems

Beijing, China, June 18 - 22, 2001

Title: Logarithmic Interations, 2-D Turbulence, and Random Matrices

Abstract:

This talk will present some recent progress in the resolution of a long-standing problem in 2-dimensional turbulence, namely the microscopic derivation of the energy spectral density in the zero viscosity limit. Physical arguments by Batchelor, Leith and Kraichnan give two power laws: a -3 law for the inverse energy cascade and a -5/3 law for the forward enstrophy cascade. These heuristic arguments are similar to those of Kolmogorov for the 3-D case.

Using a n-vortex Hamiltonian representation of the 2-D Euler equations in a unbounded domain, we will derive exact closed-form expressions for the energy spectral density, in a nonextensive continuum limit as n tends to infinity. An essential tool in this work is Wigner's observation that the dynamics and statistics of large n-body systems on the line interacting through the Coulomb logarithmic potential can be mapped onto the problem of the spectra of large random Gaussian matrices.

We use the extension by Dyson and Ginibre of this mapping to large n-body systems in the plane. In particular, Ginibre's results for the correlations of the complex spectra of large Gaussian random matrices are used to obtain one and two-body reduced distribution functions for the continuum limit of the n-vortex Hamiltonian system. These are used to obtain the energy spectral density. Random matrices have been used in theoretical physics for a long time but are just now finding its way into several hot areas in mathematics such as integrable systems, zeroes of the Riemann zeta functions, and Young's tableaux.

Program on Physics of Hydrodynamic Turbulence, Mini-Workshop on Mathematics and Fluids

Santa Barbara, California, USA, July 2 - 3, 2000

Title: Exact solution of a three constraints equilibrium statistical mechanics model for 2-D turbulence

Abstract:

This three constraints model is the third one in a family of few constraints equilibrium statistical mechanics models proposed by Turkington and Majda for the study of 2-D and quasi-geostrophic flows. This model is equivalent to the Batchelor-Lee-Kraichnan model for inverse cascades after a Fourier transform. The exact solution of this model is based on a simple but fundamental observation that the enstrophy corresponds exactly to the higher dimensional spherical constraint introduced by Kac. The exact solution is then obtained by extending Kac's and Berlin's solution of the spherical model to a long range logarithmic interaction. A negative critical temperature for phase transition to coherent structures is obtained.

Notes from this talk are available.

The talk may be heard in RealAudio streaming media audio, or one may download the entire talk.

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Last updated 29 May 2001.

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