Mehta Dhagash, Hauenstein Jonathan D, Niemerg Matthew, Simm Nicholas J, Stariolo Daniel A
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46565, USA.
Department of Chemistry, The University of Cambridge, Cambridge CB2 1EW, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Feb;91(2):022133. doi: 10.1103/PhysRevE.91.022133. Epub 2015 Feb 23.
Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.
受最近观察到的几种统计力学模型的势能景观（PEL）拓扑平凡化现象的启发，我们使用数值多项式同伦延拓和通过非厄米矩阵的重新表述对有限尺寸的2自旋球面模型进行了数值研究。延拓方法计算该模型的所有复平稳点，而矩阵方法计算实平稳点。使用这些方法，我们在改变PEL的拓扑结构以及方差的同时计算平稳点的平均数。给出了这些平稳点的直方图以及关于复平稳点的分析。这项工作将拓扑平凡化与代数几何和突变理论这两个不同的数学分支联系起来，这是进一步跨学科研究的沃土。
