Scaling analysis of random walks with persistence lengths: Application to self-avoiding walks

Area of Science:

• Statistical Physics
• Polymer Physics
• Computational Physics

Background:

• Random walks (RWs) are fundamental models in statistical physics.
• Scaling analysis of RWs is crucial for understanding polymer dynamics and phase transitions.
• Existing methods often require analyzing entire paths, which can be computationally intensive.

Purpose of the Study:

• To develop a novel approach for scaling analysis of N-step random walks (RWs).
• To introduce and define Inner Persistence Lengths (IPLs) for characterizing RWs.
• To investigate the relationship between IPLs and the mean square end-to-end distance for various RW models.

Main Methods:

• Development of a theoretical framework relating mean square end-to-end distance to IPLs.
• Analytical derivation of a relation between mean square end-to-end distance and persistence length for orthogonal transformation-invariant RWs.
• Application of series expansion and Monte Carlo simulations for self-avoiding walks (SAWs) on 2D and 3D lattices.

Main Results:

• IPLs are defined as ensemble averages of dot products between position and displacement vectors.
• A constant value for the persistence length at infinity (λ_∞) was found for SAWs.
• Scaling corrections for λ_N were identified as higher-order corrections to scaling for the mean square end-to-end distance.
• Exponents ν₀ and Δ₁ were estimated from IPL behavior, showing excellent agreement with literature values.

Conclusions:

• The study demonstrates that analyzing only ensembles of paths with the same length is sufficient for determining scaling behavior.
• The entire information needed for scaling analysis of mean square end-to-end distance is contained within the inner segments of the paths.
• The novel IPL approach offers a simplified and efficient method for RW scaling analysis.