Water and hydrogen are examples of substances proposed to exhibit a liquid-liquid critical point (LLCP) at conditions where nuclear quantum effects are relevant. The LLCP is usually accompanied by lines of maxima in density ρ and thermodynamic response functions, such as isothermal compressibility κ T and isobaric heat capacity C P , in the supercritical region of the P-T plane. In the case of water, the ρand κ T-maxima lines can be accessed in experiments, while, instead, the LLCP has not been observed due to rapid crystallization. In this work, we study the nuclear quantum effects on a monatomic liquid that exhibits waterlike anomalous properties and a LLCP. By performing path-integral Monte Carlo simulations with different values of the Planck's constant h, we are able to explore how the location of the LLCP in the P-T plane and, in particular, the maxima lines in the supercritical region, shift as the system evolves from classical, h = 0, to quantum, h > 0. We find that as the quantum nature of the liquid (as quantified by h) increases and the atoms in the liquid become more delocalized, the LLCP shifts towards higher pressures and lower temperatures while the LLCP volume remains constant. Similar shifts (towards higher pressures and lower temperatures) are found in the case of the C P-and κ T-maxima lines. Instead, the ρ-maxima line extends towards higher temperatures and expands over a wider pressure interval as the liquid becomes more quantum. It follows that the nuclear quantum effects on the location of the LLCP may be estimated from the shift in C P-and κ T-maxima lines but not on measurements of the ρ-maxima line. Interestingly, nuclear quantum effects considerably alter the slope of the liquid-liquid coexistence line and C P-maxima line in the P-T plane while the slope of the κ T-maxima line along the Widom line is barely affected. We discuss briefly the implications of our results to the case of H 2 O/D 2 O.

Using the potential energy landscape (PEL) formalism and molecular dynamics simulations, we investigate a phase transformation between two amorphous solid states of gallium, namely, a low-density amorphous solid (LDA) and a high-density amorphous solid (HDA), and compare with its equilibrium counterpart, the liquid–liquid phase transition (LLPT). It is found that on the PEL, the signatures of the out-of-equilibrium LDA–HDA transition are reminiscent of those of the equilibrium LLPT in terms of pressure, inherent structure pressure, inherent structure energy, and shape function, indicating that the LDA–HDA transformation is a first-order-like transition. However, differences are also found between the out-of-equilibrium phase transition and the equilibrium one, for example, the path from LDA to HDA on the PEL cannot be accessed by the path from LDL to HDL. Our results also suggest that the signatures of the out-of-equilibrium transition in gallium are rather general features of systems with an accessible LLPT—not only systems with pairwise interactions but also those with many-body interactions. This finding is of crucial importance for obtaining a deeper understanding of the nature of transitions in the polyamorphic family.