Domain growth on self-similar structures

The behavior of the spherical Ginzburg-Landau model on a class of nontranslationally invariant, fractal lattices is investigated in the cases of conserved and nonconserved Langevin dynamics. Interestingly, the static and dynamic properties can be expressed by means of three exponents characterizing these structures: the embedding dimensions d, the random walk exponent ${\mathrm{d}}_{\mathrm{w}}$, and the spectral dimension ${\mathrm{d}}_{\mathrm{s}}$. An order-disorder transition occurs if ${\mathrm{d}}_{\mathrm{s}}$>2. Explicit solutions show that the domain size evolves with time as R(t)∼${\mathrm{t}}^{1\mathrm{/}{\mathrm{d}}_{\mathrm{w}}}$ in the nonconserved case and as R(t)∼${\mathrm{t}}^{1\mathrm{/}2{\mathrm{d}}_{\mathrm{w}}}$ in the conserved case, whereas the height of the peak of the structure factor increases in time as ${\mathrm{t}}^{{\mathrm{d}}_{\mathrm{s}}\mathrm{/}2}$ in the first case and as ${\mathrm{t}}^{{\mathrm{d}}_{\mathrm{s}}\mathrm{/}4}$ in the second while the system orders. Finally we derive the scaling function for the nonconserved dynamics and the multiscaling function for the conserved dynamics.