声子玻尔兹曼方程的超级扩散转变
Superdiffusion transition for a phonon Boltzmann equation
AI总结：
本文研究带电粒子谐波链在磁场和随机噪声下的能量输运，量化了从$3/4$分数扩散到$5/6$分数扩散的磁场强度阈值，并描述了转变机制。
摘要：
我们考虑一个无限长的带电粒子谐波链，该链受到强度为$B$的磁场作用，并受到一个保持能量守恒的随机噪声的影响。在arXiv:0809.0177和arXiv:1402.2988中，已经证明如果$B=0$，能量输运由$3/4$分数扩散描述；而在arXiv:1808.01040中，证明了如果$B\ e 0$，则由$5/6$分数扩散描述。在arXiv:0809.0177和arXiv:1808.01040中，作者使用了两步论证，即他们首先证明了Wigner分布的动力学极限是声子玻尔兹曼方程的解，然后证明了该解收敛到指数为$3/4$（若$B=0$，见arXiv:0809.0177）或指数为$5/6$（若$B\ e 0$，见arXiv:1808.01040）的分数扩散方程的解。在本文中，我们从声子玻尔兹曼方程出发，量化了从一种宏观状态切换到另一种宏观状态所需的磁场强度。我们还描述了跨越这两个不同相位的转变机制。
英文摘要：
We consider an infinite harmonic chain of charged particles submitted to the action of a magnetic field of intensity $B$ and subject to the action of a stochastic noise conserving the energy. In arXiv:0809.0177 and arXiv:1402.2988 it has been proved that if $B=0$ the transport of energy is described by a $3/4$-fractional diffusion while it has been proved in arXiv:1808.01040 that if $B\ne 0$ it is described by a $5/6$-fractional diffusion. In arXiv:0809.0177 and in arXiv:1808.01040 the authors used a two step argument, i.e. they first proved that the kinetic limit of the Wigner distribution is the solution of a phonon Boltzmann equation and then proved that this solution converges to the solution of a fractional diffusion equation with exponent $3/4$ if $B = 0$ (see arXiv:0809.0177) and exponent $5/6$ if $B\ne 0$ (see arXiv:1808.01040). In this paper we quantify the intensity of the magnetic field required to switch from one macroscopic regime to the other one from the phonon Boltzmann equation. We also describe the transition mechanism to cross the two different phases.
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