Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Critical Points and Scaling Laws for Finite Systems

Y. Imry, David J. Bergman

Published Apr 1, 1971• 82 citations
Publisher

Abstract

We describe a new approach to the understanding of the thermodynamic properties of a physical system near a critical point. The critical point is characterized by the peculiar dependence of various quantities on the size of the system, rather than by the mathematical singularities which appear in an infinite system. We indicate a very general method of obtaining scaling laws for the critical behavior of any thermodynamic system by considering its thermodynamic fluctuations. As an example, a new scaling law is obtained for the susceptibility at a gas-liquid critical point.

DOI
10.1103/physreva.3.1416

Related papers

Universality and scaling in critical behavior

Gloria B. Lubkin

The behavior of systems near critical points is a long-standing puzzle in statistical mechanics and solid-state physics. Experimentally this behavior is characterized by divergences in a variety of thermodynamic quantities as powers of the deviation from criticality. These divergences are described by critical indices, which are the exponents in power laws; for example, the susceptibility of a ferromagnet diverges like (T−Tc)γ as T approaches Tc.

Physics Today•Mar 1, 1972•4 citations

Critical Entropy Production and Scaling Laws in Phase Transitions

Revista, Zen, PHYSICS, 10

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.

•Nov 28, 2025•0 citations

Static Phenomena Near Critical Points: Theory and Experiment

Leo P. Kadanoff, W. Götze, D.G. Hamblen +7 more

This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.

Reviews of Modern Physics•Apr 1, 1967•1,615 citations

Microcanonical analysis of small systems

Michel Pleimling, H. Behringer

The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.

Phase Transitions•Sep 1, 2005•18 citations

Universal scaling laws of fluctuations in finite systems

Robert S. Botet, Marek Płoszajczak

Universal scaling laws of fluctuations (the Δ-scaling laws) occur for any system in which second-order critical behaviour can be identified. This can be established rigorously for at-equilibrium systems when combined withfinite-size scaling analysis. For dynamic critical systems, the relations between the order parameter, the criticality and the scaling law of fluctuations can be proven numerically. Moreover, one can find general relation between the form of the scaling function and the critical exponents of the corresponding infinite system. The case of the first-order phase transition is currently under investigation. There is no background theory for precise finite-size scaling analysis, and the results are numerical. Systematic data on the q-state 2-dimensional Potts model with q > 4, shows that at the critical point, the order-parameter distribution can be splitted into two peaks - characteristic of the two coexistent phases -, each of them exhibiting a particular Δ-scaling. Conjecture is done about general occurence of this feature.

Revista Mexicana de Física•Jun 1, 2003•0 citations

Scaling laws at the critical point

S. Davatolhagh

There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.

arXiv:cond-mat/0507633•American Journal of Physics•Apr 14, 2006•2 citations
We describe a new approach to the understanding of the thermodynamic properties of a physical system near a critical point. The critical point is characterized by the peculiar dependence of various quantities on the size of the system, rather than by the mathematical singularities which appear in an infinite system. We indicate a very general method of obtaining scaling laws for the critical behavior of any thermodynamic system by considering its thermodynamic fluctuations. As an example, a new scaling law is obtained for the susceptibility at a gas-liquid critical point.
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.
This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
Universal scaling laws of fluctuations (the Δ-scaling laws) occur for any system in which second-order critical behaviour can be identified. This can be established rigorously for at-equilibrium systems when combined withfinite-size scaling analysis. For dynamic critical systems, the relations between the order parameter, the criticality and the scaling law of fluctuations can be proven numerically. Moreover, one can find general relation between the form of the scaling function and the critical exponents of the corresponding infinite system. The case of the first-order phase transition is currently under investigation. There is no background theory for precise finite-size scaling analysis, and the results are numerical. Systematic data on the q-state 2-dimensional Potts model with q > 4, shows that at the critical point, the order-parameter distribution can be splitted into two peaks - characteristic of the two coexistent phases -, each of them exhibiting a particular Δ-scaling. Conjecture is done about general occurence of this feature.
There are two independent critical exponents that describe the behavior of systems near their critical point. However, at the critical point only the exponent η, which describes the decay of the correlation function, is usually discussed. We emphasize that there is a second independent exponent η′ that describes the decay of the fourth-order correlation function. The exponent η′ is related to the exponents determining the behavior of thermodynamic functions near criticality via a fluctuation-response equation for the specific heat. We also discuss a scaling law for η′.