Quantum Fluids in Thermodynamic Geometry

TL;DR

This work applies Thermodynamic Geometry (TG) to quantum fluids, focusing on a quantum hard-sphere fluid (QHS) with Helmholtz free energy from Path-Integral Monte Carlo data and a semi-classical square-well fluid (QSW) that pairs a quantum hard-sphere repulsion with a classical attractive square-well term. Using TG metrics derived from the Helmholtz free energy, the reduced curvature $R^{*}$ is analyzed across parameter spaces defined by $\lambda_B^{*}$ and $\eta$ (and by $\lambda^*$ and $\eta$ for the SW case) to reveal how quantum effects modify effective interactions and supercritical boundaries, via curvature sign changes and Widom-line behavior. For QHS, quantum corrections partially reverse the classical BPH anomaly in a central region of the $(\lambda_B^{*},\eta)$ plane, while at low and high densities the curvature reverts to the anomalous sign; for QSW, semi-classical corrections lower the critical point relative to the classical SW and yield Widom lines whose length depends on the SW range $\lambda^*$, with near-critical lines nearly overlapping between the two models. Overall, TG proves to be a sensitive diagnostic of quantum modifications to interparticle interactions and phase-like structure in quantum fluids, offering quantitative insights for systems such as hydrogen and helium and guiding future refinements of EOS and TG analyses.

Abstract

In this work, the Thermodynamic Geometry (TG) of quantum fluids (QF) is analyzed. We present results for two models. The first one is a quantum hard-sphere fluid (QHS) whose Helmholtz free energy is obtained from Path Integrals Monte Carlo simulations (PIMC). It is found that due to quantum contributions in the thermodynamic potential, the anomaly found in TG for the classical hard-sphere fluid related to the sign of the scalar curvature, is now avoided in a considerable region of the thermodynamic space. The second model is a semi-classical square-well fluid (QSW), described by a quantum hard-sphere repulsive interaction coupled with a classical attractive square-well contribution. Behavior of the semi-classical curvature scalar as a function of the thermal de Broglie wavelength $λ_B$ is analyzed for several attractive-potential ranges, and description of the semi-classical R-Widom lines defined by the maxima of the curvature scalar, are also obtained and compared with classical results for different square-well ranges.

Figures (8)

• Figure 1: Reduced curvature for the classical and quantum hard sphere systems for different isotherms in the ($\lambda_{B}^*$,$\eta$) representation. The solid line for $\lambda_{B}^* \to 0$ represents the behavior for a classical HS fluid; as $\lambda_{B}^*$ increases (temperature decreases), the quantum contribution becomes more important in the region where the curvature becomes negative, crossing the $R=0$ line, reversing the known anomaly for the classical HS curvature.
• Figure 2: Reduced curvature for the QHS system in the ($\lambda_{B}^*$,$\eta$) representation for different values of $\eta$. The crossing through zero of the quantum curvature is also favored by increasing density. This behavior for both $\lambda_B^*$ and $\eta$ is better depicted in the three dimensional representation of $R^*$.
• Figure 3: 3D reduced curvature scalar for the QHS system (orange) as a function of $\lambda^*_B$ and $\eta$. The crossing with $R^* = 0$ plane (blue) is also presented. The main feature of this plot is the existence of a region of $(\lambda^*_B,\eta)$ for which the interpretation of the sign of the curvature scalar holds, located in the strip around the middle section of $\eta$.
• Figure 4: Contour plot for the curvature scalar for the QHS system in the $(\lambda_B^*,\eta)$ space. The region where $R^*<0$ (i.e., where the BPH anomaly is reverted) is located in the strip in the middle of the plot limited by the lower and upper contour-lines (purple section), whereas for high values of $\lambda_B^*$ and $\eta$, the value of $R^*$ becomes positive once again. A region of minimum values for the curvature also appears in the upper-middle section of the plot (violet section), which can be interpreted as the region of maximal repulsion of the QHS potential.
• Figure 5: Isotherms of the curvature $R$ of the semi-classical and classical SW fluid for a potential range of $\lambda^* = 1.5$ in the supercritical region are presented, given by solid and dashed lines, respectively. To have a better insight on the behavior of both systems, $T^*$ were chosen to be at $1.25 \ T_{\text{crit}}^*$ (bottom), $2 \ T_{\text{crit}}^*$ (middle) and $3 \ T_{\text{crit}}^*$ (upper), namely at the same percentage away from its corresponding critical value.