Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.
Critical phenomena, characterized by abrupt changes in macroscopic properties near phase transition points, exhibit universal scaling behavior independent of microscopic details. This paper explores the hypothesis that entropy serves as a fundamental universal scaling law governing these phenomena. We review the theoretical foundations of critical phenomena, including Landau theory and the renormalization group, highlighting the role of critical exponents and scaling functions. We then propose a framework where the configurational entropy, entanglement entropy, and other entropic measures exhibit universal scaling, linking them directly to the underlying critical exponents. Our methodology involves analyzing how entropy behaves under coarse-graining transformations near critical points, identifying a potential entropic fixed point in the renormalization group flow. We present theoretical arguments suggesting that the information content, as quantified by various entropy definitions, reflects the diverging correlation length and the emergent scale invariance at criticality. The discussion extends to how different entropic forms, such as Rényi and Tsallis entropies, might provide a more generalized framework for understanding critical behavior in complex and non-extensive systems. By demonstrating entropy's role as a universal scaling law, this work offers a unifying perspective on critical phenomena, deepening our understanding of phase transitions and the emergence of macroscopic order from microscopic fluctuations.
Critical phenomena, characterized by abrupt changes in the macroscopic properties of a system, exhibit a profound degree of universality, meaning that vastly different physical systems can display identical behavior near their critical points. This universality is typically understood through the lens of scaling laws and the Renormalization Group theory, which classify critical points into distinct universality classes. This paper proposes and explores the "Entropic Law of Critical Universality," an overarching principle positing that not only do thermodynamic quantities scale universally, but the fundamental information-theoretic measures of a system, particularly entropy in its various forms (thermodynamic entropy, entanglement entropy, Renyi entropy), also conform to universal scaling behaviors that uniquely characterize these universality classes. We review how entanglement entropy exhibits universal logarithmic corrections at quantum critical points, linked to the central charge of underlying conformal field theories. We also discuss how specific entropy behaves non-monotonically near quantum chromodynamics critical points and how entropic measures can identify phase transitions and their classes in various condensed matter systems, including graphene and spin models. The paper synthesizes theoretical frameworks and empirical evidence to demonstrate that entropy serves as a powerful and often more direct probe of the universal properties underlying critical phenomena, offering a comprehensive perspective on the shared information landscape of diverse critical systems.
Phase transitions represent fundamental transformations of matter, characterized by abrupt changes in macroscopic properties and often associated with singular behavior at critical points. While equilibrium aspects of phase transitions, particularly universal critical exponents and scaling laws, are well-understood, the non-equilibrium processes occurring near criticality, especially the production of entropy, remain a rich area of investigation. This paper explores the intricate relationship between critical entropy production and the emergence of scaling laws in systems undergoing phase transitions. We delve into how the rate of entropy production diverges or exhibits anomalous scaling near critical points, reflecting the system's attempts to dissipate energy and respond to external perturbations as it approaches a state of macroscopic change. By integrating concepts from non-equilibrium thermodynamics, statistical mechanics, and renormalization group theory, we aim to elucidate the underlying mechanisms that govern entropy generation in these critical regimes. We discuss theoretical frameworks that predict universal scaling behaviors for entropy production rates, analogous to those observed for equilibrium order parameters and susceptibilities. The implications of these scaling laws extend beyond theoretical physics, offering insights into complex systems, far-from-equilibrium dynamics, and the efficiency of energy conversion processes in critical environments. Our analysis highlights the critical entropy production as a unique signature of non-equilibrium dynamics during phase transitions, providing a deeper understanding of the interplay between fluctuations, dissipation, and universality near points of instability.
Abstract This chapter analyzes systems with emergent scale invariance -- fractal, self-similar behavior -- by developing the renormalization group. The renormalization group is an amazing abstraction. It describes the flow of the laws governing the system as one coarse-grains -- blurring out the short-distance or short-time details. In the huge space of possible systems (experimental and theoretical), a fixed point of the renormalization group will be the same after blurring and shrinking -- implying emergent scale invariance, a fractal self-similarity. The points which flow into the fixed point share its properties under rescaling -- implying universality, with behavior shared by theory and a wide variety of different experimental systems. The renormalization group also predicts universal power laws and universal functions, describing all behavior on long length and/or time scales. Exercises explore applications to the Ising model, the onset of lasing, superconductors, the onset of chaos, percolation, crackling noise and avalanches, earthquakes, random walks and diffusion, chemical reaction rate theory, and extreme value statistics.
The critical behavior of a system near a continuous phase transition is commonly characterized by power laws and a set of universal critical exponents. Yet universality at phase transitions has even more fundamental manifestations. It dictates that all observables follow universal scaling functions, which may extend beyond simple power laws. Furthermore, microscopic details are captured entirely by only two system-specific scale factors, that rescale any measured quantity onto its universal form. In this work, we report the experimental validation of this prediction in ultracold lattice Bose gases undergoing the superfluid transition. We extract the scale factors by collapsing the order-parameter cumulants measured at varying entropies. We show that these factors specify scaling functions governing both the order-parameter distribution and two-point correlations. By measuring these scaling functions, we also determine three critical exponents β, γ and ν. Our results provide an experimental demonstration that universal scaling is simultaneously obeyed by multiple observables in a quantum many-body system.
Overview As discussed in Chapter 16 , when we approach the critical temperature of a phase coexistence line, several fascinating things start to happen which together have come to be known as “critical phenomena.” It is observed that the spatial correlation between relevant order parameters like density, magnetization, etc. becomes infinitely large. This is reflected in the divergence of inverse susceptibility of the respective order parameters. Landau’s theory explains this in terms of the flattening of the free energy surface but fails to unravel many of the other mysteries of critical phenomena that have kept scientists engaged now for about a century. Critical phenomena are characterized by certain singular properties, like the universality of the critical exponents. Several aspects of the critical phenomena are discussed in this chapter with an aim to maintain continuity with earlier chapters. A simple theory (like that of Landau) cannot explain many of the observed features and this problem remained unsolved for a long time until K. G. Wilson could develop a sophisticated theory that explained almost everything. At the core of Wilson’s method is the renormalization group technique. Physically, this technique employs the expected invariance of the Hamiltonian due to a transformation that removes local interactions and replaces them by effective longer range interactions. Because of the existence of long range correlations, the transformation is expected to converge to a fixed point. The procedure is quite mathematical and is the only robust method to obtain the critical exponents. Here we discuss the method briefly, as an introduction to the subject.
Topological phase transitions occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and driving, and hence require a variety of techniques and concepts to describe their topological properties. For instance, topology may be accessed from single-particle Bloch wave functions, Green's functions, or many-body wave functions. We demonstrate that despite this diversity, all topological phase transitions display a universal feature: namely, a divergence of the curvature function that composes the topological invariant at the critical point. This feature can be exploited via a renormalization-group–like methodology to describe topological phase transitions. This approach serves to extend notions of correlation function, critical exponents, scaling laws and universality classes used in Landau theory to characterize topological phase transitions in a unified manner.