Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA.
Phys Rev E. 2018 Jan;97(1-1):012139. doi: 10.1103/PhysRevE.97.012139.
The field of disordered systems in statistical physics provides many simple models in which the competing influences of thermal and nonthermal disorder lead to new phases and nontrivial thermal behavior of order parameters. In this paper, we add a model to the subject by considering a disordered system where the state space consists of various orderings of a list. As in spin glasses, the disorder of such "permutation glasses" arises from a parameter in the Hamiltonian being drawn from a distribution of possible values, thus allowing nominally "incorrect orderings" to have lower energies than "correct orderings" in the space of permutations. We analyze a Gaussian, uniform, and symmetric Bernoulli distribution of energy costs, and, by employing Jensen's inequality, derive a simple condition requiring the permutation glass to always transition to the correctly ordered state at a temperature lower than that of the nondisordered system, provided that this correctly ordered state is accessible. We in turn find that in order for the correctly ordered state to be accessible, the probability that an incorrectly ordered component is energetically favored must be less than the inverse of the number of components in the system. We show that all of these results are consistent with a replica symmetric ansatz of the system. We conclude by arguing that there is no distinct permutation glass phase for the simplest model considered here and by discussing how to extend the analysis to more complex Hamiltonians capable of novel phase behavior and replica symmetry breaking. Finally, we outline an apparent correspondence between the presented system and a discrete-energy-level fermion gas. In all, the investigation introduces a class of exactly soluble models into statistical mechanics and provides a fertile ground to investigate statistical models of disorder.
统计物理学中的无序系统领域提供了许多简单的模型，这些模型中热无序和非热无序的竞争影响导致了新的相和有序参数的非平凡热行为。在本文中，我们通过考虑一个无序系统来为这个主题添加一个模型，其中状态空间由列表的各种排序组成。就像在 spin glasses 中一样，这种“排列玻璃”的无序源于哈密顿量中的一个参数取自可能值的分布，从而允许名义上的“不正确排序”比排列空间中的“正确排序”具有更低的能量。我们分析了能量成本的高斯、均匀和对称的伯努利分布，并通过运用 Jensen 不等式，得出了一个简单的条件，要求排列玻璃在低于无序系统温度的温度下始终过渡到正确排序的状态，只要这个正确排序的状态是可及的。我们反过来发现，为了使正确排序的状态是可及的，不正确排序的组件在能量上占优势的概率必须小于系统组件数的倒数。我们表明，所有这些结果都与系统的 replica symmetric ansatz 一致。我们最后通过论证这里考虑的最简单模型没有明显的排列玻璃相，并讨论如何将分析扩展到更复杂的能够产生新颖相行为和 replica symmetry breaking 的哈密顿量，来结束这篇文章。最后，我们概述了所提出的系统与离散能级费米气体之间的明显对应关系。总的来说，这项研究将一类完全可解的模型引入到统计力学中，并为研究无序的统计模型提供了肥沃的土壤。
