相互作用自避行走（ISAW）的回转半径分布
Distribution of the Radius of Gyration for an ISAW
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中文总结 AI 辅助
本研究推导了ISAW回转半径的有限尺寸概率分布显式表达式，结合累积量扩展方法将其推广至相互作用情形，经蒙特卡罗模拟验证吻合度高，还用于构建线团-球转变相图。
中文摘要 AI 辅助
我们旨在推导相互作用自避行走（Interacting Self-Avoiding Walk，ISAW）的回转半径R的有限尺寸概率分布的显式表达式，该表达式是链长N和单体-单体相互作用能ε的函数。我们首先推导非相互作用自避行走的显式自由能表达式，引入新的自然尺度变量t=ρ^g，其表示为密度ρ的幂次。随后，借助Lhuillier、Victor及其合作者提出的累积量展开方法，我们将该表达式扩展至相互作用情形，涵盖线团-球转变以及有限尺寸标度修正，包括熵贡献和表面能贡献。利用贝叶斯推断估计模型参数，我们将新自由能表达式确定的回转半径分布与三维ISAW的大量蒙特卡罗模拟结果进行对比，发现除了在Θ点附近（有限链长限制了可及的标度区域）外，二者吻合度极高。这验证了我们得到的自由能表达式，并作为应用，使我们能够构建线团-球转变的相图。
英文摘要
We aim at calculating an explicit expression for the finite-size probability distribution of the radius of gyration $R$ of an Interacting Self-Avoiding Walk (ISAW), as a function of chain length $N$ and monomer-monomer interaction energy $\varepsilon$. We first derive the explicit free energy expression for a non-interacting Self-Avoiding Walk, introducing a new natural scale variable $t = ρ^g$ expressed as a power of the density $ρ$. Then, thanks to a cumulant expansion approach introduced by Lhuillier, Victor and coworkers, we extend it to the interacting case, capturing both the coil-globule transition and finite-size corrections to scaling, including entropic and surface-energy contributions. The radius of gyration distribution determined by the new free energy expression is then compared, using Bayesian inference to estimate the model parameters, to results from extensive Monte Carlo simulations of three-dimensional ISAWs, showing excellent agreement except very close to the $Θ$-point, where finite chain lengths limit the accessible scaling regime. This validates our resulting free energy expression and allows us, as an application, to construct a phase diagram of the coil-globule transition.
