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Generalized similarity, renormalization groups, and nonlinear clocks for multiscaling.

M Park¹, D O'Malley², J H Cushman³

¹Department of Mathematics and Department of Earth, Atmospheric, and Planetary Sciences (EAPS), Purdue University, West Lafayette Indiana, 47907, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|May 16, 2014

Summary

This study generalizes p-self-similar processes using renormalization group operators, introducing (F,G)-self-similar and (F,G)-X(t)-similar processes. These methods are applied to model diffusion in nanopores, offering a new way to represent multiscaling processes.

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Area of Science:

Statistical Mechanics
Stochastic Processes
Complex Systems

Background:

Introduces p-self-similar processes defined by a renormalization group operator Rp,r.
Highlights the need for a generalized framework for self-similar stochastic processes.

Purpose of the Study:

Generalize the concept of p-self-similar processes using a broader renormalization group operator RF,G.
Define and analyze (F,G)-self-similar and (F,G)-X(t)-similar processes.
Apply these generalized processes to model physical phenomena like diffusion in nanopores.

Main Methods:

Definition of the generalized renormalization group operator RF,G, where F and G are bijections.
Analysis of fixed points of RF,G to define (F,G)-self-similar processes.
Introduction of (F,G)-X(t)-similar processes via RF,GX(t)=Y(t) in distribution.
Development of a power law multiscaling process with a multipower-law clock.

Main Results:

Derived exit time distributions and finite-size Lyapunov exponents for (F,G)-X(t)-similar processes.
Successfully employed the power law multiscaling process to statistically represent diffusion in nanopores.
Demonstrated the applicability to monolayer fluids confined between structured surfaces.

Conclusions:

The generalized framework provides a robust method for analyzing self-similar stochastic processes.
The developed multiscaling process offers a powerful tool for modeling complex diffusion phenomena.
The presented techniques offer a straightforward approach to statistically represent any multiscaling process in time.