### Abstract

The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in thermal equilibrium are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting renormalized nonlinear waves that possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized β -FPU chain is found to be in very good agreement with its numerically measured value.

Original language | English (US) |
---|---|

Article number | 046603 |

Journal | Physical Review E |

Volume | 75 |

Issue number | 4 |

DOIs | |

State | Published - Apr 10 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E*,

*75*(4), [046603]. https://doi.org/10.1103/PhysRevE.75.046603

**Interactions of renormalized waves in thermalized Fermi-Pasta-Ulam chains.** / Gershgorin, Boris; Lvov, Yuri V.; Cai, David.

Research output: Contribution to journal › Article

*Physical Review E*, vol. 75, no. 4, 046603. https://doi.org/10.1103/PhysRevE.75.046603

}

TY - JOUR

T1 - Interactions of renormalized waves in thermalized Fermi-Pasta-Ulam chains

AU - Gershgorin, Boris

AU - Lvov, Yuri V.

AU - Cai, David

PY - 2007/4/10

Y1 - 2007/4/10

N2 - The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in thermal equilibrium are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting renormalized nonlinear waves that possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized β -FPU chain is found to be in very good agreement with its numerically measured value.

AB - The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in thermal equilibrium are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting renormalized nonlinear waves that possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized β -FPU chain is found to be in very good agreement with its numerically measured value.

UR - http://www.scopus.com/inward/record.url?scp=34147168606&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34147168606&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.75.046603

DO - 10.1103/PhysRevE.75.046603

M3 - Article

AN - SCOPUS:34147168606

VL - 75

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 4

M1 - 046603

ER -