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arXiv:cond-mat/0507633 · American Journal of Physics

Scaling laws at the critical point

S. Davatolhagh

Published Apr 14, 2006• 2 citations
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Abstract

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′.

DOI
10.1119/1.2173273

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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′.
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We discuss the universal scaling laws of order-parameter fluctuations in any system in which a second-order critical behavior can be identified. These scaling laws can be derived rigorously for equilibrium systems when combined with a finite-size scaling analysis. The relation between the order parameter, the criticality, and the scaling law of fluctuations has been established, and the connection between the scaling function and the critical exponents has been found. We give examples in out-of-equilibrium aggregation models such as the Smoluchowski kinetic equations, or at-equilibrium Ising and percolation models.
Universal scaling laws of fluctuations (the Delta\\DeltaDelta-scaling laws) can be derived for equilibrium and off-equilibrium systems when combined with the finite-size scaling analysis. In any system in which the second-order critical behavior can be identified, the relation between order parameter, criticality and scaling law of fluctuations has been established and the relation between the scaling function and the critical exponents has been found.
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Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.