热化Fermi-Pasta-Ulam链的性质
Characterization of thermalized Fermi-Pasta-Ulam chains
摘要：
从波相互作用和粒子相互作用的角度研究了\textit{thermal equilibrium}中粒子的Fermi-Pasta-Ulam（FPU）链。结果表明，即使在强非线性区，链在热平衡可以有效地描述由弱相互作用的\textit{renormalized}非线性波系统。这些波具有（i）瑞利-金斯分布和（ii）波之间的零相关性，就像无相互作用的自由波一样。这种重正化是通过一组正则变换实现的。这些重整化波的重整化线性色散得到，并示出在良好的协议与数值实验。此外，通过自洽平均场论证提供了色散关系重整化的动力学解释。事实证明，这种重正化主要来自平凡的共振波相互作用，即，没有动量交换的相互作用此外，使用多时间尺度，统计平均方法，我们表明，近共振波的相互作用引起的共振峰的频谱的重整化模式的加宽。对$β$ -FPU链的共振宽度的理论预测与数值测量值吻合得很好。此外，我们表明，热化的$β$ -FPU链的动力学方案是空间高度本地化的离散呼吸子骑在空间扩展的，重整化波混沌。我们提出了数值证据的存在离散呼吸热平衡。
英文摘要：
The Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal equilibrium} are studied from both wave-interaction and particle-interaction points of view. 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 \textit{renormalized} nonlinear waves. These waves 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. Moreover, we show that the dynamical scenario for thermalized $β$-FPU chains is spatially highly localized discrete breathers riding chaotically on spatially extended, renormalized waves. We present numerical evidence of existence of discrete breathers in thermal equilibrium.
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