Abstract

We consider two-dimensional systems of point particles located on rectangular lattices and interacting via pairwise potentials. The goal of this paper is to investigate the phase transitions (and their nature) at fixed density for the minimal energy of such systems. The 2D rectangle lattices we consider have an elementary cell of sides a and b, the aspect ratio is defined as \(\Delta =b/a\) and the inverse particle density \(A = a b\) ; therefore, the “symmetric” state with \(\Delta =1\) corresponds to the square lattice and the “non-symmetric” state to the rectangular lattice with \(\Delta \ne 1\) . For certain types of the interaction potential, by changing continuously the particle density, such lattice systems undertake at a specific value of the (inverse) particle density \(A^*\) a structural transition from the symmetric to the non-symmetric state. The structural transition can be either of first order ( \(\Delta \) unstick from its symmetric value \(\Delta =1\) discontinuously) or of second order ( \(\Delta \) unstick from \(\Delta =1\) continuously); the first and second-order phase transitions are separated by the so-called tricritical point. We develop a general theory on how to determine the exact values of the transition densities and the location of the tricritical point. The general theory is applied to the double Yukawa and Yukawa–Coulomb potentials.

中文翻译：

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相互作用晶格系统中的结构转变

摘要

我们考虑位于矩形晶格上并通过成对势相互作用的点粒子的二维系统。本文的目标是研究此类系统在固定密度下的最小能量下的相变（及其性质）。我们考虑的二维矩形晶格具有边a和b的基本单元，长宽比定义为\(\Delta =b/a\)和逆粒子密度\(A = ab\)；因此， \(\Delta =1\)的“对称”状态对应于方形晶格，而“非对称”状态对应于\(\Delta \ne 1\) 的矩形晶格。对于某些类型的相互作用势，通过连续改变粒子密度，这种晶格系统在特定的（逆）粒子密度值\(A^*\)下进行从对称状态到非对称状态的结构转变。结构转变可以是一阶（\(\Delta \)不连续地脱离其对称值\(\Delta =1\)）或二阶（\(\Delta \)脱离\(\Delta =1\) \)连续); 一级和二级相变由所谓的三临界点分开。我们开发了关于如何确定转变密度的精确值和三临界点位置的一般理论。一般理论适用于双汤川势和汤川-库仑势。