Scaling, fractal dynamics, and critical exponents: Application in a noninteger-dimensional Ising model.

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Mathematical Physics

Background:

• Correlation functions are crucial for analyzing complex systems, particularly in statistical mechanics.
• Fisher's autocorrelation function is key for understanding equilibrium second-order phase transitions but is limited to Euclidean dimensions.
• Recent work highlights the necessity of fractal analysis for correlation functions at critical temperatures (T=T_c).

Purpose of the Study:

• To investigate the interplay between scaling behavior, critical exponents, and fractal geometry in phase transitions.
• To develop a more comprehensive mathematical framework for correlation functions beyond Euclidean limitations.
• To obtain exact expressions for critical exponents and validate scaling relations.

Main Methods:

• Application of modern fractional differentials to derive an equation for the correlation function.
• Analysis of scaling behavior and critical exponents in the context of fractal geometry.
• Examination of the Rushbrooke scaling relation using results from the Ising model.

Main Results:

• An exact expression for the Fisher exponent (η) was derived.
• The proposed method using fractional differentials successfully recovers correct critical exponents below the upper critical dimension.
• The Rushbrooke scaling relation was confirmed, even for non-integer dimensions.

Conclusions:

• Fractional differentials provide a powerful tool for describing correlation functions in phase transitions, incorporating fractal geometry.
• The study validates fundamental scaling laws in statistical mechanics, extending their applicability.
• This work offers a refined understanding of critical phenomena and their mathematical underpinnings.