Fisher’s scaling law is another important relation in the field of critical phenomena, named after the physicist Michael E. Fisher, who made significant contributions to the understanding of phase transitions and critical points. This scaling law relates the critical exponent (\gamma), which describes the divergence of the susceptibility near the critical point, to the critical exponents (\nu) and (\eta), which describe the behavior of the correlation length and the decay of correlations at the critical point, respectively.
Critical Exponents in Fisher’s Scaling Law
- (\nu) describes how the correlation length (\xi), the measure of how far microscopic interactions extend spatially in the system, diverges as the system approaches the critical point: (\xi \sim |T - T_c|^{-\nu}).
- (\gamma) characterizes the divergence of the susceptibility (e.g., magnetic susceptibility in a ferromagnet) near the critical point: (\chi \sim |T - T_c|^{-\gamma}).
- (\eta) describes the anomalous dimension of the order parameter field, altering the expected decay of the correlation function at the critical point from a simple exponential decay. Specifically, at the critical point, the correlation function (G(r)) between two points separated by distance (r) decays as (G(r) \sim r^{-d+2-\eta}) in (d) dimensions, where the typical expectation without critical fluctuations would be (r^{-d+2}) for a simple exponential decay.
Fisher’s Scaling Law
Fisher’s scaling law provides a relationship between these exponents:
[ \gamma = \nu(2 - \eta) ]
This equation indicates that the divergence of the susceptibility ((\gamma)) is directly related to the behavior of the correlation length ((\nu)) and the anomalous dimension ((\eta)), offering a deep insight into the interplay between spatial correlations in a system and its response to external fields near the critical point.
Implications and Significance
- Universality and Scaling: Fisher’s scaling law underscores the universality of critical phenomena, showing that the critical behavior of systems within the same universality class can be described by the same set of critical exponents, despite differences in their microscopic details. This law is a key part of the scaling hypothesis that aims to describe the singular behavior of thermodynamic quantities near critical points.
- Experimental and Theoretical Verification: Like other scaling laws, Fisher’s scaling law serves as a crucial benchmark for theoretical models of phase transitions. Agreement between experimentally measured exponents and those predicted by theory validates the universality class and the scaling hypothesis for the system under study.
- Renormalization Group (RG) Theory: Fisher’s scaling law is consistent with the predictions of RG theory, which explains how physical quantities scale near critical points and provides a framework for calculating critical exponents. RG theory has been instrumental in demonstrating the applicability of scaling laws across different physical systems.
Fisher’s scaling law, together with other scaling laws like those of Rushbrooke and Widom, forms an integral part of the theoretical framework that describes critical phenomena. These laws have facilitated a deeper understanding of phase transitions, enabling scientists to predict and categorize the behavior of a wide range of systems near criticality based on their symmetry properties, dimensionality, and dynamical rules.
η describes the anomalous dimension of the order parameter field, altering the expected decay of the correlation function at the critical point from a simple exponential decay.