Ballistic and superdiffusive scales in the macroscopic evolution of a chain of oscillators
Ballistic and superdiffusive scales in the macroscopic evolution of a chain of oscillators
复制标题
振子链宏观演化中的弹道尺度和超扩散尺度
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
S. Olla
中科院分区：
文献类型：
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作者：
T. Komorowski;S. Olla
我们考虑一维无限长的谐振子声学链，其动力学受到随机速度交换的扰动，使得链的能量和动量守恒。因此，系统的演化只有三个守恒量：体积、动量和能量。我们表明存在两个时空尺度上的系统的能量演变。在双曲标度（t −1，x −1）上？>守恒量的极限满足欧拉方程组，而宏观能量分布的热部分保持稳定。热能开始在较长的时间尺度上演化，对应于超扩散标度（t −3/2，x −1）？>，并遵循分数热方程。我们还证明了扩散标度限制的黎曼不变量，所谓的正常模式，对应于线性双曲传播。
We consider a one dimensional infinite acoustic chain of harmonic oscillators whose dynamics are perturbed by a random exchange of velocities, such that the energy and momentum of the chain are conserved. Consequently, the evolution of the system has only three conserved quantities: volume, momentum and energy. We show the existence of two space–time scales on which the energy of the system evolves. On the hyperbolic scale (tϵ−1,xϵ−1) ?> the limits of the conserved quantities satisfy a Euler system of equations, while the thermal part of the macroscopic energy profile remains stationary. Thermal energy starts evolving at a longer time scale, corresponding to superdiffusive scaling (tϵ−3/2,xϵ−1) ?>, and follows a fractional heat equation. We also prove the diffusive scaling limit of the Riemann invariants—the so-called normal modes, corresponding to linear hyperbolic propagation.