统计力学更新模型中的临界涨落
Critical Fluctuations in Renewal Models of Statistical Mechanics
AI总结：
本文研究统计力学更新模型在临界钉扎—脱钉相变处广延可观测量的大偏差，在等待时间次指数修正正则变化条件下建立精确大偏差原理，并刻画更新次数涨落。
摘要：
我们研究统计力学更新模型中广延可观测量大涨落在临界点处的尖锐渐近行为，例如 DNA 变性的 Poland-Scheraga 模型、流体的 Fisher-Felderhof 模型、蛋白质折叠的 Wako-Saitô-Muñoz-Eaton 模型以及应变异质外延的 Tokar-Dreyssé 模型。这些模型等价于经典更新过程的 Gibbs 测度变换，并可等同于聚合物的约束钉扎模型。进入热力学描述的广延可观测量最终对应于累积奖励，而这些累积奖励来自由等待时间唯一确定且增长不快于等待时间的确定性奖励。其涨落概率随系统规模的衰减在临界点处由指数型转变为次指数型，这一区间对应于不连续的钉扎—脱钉相变。我们在等待时间分布的次指数修正项具有正则变化性的假设下，通过提出一个精确的大偏差原理来描述这种衰减。该原理尤其用于刻画更新次数的涨落；在 Poland-Scheraga 模型中更新次数度量与 DNA 结合的单体，在 Fisher-Felderhof 模型和 Tokar-Dreyssé 模型中度量粒子，而在 Wako-Saitô-Muñoz-Eaton 模型中度量天然肽键。
英文摘要：
We investigate the sharp asymptotic behavior at criticality of the large fluctuations of extensive observables in renewal models of statistical mechanics, such as the Poland-Scheraga model of DNA denaturation, the Fisher-Felderhof model of fluids, the Wako-Saitô-Muñoz-Eaton model of protein folding, and the Tokar-Dreyssé model of strained epitaxy. These models amount to Gibbs changes of measure of a classical renewal process and can be identified with a constrained pinning model of polymers. The extensive observables that enter the thermodynamic description turn out to be cumulative rewards corresponding to deterministic rewards that are uniquely determined by the waiting time and grow no faster than it. The probability decay with the system size of their fluctuations switches from exponential to subexponential at criticality, which is a regime corresponding to a discontinuous pinning-depinning phase transition. We describe such decay by proposing a precise large deviation principle under the assumption that the subexponential correction term to the waiting time distribution is regularly varying. This principle is in particular used to characterize the fluctuations of the number of renewals, which measures the DNA-bound monomers in the Poland-Scheraga model, the particles in the Fisher-Felderhof model and the Tokar-Dreyssé model, and the native peptide bonds in the Wako-Saitô-Muñoz-Eaton model.
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