AI 中文总结
该研究在热力学几何中引入标量曲率密度$\boldsymbol{\textit{R}}$作为新不变量，对比两种Ruppeiner度规的临界行为，发现其与标量曲率的标度规律差异，定义了两条新曲线，明确度规选择对相变描述的重要性。
AI 中文摘要
我们通过对标量曲率$R$进行分析，并引入标量曲率密度$\boldsymbol{\textit{R}} = \boldsymbol{\textit{|g|}}\boldsymbol{\textit{R}}$作为补充几何不变量，比较了在固定体积和固定粒子数条件下构建的两种Ruppeiner度规（$g_{_V}$和$g_{_N}$）。研究采用了三种物理真实度递增的流体模型：范德瓦尔斯模型、伦纳德-琼斯模型，以及通过多参数状态方程描述的氩模型，该模型可正确重现与伊辛普适类一致的非平均场临界行为。我们发现$R$和$\boldsymbol{\textit{R}}$呈现出不同的临界标度：$R \boldsymbol{\textit{t}}^{-d\nu}$由关联长度指数控制，而$\boldsymbol{\textit{R}} \boldsymbol{\textit{t}}^{-(1+\beta)}$仅与序参量指数$\beta$相关，该结果通过超标度关系和Rushbrooke关系解析得出，且与普适类无关。在远离临界点的气液共存曲线重构中，$N$-度规始终优于$V$-度规；由$R_{_V}$、$R_{_N}$、$\boldsymbol{\textit{R}}_{_V}$和$\boldsymbol{\textit{R}}_{_N}$的最小值定义的四条Widom线，呈现出在所有三个模型中均重复出现的特征四重结构。两种表示给出相同几何描述的轨迹定义了两个新对象：曲率相等曲线（CEC，$R_{_V} = R_{_N}$）和曲率密度相等曲线（CDEC，$\boldsymbol{\textit{R}}_{_V} = \boldsymbol{\textit{R}}_{_N}$），且在所有情况下CDEC包围的相图区域都显著更大。这些结果确立了$\boldsymbol{\textit{R}}$作为热力学几何中$R$的有意义补充，并强调了度规选择在相变和超临界行为描述中的非平凡作用。
英文摘要
We compare two Ruppeiner metrics constructed under fixed volume and fixed particle number conditions ($g_{_V}$ and $g_{_V}$) by analyzing the scalar curvature $R$ and introducing the scalar curvature density $\mathcal{R} = \sqrt{|g|}\,R$ as a complementary geometric invariant. Three fluid models of increasing physical realism are considered: van der Waals, Lennard-Jones, and argon described by a multiparameter equation of state that correctly reproduces non-mean-field critical behavior consistent with the Ising universality class. We find that $R$ and $\mathcal{R}$ exhibit distinct critical scaling: $R \sim t^{-dν}$ is governed by the correlation length exponent, whereas $\mathcal{R} \sim t^{-(1+β)}$ scales solely with the order parameter exponent $β$, a result that follows analytically from hyperscaling and the Rushbrooke relation independently of the universality class. The $N$-metric consistently outperforms the $V$-metric in reconstructing the vapor-liquid coexistence curve away from criticality, and the four Widom lines defined by the minima of $R_{_V}$, $R_{_N}$, $\mathcal{R}_{_V}$, and $\mathcal{R}_{_N}$ display a characteristic fourfold structure reproduced across all three models. The loci where both representations yield identical geometric descriptions define two new objects: the Curvature Equality Curve (CEC, $R_{_V} = R_{_N}$) and the Curvature-Density Equality Curve (CDEC, $\mathcal{R}_{_V} = \mathcal{R}_{_N}$), with the CDEC enclosing a substantially larger region of the phase diagram in all cases. These results establish $\mathcal{R}$ as a meaningful complement to $R$ in thermodynamic geometry and highlight the nontrivial role of metric choice in the description of phase transitions and supercritical behavior.
