# Time Scale to Ergodicity in the FPU System

### J. De Luca, Allan J. Lichtenberg and Michael A. Lieberman

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M93/92

1993

We study the approach to near-equipartition in an N-dimensional FPU Hamiltonian. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy, one can reach near-equipartition if one waits for a time proportional N2. Below this critical energy the time is exponentially long. We develop a theory to understand the time evolution and deformation of the actions of the oscillators based on a normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for near-equipartition, the time scale to near-equipartition and the deformation of the actions below equipartition, in qualitative agreement with the numerical results.

BibTeX citation:

@techreport{De Luca:M93/92, Author = {De Luca, J. and Lichtenberg, Allan J. and Lieberman, Michael A.}, Title = {Time Scale to Ergodicity in the FPU System}, Institution = {EECS Department, University of California, Berkeley}, Year = {1993}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2473.html}, Number = {UCB/ERL M93/92}, Abstract = {We study the approach to near-equipartition in an N-dimensional FPU Hamiltonian. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy, one can reach near-equipartition if one waits for a time proportional N2. Below this critical energy the time is exponentially long. We develop a theory to understand the time evolution and deformation of the actions of the oscillators based on a normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for near-equipartition, the time scale to near-equipartition and the deformation of the actions below equipartition, in qualitative agreement with the numerical results.} }

EndNote citation:

%0 Report %A De Luca, J. %A Lichtenberg, Allan J. %A Lieberman, Michael A. %T Time Scale to Ergodicity in the FPU System %I EECS Department, University of California, Berkeley %D 1993 %@ UCB/ERL M93/92 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2473.html %F De Luca:M93/92