Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals
Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals
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DOI：
10.1002/cpa.21570
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发表时间：
2013-07
影响因子：
3
通讯作者：
N. Rougerie;S. Serfaty
中科院分区：
文献类型：
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作者：
N. Rougerie;S. Serfaty
我们考虑了任意维d≥2外闭束势中n个带电粒子的经典系统。粒子之间通过成对排斥库仑力相互作用，耦合参数为n−1阶(平均场标度)。通过对哈密顿量的适当分裂，我们提取了超出平均场极限的基态能量的次前导阶项。我们证明了表征系统涨落的这一次级项是由一个新的“重整化能量”泛函控制的，它提供了一种计算凝胶体(即由均匀中和背景遮挡的无限多个点电荷)在任何维度上的总库仑能量的方法。切断无限能量部分的重整化是通过在小范围内抹掉点电荷来实现的，就像在Onsager引理中那样。我们得到了库仑气体统计力学的结果：自由能或配分函数的次前导阶渐近展开式，吉布斯度量的刻画，局部电荷涨落的估计，以及约化密度的因式分解估计。这通过另一种方法将Sandier和Serfaty的结果扩展到高于2的维度。©2016 Wiley期刊，Inc.
We consider a classical system of n charged particles in an external confining potential in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n−1 (mean‐field scaling). By a suitable splitting of the Hamiltonian, we extract the next‐to‐leading‐order term in the ground state energy beyond the mean‐field limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new “renormalized energy” functional providing a way to compute the total Coulomb energy of a jellium (i.e., an infinite set of point charges screened by a uniform neutralizing background) in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager's lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next‐to‐leading‐order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations, and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than 2 by an alternative approach. © 2016 Wiley Periodicals, Inc.