Extended Vicsek fractals: Laplacian spectra and their applications

Extended Vicsek fractals (EVF) are the structures constructed by introducing linear spacers into traditional Vicsek fractals. Here we study the Laplacian spectra of the EVF. In particularly, the recurrence relations for the Laplacian spectra allow us to obtain an analytic expression for the sum of all inverse nonvanishing Laplacian eigenvalues. This quantity characterizes the large-scale properties, such as the gyration radius of the polymeric structures, or the global mean-first passage time for the random walk processes. Introduction of the linear spacers leads to local heterogeneities, which reveal themselves, for example, in the dynamics of EVF under external forces.

Figure 1

A schematic representation of the construction procedure for extended Vicsek fractals (EVF). An EVF of generation $g$ is constructed from the normal Vicsek fractal (VF) of generation $g$ by replacing each nonbranching node [see blue beads on (a) for VF of $g=1$ and on (b) for VF of $g=2$] through a linear chain of $k$ nodes [see yellow beads on (c) and (d), here $k=2$]. Both for VF and EVF the functionality parameter $f$ denotes the number of nearest neighbors of branching nodes (red beads, here $f=4$).

Figure 2

Gyration radius of EVF as a function of total number of beads $N$ for different values of the parameters $f$ and $k$. Filled symbols represent the results based on the exact calculations and open symbols show the result of the approximate Eq. (36). See text for details.

Figure 3

Structure-averaged bead displacement $\langle Y\left(t\right)\rangle$ under a constantly acting force for EVF of generation $g=10$, spacer parameter $k=20$, and different functionalities $f$. The inset represents the curves for different values of the spacer length $k$. The ratio between parameters $F$ and $\zeta$ is $F/\zeta =1$. See text for details.

Figure 4

Dynamical shear modulus $\left[G\right(t\left)\right]$ for EVF of generation $g=10$, spacer parameter $k=20$, and different functionalities $f$. The inset represents the curves for different values of the spacer length $k$. See text for details.