Self-Similar Bridge between Regular and Critical Regions

TL;DR

The paper shows that regular and critical regions, typically treated as distinct, are connected through self-similar approximation theory. By embedding finite-sum expansions into a self-similar cascade and constructing factor approximants, it is possible to extrapolate from the regular region to the critical point and vice versa, predicting critical points and exponents from limited information. The authors illustrate this with a hard-disk equation of state, where the regular-region virial expansion yields a predicted critical packing $x_c$ and exponent $\alpha$ in agreement with numerical results, and with systems displaying discrete scale invariance, where log-periodic oscillations in the critical region can be extrapolated toward the regular region. Overall, the approach provides a unified, quantitative framework for bridging diverse critical phenomena and reveals that log-periodic structures can persist far from criticality, with implications for earthquakes, material fracture, and financial crashes.

Abstract

In statistical and nonlinear systems, two qualitatively distinct parameter regions are typically identified: the regular region, characterized by smooth behavior of key quantities, and the critical region, where these quantities exhibit singularities or strong fluctuations. Due to their starkly different properties, these regions are often perceived as being weakly related, if at all. However, we demonstrate that these regions are intimately connected, a relationship that can be explicitly revealed using self-similar approximation theory. This framework enables the prediction of observable quantities near the critical point based on information from the regular region and vice versa. Remarkably, the method relies solely on asymptotic expansions with respect to a parameter, regardless of whether the expansion originates in the regular or critical region. The mathematical principles of self-similar theory remain consistent across both cases. We illustrate this connection by extrapolating from the regular region to predict the existence, location, and critical indices of a critical point of an equation of state for a statistical system, even when no direct information about the critical region is available. Conversely, we explore extrapolation from the critical to the regular region in systems with discrete scale invariance, where log-periodic oscillations in observables introduce additional complexity. Our findings provide insights and solutions applicable to diverse phenomena, including material fracture, stock market crashes, and earthquake forecasting.

Figures (12)

• Figure 1: Functions $f(x)$ (solid line) and $f_0(x)$ (dashed-dotted line) for the fixed parameters $\alpha=1$, $a=-3$, $b=-1$, $\varphi=0.1$, and $\omega$ being varied: (a) $\omega=10$; (b) $\omega=50$;
• Figure 2: Functions $f(x)$ (solid blue line) for $\varphi=0.1$, $f(x)$ (dashed-dotted green line) for $\varphi=10$, and function $f_0(x)$ (dashed red line) are presented for the fixed parameters $\alpha=1$, $\beta=-1$, $a=-3$, $b=-1$, and $\omega=10$. Note that $f_0(x)$ does not depend on $\varphi$.
• Figure 3: Functions $f(x)$ (oscillating solid blue line) and $f_0(x)$ (monotonic solid red line) for the fixed parameters $a=-5$, $b=-2$, $\omega=10$, $\varphi=0.1$, and $\alpha$ being varied: (a) $\alpha=1$; (b) $\alpha=3$;
• Figure 4: Functions $f(x)$ (solid blue line) and $f_0(x)$ (dashed-dotted red line) for the fixed parameters $\alpha=1$, $b=-1$, $\omega=25$, $\varphi=5$, and $a$ being varied: (a) $a=-3$; (b) $a=-5$;
• Figure 5: Functions $f(x)$ (solid blue line) and $f_0(x)$ (dashed-dotted red line) for the fixed parameters $\alpha=1$, $a=-3$, $\omega=10$, $\varphi=1$, and $b$ being varied: (a) $b=-1$; (b) $b=-2$;