晶体经典模型动力学涨落中的Tsallis分布、弛豫和$Δt \cdot ΔE \simeq h$关系
Tsallis distributions, their relaxations and the relation $Δt \cdot ΔE \simeq h$, in the dynamical fluctuations of a classical model of a crystal
摘要：
我们报告的结果进行了数值研究，在动力系统理论的框架内，为一个现实的模型的离子晶体，由于存在的远程库仑相互作用，吉布斯分布没有很好地定义。取模式能量为Maxwell-Boltzmann分布的初始数据，研究了模式能量的动力学涨落，计算了模式能量变化的模$E_k$。主要的结果是，他们遵循Tsallis分布，放松到接近麦克斯韦-玻尔兹曼的分布;迹象也给出了系统仍然相关。弛豫时间$τ$取决于比能$\varepsilon$，对于曲线$τ$ vs，$\varepsilon$，有两个结果。首先，存在一个能量阈值$\varepsilon_0$，在该阈值以上，曲线的形式为$$ τ\cdot \varepsilon \simeq h\ , $$，在此，出乎意料地，普朗克常数$h$出现了。根据模式能量的标准偏差$ΔE$（对于它有$ΔE=\varepsilon$），用$Δt$表示弛豫时间$τ$，关系式读作$Δt \cdot ΔE \simeq h$，这使人想起海森堡测不准关系。此外，阈值对应于零点能量。实际上，后者的量子值是$hν/2$（其中$ν$是系统的特征红外频率），而我们发现$\varepsilon \simeq hν/4$，因此只有因子2的差异。因此，在统计热力学中，缺乏完全的混沌性似乎是通过类似量子的现象来表现的。
英文摘要：
We report the results of a numerical investigation, performed in the frame of dynamical systems' theory, for a realistic model of a ionic crystal for which, due to the presence of long--range Coulomb interactions, the Gibbs distribution is not well defined. Taking initial data with a Maxwell-Boltzmann distribution for the mode-energies $E_k$, we study the dynamical fluctuations, computing the moduli of the the energy-changes $|E_k(t)-E_k(0)|$. The main result is that they follow Tsallis distributions, which relax to distributions close to Maxwell-Boltzmann ones; indications are also given that the system remains correlated. The relaxation time $τ$ depends on specific energy $\varepsilon$, and for the curve $τ$ vs, $\varepsilon$ one has two results. First, there exists an energy threshold $\varepsilon_0$, above which the curve has the form $$ τ\cdot \varepsilon \simeq h\ , $$ where, unexpectedly, Planck's constant $h$ shows up. In terms of the standard deviation $ΔE$ of a mode-energy (for which one has $ΔE=\varepsilon$), denoting by $Δt$ the relaxation time $τ$, the relation reads $Δt \cdot ΔE \simeq h$, which reminds of the Heisenberg uncertainty relation. Moreover, the threshold corresponds to zero-point energy. Indeed, the quantum value of the latter is $hν/2$ ( where $ν$ is the characterisic infrared frequency of the system), while we find $\varepsilon \simeq hν/4$, so that one only has a discrepancy of a factor 2. So it seems that lack of full chaoticity manifests itself, in Statistical Thermodynamics, through quantum-like phenomena.
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