Phase transition revealed by eigen microstate entropy

TL;DR

This work introduces eigen microstate entropy $S_{ ext{EM}}$, a data-driven entropy defined as $S_{ ext{EM}}=-\sum_I p_I\ln p_I$ with $p_I=\sigma_I^2$ from the SVD of ensemble data, enabling a dimensionality-reduced, interpretable description of complex systems. It establishes finite-size scaling for $S_{ ext{EM}}$, linking regular and singular contributions and deriving universal scaling forms for critical behavior. Applying $S_{ ext{EM}}$ to equilibrium models (mean spherical, Ising, frustrated Ising, Potts) validates its ability to capture universality classes and critical exponents through scaling of $S_{ ext{EM}}$ and its derivative. The method is then demonstrated on non-equilibrium real-world systems: liquid-liquid phase separation in living cells, where $S_{ ext{EM}}$ shows a precursor rise preceding condensate formation, and El Niño events, where entropy increases months before onset, highlighting $S_{ ext{EM}}$ as a versatile precursor detector and interpretive tool for phase transitions in complex systems.

Abstract

We introduce the eigen microstate entropy ($S_{\text{EM}}$), a novel metric of complexity derived from the probabilities of statistically independent eigen microstates. After establishing its scaling behavior in equilibrium systems and demonstrating its utility in critical phenomena (mean spherical, Ising, and Potts models), we apply $S_{\text{EM}}$ to non-equilibrium complex systems. Our analysis reveals a consistent precursor signal: a significant increase in $S_{\text{EM}}$ precedes major phase transitions. Specifically, we observe this entropy rise before biomolecular condensate formation in liquid-liquid phase separation in living cells and months ahead of El Niño events. These findings position $S_{\text{EM}}$ as a general framework for detecting and interpreting phase transitions in non-equilibrium systems.

Figures (4)

• Figure 1: Critical behaviors of $S_{\text{EM}}$ and its temperature derivative in Ising models. (a) Scaling relation $S_{\text{EM}} \approx S_{\text{ns}}(L) \propto 1.771\ln{L}$ at $T_c$ in the 2D Ising model. Red dashed line represents a fit to the last six data points. (b) Data collapse of $\partial_t{S}_{\text{EM}} L^{(2\beta-1)/\nu}$ versus $tL^{1/\nu}$ in the 2D Ising model with critical exponents $\nu=1$, $\beta=1/8$. (c) Hidden scaling behavior of ${S}_{\text{EM}}$ versus $L / \xi$ in the 1D Ising model. (d) Data collapse of $\tilde{g} \partial_t{S}_{\text{EM}}$ versus $L / \xi$ in the 1D Ising model. Solid lines are guides for the eye; error bars are smaller than symbol sizes.
• Figure 2: Eigen microstates and $S_{\text{EM}}$ in frustrated Ising models. (a) Data collapse of $\partial_t{S}_{\text{EM}} L^{(2\beta-1)/\nu}$ versus $tL^{1/\nu}$ with critical exponents $\nu=0.78$, $\beta=0.1$ for the frustrated Ising model with $\kappa = 0.8$. (b) Spatial patterns of the three dominant eigen microstates at high ($T=4$), critical ($T=1.568$), and low ($T=0.8$) temperatures for the frustrated Ising model with $\kappa = 0.8$. (c) Data collapse of $\partial_t{S}_{\text{EM}} L^{(2\beta-1)/\nu}$ versus $tL^{1/\nu}$ with critical exponents $\nu=1$, $\beta=0.125$ for the frustrated Ising model with $\kappa = 0.2$. (d) Spatial patterns of the three dominant eigen microstates at high ($T=4$), critical ($T=1.608$), and low ($T=0.8$) temperatures for the frustrated Ising model with $\kappa = 0.2$.
• Figure 3: The phase transition in LLPS is revealed by $S_{\text{EM}}$. (a) Imaging of a U2OS cell with diffusing G3BP1-EGFP. Bright spots correspond to diffusing G3BP1-EGFP molecules imaged at different time points. Camera exposure time is 200 ms. (b) $S_{\text{EM}}$ versus time in living cells (10 samples). The black line gives the mean over all samples, and its shaded region denotes the standard deviation. The inset figure presents the time derivative of entropy versus time. (c) Spatial pattern of the three largest eigen microstates at different times. (d) The probability weights of the three largest eigen microstates versus time.
• Figure 4: $S_{\text{EM}}$ serves as a precursor signal for the 1997/1998 El Niño event. (a) Temporal evolution of $S_{\text{EM}}$ (the red line) alongside the ONI (the black line) of the 1997/1998 El Niño event. Blue shading indicates El Niño periods. (b) Spatial pattern of the three largest eigen microstates at 10 months before (July 1996), 5 months before (December 1996), at the $S_{\text{EM}}$ peak (May 1997), and at the ONI peak (November 1997) of the 1997/1998 El Niño event. (c) The probability weights associated with the six leading eigenvalues as a function of time.