Thermodynamic constraints and pseudotransition behavior in a one-dimensional water-like system

TL;DR

The paper analyzes a one-dimensional water-like lattice with van der Waals and hydrogen-bond interactions, allowing particle-number fluctuations via a chemical potential. Using a transfer-matrix approach in the grand-canonical ensemble, it derives the exact thermodynamics from the leading eigenvalue and identifies three ground-state phases, revealing pseudotransitions with sharp but analytic changes in thermodynamic quantities. It then shows how enforcing a fixed density through a Legendre transform smooths these anomalies, yielding density-dependent residual entropy and a finite, non-divergent jump in the specific heat. The results demonstrate the ensemble dependence of pseudotransitions in a simple 1D model and provide insight into the thermodynamics of confined water-like systems, where similar quasi-critical behaviors may arise without true phase transitions.

Abstract

We investigate a one-dimensional water-like lattice model with Van der Waals and hydrogen-bond interactions, allowing for particle number fluctuations through a chemical potential. The model, defined on a chain with periodic boundary conditions, exhibits three ground-state phases: gas, bonded liquid, and dense liquid, separated by sharp phase boundaries in the chemical potential and temperature plane. Using the transfer matrix method, we derive exact analytical results within the grand-canonical ensemble and examine the finite-temperature behavior. The system exhibits clear pseudotransition features, including sharp but analytic changes in entropy, density, and internal energy, along with finite peaks in specific heat and correlation length. To assess the role of thermodynamic constraints, we consider the behavior under fixed density through a Legendre transformation. This constrained analysis reveals smoother anomalies, such as entropy kinks and finite jumps in specific heat, contrasting with the sharper grand-canonical signatures. These results underscore the ensemble dependence of pseudotransitions and show how statistical constraints modulate critical-like behavior. We also verify that the residual entropy continuity criterion holds in the grand-canonical ensemble but is violated when the system is constrained. Our findings illustrate how even a simple one-dimensional model can mimic water-like thermodynamic anomalies.

Figures (9)

• Figure 1: Phase diagram $\epsilon_{v}/\epsilon_{h}$ against $\mu/\epsilon_{h}$. (Left) The background density plot corresponds to the entropy ${\cal S}/k_{B}$ for a low temperature value $k_{B}T/\epsilon_{h}=0.1$. (Right) The density $\rho$ plot describes the molecule density at $k_{B}T/\epsilon_{h}=0.1$.
• Figure 2: Pseudo-critical temperature $k_{B}T_{p}/\epsilon_{h}$ as a function of the chemical potential $\mu/\epsilon_{h}$, calculated using the expression given in \ref{['eq:Tp-mu']}.
• Figure 3: (a) Density as a function of chemical potential, assuming $\epsilon_{v}/\epsilon_{h}=1$. (b) Density as a function of chemical potential, for $\epsilon_{v}/\epsilon_{h}=0.5$ . (c) Entropy as a function of chemical potential, for fixed $\epsilon_{v}/\epsilon_{h}=1$. (d) Entropy as a function of chemical potential, considering $\epsilon_{v}/\epsilon_{h}=0.5$.
• Figure 4: (a) Density as a function of temperature, assuming $\epsilon_{v}/\epsilon_{h}=1$. (b) Entropy as a function of temperature, for the same set of parameters and conditions as in panel (a). (c) Correlation length $\xi$ as a function of temperature, under the same conditions as in panel (a). (d) Specific heat $C$ as a function of temperature, using the same set of parameters as in the previous panels.
• Figure 5: Phase diagram $\epsilon_{v}/\epsilon_{h}$ versus $\rho$. The dotted line corresponds to the DF phase, the short-dashed line represents the G phase, and the long-dashed line denotes the BF phase. (Left) The background density plot shows the entropy ${\cal S}/k_{B}$ for $k_{B}T/\epsilon_{h}=0.01$. (Right) The background shows $\mu/\epsilon_{h}$ for the same temperature.