AI 中文总结
研究由非凸限制势和合作相互作用驱动的相变类型，通过叉形分岔定理和鞍结分岔定理分别刻画连续与不连续相变，还将道森相变点准则扩展到\(n\)维空间，并应用于多种模型。
AI 中文摘要
众所周知，具有对称双阱势的麦克凯恩-弗拉索夫随机微分方程呈现连续相变。相比之下，对于非对称双阱势，系统经历不连续相变，其中一个不变测度在较浅阱中突然出现，并随着温度降低随后分裂为两个。在这项工作中，我们系统地研究了由非凸限制势和\(\mathbb{R}^n\)中的合作相互作用驱动的相变类型。我们的主要结果包括一个刻画连续相变的叉形分岔定理和一个支配不连续相变的鞍结分岔定理。此外，尽管相变具有维度依赖性，我们能够将最初为一维空间中的二次相互作用制定的道森相变点准则扩展到\(n\)维空间。这种扩展将临界参数和分岔方向分别直接与临界协方差矩阵的特征值和特征向量相关联。最后，我们将理论结果应用于上述双阱模型、二维欧几里得空间中表现出多个相变的一类四阱模型以及一个吸引性高斯相互作用模型。
英文摘要
It is well known that the McKean-Vlasov stochastic differential equation with a symmetric double-well potential exhibits a continuous phase transition. In contrast, for an asymmetric double-well potential, the system undergoes a discontinuous phase transition, in which an invariant measure abruptly appears in the shallower well and subsequently splits into two as the temperature decreases. In this work, we systematically investigate the types of phase transitions driven by non-convex confining potentials and cooperative interactions in $\mathbb{R}^n$. Our main results consist of a pitchfork bifurcation theorem characterizing continuous phase transitions and a saddle-node bifurcation theorem governing discontinuous phase transitions. In addition, despite the dimension-dependent nature of phase transitions, we are able to extend Dawson's criterion for phase transition points -- originally formulated for quadratic interactions in $1$-dimensional space -- to $n$-dimensional space. This extension directly relates the critical parameter and bifurcation direction to the eigenvalue and eigenvector of the critical covariance matrix, respectively. Finally, we apply our theoretical results to the aforementioned double-well models, to a class of four-well models in two-dimensional Euclidean space that exhibit multiple phase transitions, and to an attractive Gaussian interaction model.
