Generalized similarity, renormalization groups, and nonlinear clocks for multiscaling

Fixed points of the renormalization group operator $R_{p,r}X\left(t\right)\equiv X\left(rt\right)/r^p$ are said to be $p$-self-similar. Here $X\left(t\right)$ is an arbitrary stochastic process. The concept of a $p$-self-similar process is generalized via the renormalization group operator ${\mathcal{R}}_{F,G}X\left(t\right)=F\left[X\left(G\left(t\right)\right)\right]$, where $F$ and $G$ are bijections on $(-\infty ,\infty )$ and $[0,\infty )$, respectively. If $X\left(t\right)$ is a fixed point of ${\mathcal{R}}_{F,G}$, then $X\left(t\right)$ is said to be $(F,G)$-self-similar. We say $Y\left(t\right)$ is $(F,G)$-$X\left(t\right)$-similar if ${\mathcal{R}}_{F,G}X\left(t\right)=Y\left(t\right)$ 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.

Figure 1

Plots of the logarithmic mean $a$ time $\langle T_a\left(r\right)\rangle$ for two $G\left(t\right)$ in Eq. (27): solid lines have $q_1=0.5$ for $t\in (0,2]$ and $q_2=1$ for $t\in (2,\infty )$, while dashed lines have $q_1=2$ for $t\in (0,2]$, $q_2=1$ for $t\in (2,9]$, and $q_3=0.1$ for $t\in (9,\infty )$.

Figure 2

MSD with $u\left(t\right)=x\left(t\right)$ or $y\left(t\right)$: triangles, squares, and diamonds are the data in Fig. 4 of Su et al. [36] and solid lines are from $B\left(G\right(t\left)\right)$ using Table 1.

Figure 3

(Adapted from Fig. 6 in Su et al. [36].) Simulation trajectories for OMCTS. The $\times$ and $\circ$ symbols mark the positions of the upper and lower surface oxygen atoms, respectively. In the left panel the surfaces are aligned directly over one another $({\alpha }_x=0.0)$ and in the right panel the surfaces are arranged to form channels in the $x$- direction $({\alpha }_x=0.5)$.

Figure 4

Simulation trajectories from $t=4.8$ to $t=7.0$: (a) ${\alpha }_x=0$ and ${\alpha }_y=0$; (b) ${\alpha }_x=0.5$ and ${\alpha }_y=0$.