Coarsening of bicontinuous microstructures is observed in a variety of systems, such as nanoporous metals and mixtures that have undergone spinodal decomposition. To better understand the morphological evolution of these structures during coarsening,
we compare the morphologies resulting from two different coarsening mechanisms, surface and bulk diffusion. We perform phase-field simulations of coarsening via each mechanism in a two-phase mixture at nominal volume fractions of 50%-50% and 36%-64%, and the simulated structures are characterized in terms of topology (genus density), the interfacial shape distribution, structure factor, and autocorrelations of phase and mean curvature. We observe self-similar evolution of morphology and topology and agreement with the expected power laws for dynamic scaling, in which the characteristic length scale increases over time proportionally to $t^{1/4}$ for surface diffusion and $t^{1/3}$ for bulk diffusion. While we observe the expected difference in the coarsening kinetics, we find that differences in self-similar morphology due to coarsening mechanism are relatively small, although typically they are larger at 36% volume fraction than at 50% volume fraction. In particular, we find that bicontinuous structures coarsened via surface diffusion have lower scaled genus densities than structures coarsened via bulk diffusion. We also compare the self-similar morphologies to those in literature and to two model bicontinuous structures, namely, constant-mean-curvature surfaces based on the Schoen G minimal surface and random leveled-wave structures. The average scaled mean curvatures of these model structures agree reasonably with those of the coarsened structures at both 36% and 50%, but we find substantial disagreements in the scaled genus densities and the standard deviations of mean curvature.
We describe a novel algorithm for random sampling of freely reduced words equal to the identity in a finitely presented group. The algorithm is based on Metropolis Monte Carlo sampling. The algorithm samples from a stretched Boltzmann distribution be
gin{align*}pi(w) &= (|w|+1)^{alpha} beta^{|w|} cdot Z^{-1} end{align*} where $|w|$ is the length of a word $w$, $alpha$ and $beta$ are parameters of the algorithm, and $Z$ is a normalising constant. It follows that words of the same length are sampled with the same probability. The distribution can be expressed in terms of the cogrowth series of the group, which then allows us to relate statistical properties of words sampled by the algorithm to the cogrowth of the group, and hence its amenability. We have implemented the algorithm and applied it to several group presentations including the Baumslag-Solitar groups, some free products studied by Kouksov, a finitely presented amenable group that is not subexponentially amenable (based on the basilica group), and Richard Thompsons group $F$.
We compute the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N > $, which we show to be D-finite. It follows that their cogrowth rates are algebraic numbers.
We propose a numerical method for studying the cogrowth of finitely presented groups. To validate our numerical results we compare them against the corresponding data from groups whose cogrowth series are known exactly. Further, we add to the set of
such groups by finding the cogrowth series for Baumslag-Solitar groups $mathrm{BS}(N,N) = < a,b | a^N b = b a^N >$ and prove that their cogrowth rates are algebraic numbers.
