1D log gases and the renormalized energy: crystallization at vanishing temperature
1D log gases and the renormalized energy: crystallization at vanishing temperature
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DOI：
10.1007/s00440-014-0585-5
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发表时间：
2015-08-01
影响因子：
2
通讯作者：
Serfaty, Sylvia
中科院分区：
文献类型：
--
作者：
Sandier, Etienne;Serfaty, Sylvia
我们研究了具有一般势和任意温度倒数的一维对数气体或系综的统计力学，根据我们在Sandier和Serfaty（Ann Probab，2014）中介绍的二维库仑气体的方法。在某些特定情况下，此类集合对应于随机矩阵模型。形式上的限制对应于“加权Fekete集”，也被处理。我们引入Sandier和Serfaty的“重整化能量”的一维版本（Commun Math Phys 313（3）：635-743，2012），测量在均匀中和背景中的真实的线上的无限多个点的总对数相互作用。我们表明，这种能量是最小化的点时，在一个晶格。通过一个适当的分裂的哈密顿量，我们连接到这个重整化能量的完整的统计力学问题，这使我们能够获得新的结果在微观尺度上的点的分布：特别是我们表明，配置的是高于一定的阈值（这往往是）有指数小的概率。这表明，随着温度降至零，构型具有增加的顺序和结晶。
We study the statistical mechanics of a one-dimensional log gas or -ensemble with general potential and arbitrary , the inverse of temperature, according to the method we introduced for two-dimensional Coulomb gases in Sandier and Serfaty (Ann Probab, 2014). Such ensembles correspond to random matrix models in some particular cases. The formal limit corresponds to "weighted Fekete sets" and is also treated. We introduce a one-dimensional version of the "renormalized energy" of Sandier and Serfaty (Commun Math Phys 313(3):635-743, 2012), measuring the total logarithmic interaction of an infinite set of points on the real line in a uniform neutralizing background. We show that this energy is minimized when the points are on a lattice. By a suitable splitting of the Hamiltonian we connect the full statistical mechanics problem to this renormalized energy , and this allows us to obtain new results on the distribution of the points at the microscopic scale: in particular we show that configurations whose is above a certain threshold (which tends to as ) have exponentially small probability. This shows that the configurations have increasing order and crystallize as the temperature goes to zero.