From deterministic dynamics to thermodynamic laws II: Fourier's law and mesoscopic limit equation

Local Laws and a Mesoscopic CLT for $β$-ensembles

We study the statistical mechanics of the log-gas, or $β$-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations of linear statistics that is valid at all mesoscales. Using these local laws, we are able to exhibit for the first time a CLT at arbitrary mesoscales, improving upon a previous result of Be…

含稳定分布的幂律的Fourier和Cauchy-斯蒂尔切斯变换

Fourier and Cauchy-Stieltjes transforms of power laws including stable distributions

引入了一类概率测度，其在无穷远处的密度是Pareto分布的混合.这类变换可以用傅里叶变换来表征，它具有包括实幂而不仅仅是整数幂的幂级数展开。这类包括概率中的稳定分布和非交换概率论。我们还刻画了类的Cauchy-斯蒂尔切斯变换和Voiculescu变换。如果稳定性指数大于1，则概率论中的稳定分布不属于这类分布，而非交换概率中的稳定分布属于这类分布。

傅立叶定律、相变和定态Stefan问题

Fourier law, phase transitions and the stationary Stefan problem

本文研究了由Giacomin和Lebowitz从Kawasaki动力学导出的具有Kac势的Ising系统\cite{GiacominLebowitz}中的积分微分方程的一维定态解.我们构建固定的解决方案与非零电流和证明的傅立叶定律的热力学极限的有效性表明，低于临界温度的极限平衡剖面具有不连续性（定义的接口的位置），并满足一个固定的自由边界Stefan问题。过冷和过热的影响也进行了研究。我们表明，如果亚稳态值施加在边界处，那么介观固定配置文件不再是单调的，因此傅立叶定律是不满意的。然而，它在热力学极限中恢复其有效性，其中极限轮廓再次远离界面单调。

纳维-斯托克斯-傅里叶流的标度律及宏观动力学在分子层面的不可思议有效性

Scaling laws for stationary Navier-Stokes-Fourier flows and the unreasonable effectiveness of hydrodynamics at the molecular level

本文通过纳维-斯托克斯-傅里叶方程研究了稳态流的标度律，揭示了宏观动力学在分子层面的高准确性。

New decay rates for a Cauchy thermelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws

The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on one component modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy the same polynomial stability estimates in the L2 -norm of the solution and its higher order derivatives…

A microscopic view on the Fourier law

The Fourier law of heat conduction describes heat diffusion in macroscopic systems. This physical law has been experimentally tested for a large class of physical systems. A natural question is to know whether it can be derived from the microscopic models using the fundamental laws of mechanics.