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

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.