AI 中文总结
该研究为真实流体引入统一熵框架，用几何热力学通过标量曲率探索微观作用，揭示临界现象等特征，引入无量纲临界振幅比，还用贝叶斯推理和MCMC方法重构零曲率曲线，证明GTD流形能编码热力学模型信息。
AI 中文摘要
我们为真实流体引入了一个统一的熵框架，该框架在共同的热力学描述中涵盖了范德瓦尔斯、贝塞罗、雷德利希 - 邝和彭 - 罗宾逊状态方程。然后通过几何热力学（GTD）利用平衡流形的标量曲率\(\mathcal{R}\)探索相应的微观相互作用。我们表明曲率奇点准确地再现了宏观临界现象，而曲率\(\mathcal{R}=0\)确定了分子间吸引力和排斥力有效平衡的特定热力学状态。此外，我们引入了一组无量纲临界振幅比\(Q^i_{j}\)，揭示了临界区域的普遍几何特征。尽管单个临界振幅对系统大小呈现对数依赖性，但这些不变比根据临界强度对不同分子种类进行组织并编码普遍缩放特征。最后，采用贝叶斯推理和马尔可夫链蒙特卡罗（MCMC）方法，我们对零曲率曲线进行了统计重构。后验分布支持几何缩放行为的一致性，表明GTD流形编码了有关基础热力学模型的重要信息。
英文摘要
We introduce a unified entropic framework for real fluids that encompasses the van der Waals, Berthelot, Redlich Kwong, and Peng Robinson equations of state within a common thermodynamic description. The corresponding microscopic interactions are then explored using Geometrothermodynamics, GTD, through the scalar curvature $mathcal{R}$ of the equilibrium manifold. We show that curvature singularities accurately reproduce macroscopic critical phenomena, while vanishing curvature $\mathcal{R}=0$ identifies specific thermodynamic states where attractive and repulsive intermolecular forces effectively balance. Furthermore, we introduce a set of dimensionless critical-amplitude ratios $Q^i_{j}$, which reveal universal geometric features of the critical regime. Although individual critical amplitudes exhibit a logarithmic dependence on the system size, these invariant ratios organize different molecular species according to the strength of criticality and encode universal scaling features, suggesting their potential as robust classification parameters. Finally, employing Bayesian inference and Markov Chain Monte Carlo, MCMC methods, we statistically reconstruct the zero-curvature curves. The posterior distributions support the consistency of the geometric scaling behavior, demonstrating that the GTD manifold encodes non-trivial information about the underlying thermodynamical models.
