Widom's scaling law

Widom’s scaling law is a fundamental relation in the theory of critical phenomena that connects several critical exponents which characterize the behavior of physical systems near phase transitions. This scaling law is named after Benjamin Widom, a chemist who made significant contributions to the understanding of critical phenomena and phase transitions during the 1960s. Widom’s scaling law provides a link between the critical exponents (\beta), (\gamma), and (\delta), which describe how the order parameter, susceptibility, and response to an external field, respectively, behave near the critical point.

Critical Exponents Involved in Widom’s Scaling Law

• (\beta) describes the behavior of the order parameter (e.g., magnetization in a ferromagnetic system) as the critical point is approached from below: (M \sim (T_c - T)^\beta) for (T < T_c).
• (\gamma) characterizes the divergence of the susceptibility (e.g., magnetic susceptibility) near the critical point: (\chi \sim |T - T_c|^{-\gamma}).
• (\delta) describes how the order parameter responds to an external field (h) at the critical temperature: (M \sim h^{1/\delta}) for (T = T_c).

Widom’s Scaling Law

Widom’s scaling law states that the critical exponents (\beta), (\gamma), and (\delta) are not independent but are related by the equation:

[ \gamma = \beta(\delta - 1) ]

This relationship arises from the assumption of a homogeneous and universal form of the free energy function near the critical point and is a manifestation of the scale invariance that characterizes critical phenomena.

Implications and Significance

• Universality: Widom’s scaling law reinforces the concept of universality in critical phenomena, indicating that systems belonging to the same universality class (defined by their dimensionality and symmetry properties) will share the same set of critical exponents, despite possibly having different microscopic details.
• Experimental and Theoretical Verification: Like other scaling laws, Widom’s scaling law provides a testable prediction that can be used to verify theoretical models of critical phenomena. Experimental measurements of the critical exponents that satisfy this scaling law lend support to the universality hypothesis and the validity of the models used to describe the system.
• Scaling Functions: The scaling law is part of a broader framework that includes scaling functions, which describe how physical quantities depend on the temperature and external fields near the critical point. These functions further elucidate the behavior of systems undergoing phase transitions.

Broader Context

Widom’s scaling law, along with other scaling laws like Rushbrooke’s equality and Fisher’s, form the cornerstone of the modern understanding of phase transitions and critical phenomena. These laws, derived and validated within the framework of the Renormalization Group (RG) theory, have profound implications for statistical physics, condensed matter physics, and materials science, providing a unified approach to studying systems as diverse as magnetic materials, liquid-gas transitions, and binary fluid mixtures near their critical points.