Deformed random walk: Suppression of randomness and inhomogeneous diffusion

Area of Science:

• Statistical Mechanics
• Mathematical Physics
• Nonextensive Statistics

Background:

• Standard random walks (RW) are foundational in modeling diffusion and stochastic processes.
• Nonextensive statistics, based on q-algebra, offers a framework for systems with long-range correlations.
• Generalizing RW is crucial for understanding complex systems beyond standard assumptions.

Purpose of the Study:

• To introduce and analyze a deformed random walk (DRW) model based on q-algebra.
• To investigate the mathematical properties and physical implications of this DRW, including path convergence and diffusion behavior.
• To extend the DRW framework to two dimensions and compare it with existing models.

Main Methods:

• Generalization of the standard random walk using a deformed unitary step derived from q-algebra.
• Analysis of the associated deformed Pascal triangle and inhomogeneous diffusion processes.
• Derivation and solution of the master equation and Fokker-Planck equations in continuous space and for 2D cases.

Main Results:

• DRW paths exhibit convergence to a fixed point, contrasting with the divergent paths of standard RW in deformed space.
• A van Kampen inhomogeneous diffusion equation with exponential hyperdiffusion predicts particle localization.
• The 2D DRW shows path convergence and inhomogeneous diffusion controlled by two deformation parameters.

Conclusions:

• The deformed random walk provides a novel framework for studying diffusion in complex systems with nonextensive statistical properties.
• The model demonstrates particle localization and suppressed randomness, offering insights into phenomena not captured by standard random walks.
• The generalization is robust, extending to two dimensions and showing symmetry properties under parameter transformation.