Fluctuations in Superdense and Supercritical Systems: A van der Waals Perspective

We present new insights into the van der Waals (vdW) model, focusing on two key aspects: 1) the concept of maximum possible density (MPD) n=1/b, and 2) the Fano factor’s role in analyzing supercritical fluids and also in determining limits on the applicability of the van der Waals approach. Our aim in these investigations is to bridge the complexity of statistical mechanics with practical applications, targeting researchers and engineers in thermodynamic modeling and supercritical fluid systems. Our MPD, derived from the vdW excluded volume parameter b, offers a theoretical framework to explore packing constraints imposed by molecular interactions. At n=1/b, we observe a vanishing Fano factor ω, signifying suppressed particle number fluctuations. This behavior highlights the reduced configurational complexity due to geometric constraints, with implications for dense granular systems and high-pressure materials design. We use an auxiliary temperature Tw that emerges as a critical diagnostic tool, reflecting the scaling of fluctuations and bridging microscopic interactions with macroscopic stability. In supercritical fluids, we show that the Fano factor provides a sensitive parameter to detect instability thresholds near the spinodal line, revealing the intricate transitions between liquid-like and gas-like states. Our findings underscore the importance of fluctuations in modeling the nuanced thermodynamics of confined fluids and industrial supercritical reservoirs. Our results advance the vdW framework by connecting fundamental thermodynamic principles to actionable insights for practical technologies, particularly in the optimization of supercritical extraction and drying processes. This dual emphasis enhances our understanding of complex systems, marking a meaningful contribution to both theoretical and applied domains. We will see that a putative solid phase is predicted by the van der Waals equation at high densities. Importantly enough, we find that the Fano factor is able to detect the limits of applicability for the van der Waals method.