Critical slowing down exponents of mode coupling theory

Area of Science:

• Statistical Physics
• Theoretical Condensed Matter Physics
• Complex Systems Analysis

Background:

• Critical slowing down is a key phenomenon in phase transitions, often analyzed using dynamic approaches within mode coupling theory.
• Existing methods for calculating dynamic exponents can be computationally intensive and dependent on specific dynamic assumptions.
• A need exists for alternative, potentially simpler, methods to determine these crucial parameters.

Purpose of the Study:

• To develop and present a novel method for computing the exponent parameter (λ) related to critical slowing down in mode coupling theory.
• To establish a method independent of dynamic approaches, utilizing an effective static field theory.
• To provide explicit expressions for λ and validate the method across various mean-field models.

Main Methods:

• Formulation of an effective static field theory.
• Derivation of expressions for the exponent parameter λ in terms of third-order action coefficients or six-point cumulants.
• Application and validation of the method to diverse mean-field models, including those with hard/soft variables and varying interaction connectivities.

Main Results:

• A static field theory approach successfully computes the dynamic exponent λ, independent of dynamic calculations.
• Explicit formulas for λ are derived using static properties (third-order coefficients or six-point cumulants).
• The method shows agreement with existing results for various models like Potts glass, random orthogonal, and Sherrington-Kirkpatrick models.

Conclusions:

• The proposed static field theory method offers a robust and versatile alternative for calculating critical slowing down exponents.
• This approach simplifies the analysis of dynamic critical phenomena by relying solely on static properties.
• The findings provide a valuable tool for theoretical studies of phase transitions and complex systems.