Generalized similarity, renormalization groups, and nonlinear clocks for multiscaling.
Fixed points of the renormalization group operator Rp,rX(t)≡X(rt)/rp are said to be p-self-similar. Here X(t) is an arbitrary stochastic process. The concept of a p-self-similar process is generalized via the renormalization group operator RF,GX(t)=F[X(G(t))], where F and G are bijections on (-∞,∞) and [0,∞), respectively. If X(t) is a fixed point of RF,G, then X(t) is said to be (F,G)-self-similar. We say Y(t) is (F,G)-X(t)-similar if RF,GX(t)=Y(t) in distribution. Exit time distributions and finite-size Lyapunov exponents were obtained for these latter processes. A power law multiscaling process is defined with a multipower-law clock. This process is employed to statistically represent diffusion in a nanopore, a monolayer fluid confined between atomically structured surfaces. The tools presented provide a straightforward method to statistically represent any multiscaling process in time.
Department of Mathematics and Department of Earth, Atmospheric, and Planetary Sciences (EAPS), Purdue University, West Lafayette Indiana, 47907, USA.
M
M
Park
Computational Earth Science Group, Earth and Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA.
D
D
O'Malley
Department of Mathematics and Department of Earth, Atmospheric, and Planetary Sciences, Purdue University, West Lafayette Indiana, 47907, USA.
J H
JH
Cushman