Eigen Microstate Condensation and Critical Phenomena in the Lennard-Jones Fluid

TL;DR

This work tackles the challenge of locating the liquid-gas critical point and exponents for the Lennard-Jones fluid by applying eigen microstate theory (EMT) in the canonical ensemble. By constructing an ensemble matrix from density fluctuations and performing finite-size scaling of the leading eigen microstate amplitudes $\lambda_I$, the authors determine $T_c$ and $\rho_c$ simultaneously and extract critical exponents $\beta$ and $\nu$ that agree with the 3D Ising universality class. EMT also reveals mesoscopic spatial patterns through the leading eigen microstates, providing a structural picture of the critical region that raw microstates do not display. The approach avoids reliance on higher-order moments or Binder cumulants and offers a flexible, parameter-free framework for studying critical phenomena in complex fluids and potentially in confined or mixed systems.

Abstract

Despite extensive study of the liquid-gas phase transition, accurately determining the critical point and the critical exponents in fluid systems through direct simulation remains a challenge. We employ the eigen microstate theory (EMT) to investigate the liquid-gas continuous phase transition in the Lennard-Jones (LJ) fluid within the canonical ensemble. In EMT, the probability amplitudes of eigen microstates serve as the order parameter. Using finite-size scaling of probability amplitudes, we simultaneously determine the critical temperature, $T_c = 1.188(2)$, and critical density, $ρ_c = 0.320(4)$. Furturemore, we obtain critical exponents of the LJ fluid, $β= 0.32(2)$ and $ν= 0.64(3)$, which demonstrate a great agreement with the Ising universality class. This method also reveals the mesoscopic structure of the emergent phase, characterizing the three-dimensional (3D) spatial configuration of the fluid in the critical region. This work also confirms the finite-size scaling behavior of the probability amplitudes of the eigen microstates in the critical region. The EMT provides a powerful tool for studying the critical phenomena of complex fluid system.

Figures (7)

• Figure 1: The first two probability amplitudes, $\lambda_1$ and $\lambda_2$, near the critical region at the largest simulation scale $L_\sigma = 35$. The error bars are smaller than the symbol size.
• Figure 2: (a-e) Log-log plot of $\lambda_1$ against system size $L_\sigma$ at different target densities: (a) $\rho=0.312$, (b) $\rho=0.316$, (c) $\rho=0.320$, (d) $\rho=0.324$, (e) $\rho=0.328$. For each density, data are shown for a series of temperatures near the critical region. Across all densities, the double-logarithmic plots of $\lambda_1$ versus system size $L_\sigma$ exhibit the most robust linear behavior at $T=1.188$ (solid lines), compared to the clear deviations observed at other temperatures. (f) fixes the temperature at $T=1.188$ and compares different densities. At this stage, it is not possible to effectively distinguish the critical density.
• Figure 3: Determination of the critical point through finite-size scaling of probability amplitude ratios. (a-e) The ratio of the first two probability amplitudes $R(T, \rho, L_\sigma)= \lambda_2/\lambda_1$, as a function of temperature for different system sizes $L_\sigma$ at densities: (a) $\rho=0.312$, (b) $\rho=0.316$, (c) $\rho=0.320$, (d) $\rho=0.324$, (e) $\rho=0.328$. The curves for different $L_\sigma$ intersect at a common point only when both the density and temperature reach their critical values. (f) The ratio $R(T, \rho, L_\sigma)$ as a function at the fixed temperature $T=1.188$. The curves for different $L_\sigma$ intersect precisely at $\rho=0.320$, confirming the critical density $\rho_c=0.320(4)$. These consistent crossing behaviors collectively determine the critical point at $T_c=1.188(2), \rho_c=0.320(4)$.
• Figure 4: (a) According to Eq. (\ref{['eq:nu']}), the slope of solid black line corresponds to $1/\nu$, where $\nu$ is the critical exponent associated with the correlation length. The fitted slope gives $\nu = 0.64(3)$. (b) Finite-size scaling form of $R(T, \rho_c, L_\sigma)$ when $\nu=0.64$.
• Figure 5: Scaling behavior of the first two probability amplitude near the critical point across different system sizes. (a) The first probability amplitude $\lambda_1$ and (b) its finite-size scaling form by $\nu=0.64$ and $\beta / \nu = 0.513$. (c) Shows the second probability amplitude $\lambda_2$ and (d) its finite-size scaling form by the same critical exponents.