论文标题
Fermi-Pasta-ulam-tsingou链中的汉堡湍流
Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
论文作者
论文摘要
我们通过分析证明并在数值上证明了Fermi-Pasta-Pasta-ulam-tsingou链的动力学的特征是瞬时汉堡湍流方案在较大的时间和能量尺度上。该机制以长波长和能量的每个粒子足够小，以至于无法在快速时间尺度上达到电气。在此范围内，我们证明了热量的驱动机制是可以使用一对广义汉堡方程来预测的冲击的形成。我们以每个粒子的少量能量进行扰动计算，证明链条$ e_k $的能量光谱是幂律，$ e_k \ sim k^{ - ζ（t）} $，在大量的vavenumbers $ k $上。我们预测，$ζ（t）$首先将值$ 8/3 $在汉堡冲击时间中占用，然后在两个冲击时间内达到接近$ 2 $的值。指数$ζ= 2 $的价值在系统最终放松到设置之前一直持续几次。在这个广泛的时光窗口中，与先前的结果一致，在$ k $中观察到频谱中的指数截止。事实证明，这种情况是通用的，即独立于表征系统和初始条件的参数，一旦以冲击时间为单位测量时间。
We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterised by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast time scale. In this range, we prove that the driving mechanism to thermalisation is the formation of a shock that can be predicted using a pair of generalised Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain $E_k$ decays as a power law, $E_k\sim k^{-ζ(t)}$, on an extensive range of wavenumbers $k$. We predict that $ζ(t)$ takes first the value $8/3$ at the Burgers shock time, and then reaches a value close to $2$ within two shock times. The value of the exponent $ζ=2$ persists for several shock times before the system eventually relaxes to equipartition. During this wide time-window, an exponential cut-off in the spectrum is observed at large $k$, in agreement with previous results. Such a scenario turns out to be universal, i.e. independent of the parameters characterising the system and of the initial condition, once time is measured in units of the shock time.
