Micro–macro transition in the atomic chain via Whitham's modulation equation
Micro–macro transition in the atomic chain via Whitham's modulation equation
复制标题
通过 Whitham 调制方程实现原子链中的微观-宏观转变
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Alexander Mielke
中科院分区：
文献类型：
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作者：
Wolfgang Dreyer;Michael Herrmann;Alexander Mielke
本文的研究对象是非线性原子链随温度变化的热力学描述。出于这个原因，我们考虑牛顿方程的特殊近似解，其中原子以调制行波的形式进行微观振荡。我们从任意大振幅的周期行波的存在性结果开始，并研究了几个例子，包括谐波链，硬球模型和小振幅近似。然后讨论了行波的热力学性质，导出了相应的吉布斯方程。然后，我们集中在调制行波的宏观演化。为此，我们将Whitham的调制理论应用于原子链，导出了调制方程，它是一个由四个宏观守恒律组成的系统。最后一部分是证明问题：我们陈述了一般情况下的一个猜想，并证明了这个猜想的调和链和硬球模型。
The subject matter of this paper is the thermodynamic description of the nonlinear atomic chain with temperature. For this reason we consider special approximate solutions of Newton's equations, in which the atoms perform microscopic oscillations in the form of modulated travelling waves. We start with an existence result for periodic travelling waves with arbitrary large amplitudes and study several examples including the harmonic chain, the hard sphere model and the small-amplitude approximation. Then we discuss the thermodynamic properties of travelling waves and derive the corresponding Gibbs equation. Afterwards we focus on the macroscopic evolution of modulated travelling waves. For this purpose we apply Whitham's modulation theory to the atomic chain and derive the modulation equation, which turns out to be a system of four macroscopic conservation laws. The last part is devoted to the justification problem: we state a conjecture for the general case and prove this conjecture for the harmonic chain and the hard sphere model.