A connection between the global roughness exponent and the fractal dimension of a rough interface, whose dynamics are expected to be described by stochastic continuum models, still needs more support from simulations in lattice models, which are key to provide completeness for the characterization of a given universality class. In this work, we investigate the asymptotic fractal dimension of interfaces that evolve according to some specific lattice models in d + 1 dimensions ( d = 1, 2), which are expected to belong to the Edwards–Wilkinson or Kardar–Parisi–Zhang universality classes. Our results, based on the Higuchi method and on the extrapolation of the effective fractal dimension, allow one to achieve dependence between the asymptotic fractal dimension and global roughness exponent, in which the latter is expected to be hardly accessible for experimentalists. Conversely, we also use a two-points correlation function, which gives the time evolution of the local roughness exponent. As a byproduct, our results suggest that, for d = 1, the fractal dimension converges faster than the global roughness exponents to the asymptotic ones. Therefore, the analysis of the fractal dimension, for d = 1, is suggested to be more accessible than the global roughness exponents to determine the universality class. Corrections for the fractal dimensions in d = 2 were found to be stronger than for d = 1.

Growth in crystals can be usually described by field equations such as the Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can be characterized by Euclidean geometry with its peculiar symmetries, the growth dynamics creates a fractal structure at the interface of a crystal and its growth medium, which in turn determines the growth. Recent work by Gomes-Filho et al. (Results in Physics, 104,435 (2021)) associated the fractal dimension of the interface with the growth exponents for KPZ and provides explicit values for them. In this work, we discuss how the fluctuations and the responses to it are associated with this fractal geometry and the new hidden symmetry associated with the universality of the exponents.

We use a decimation procedure in order to obtain the dynamical renormalization group transformation (RGT) properties of random walk distribution in a 1+1 lattice. We obtain an equation similar to the Chapman-Kolmogorov equation. First we show the existence of invariants through the RGT. We also show the existence of functions which are semi-invariants through the RGT. Second, we show as well that the distribution $R_q\mathrm{}\left(x\right)=[1+b(q-{1)x}^2{]}^{1/\left(1-q\right)}\hspace{0.5em}\mathrm{}(q>\mathrm{}1\mathrm{}),$ which is an exact solution of a nonlinear Fokker-Planck equation, is a semi-invariant for RGT. We obtain the map $q^{\prime }=f\left(q\right)\mathrm{}$ from the RGT and we show that this map has two fixed points: $q=1,$ attractor, and $q=2,$ repellor, which are the Gaussian and the Lorentzian, respectively. We show the connections between these result and the Levy flights.

A computational model is proposed to investigate drug delivery systems in which erosion and diffusion mechanisms are participating in the drug release process. Our approach allowed us to analytically estimate the crossover point between those mechanisms through the value of the parameter $b$ $(b_c=1)$ and the scaling behavior of parameter $\tau$ on the Weibull function, $exp[-{(t/\tau )}^b]$, used to adjust drug release data in pharmaceutical literature. Numerical investigations on the size dependence of the characteristic release time $\tau$ found it to satisfy either linear or quadratic scaling relations on either erosive or diffusive regimes. Along the crossover, the characteristic time scales with the average coefficient observed on the extreme regimes (i.e., $\tau \sim L^{3/2}$), and we show that this result can be derived analytically by assuming an Arrhenius relation for the diffusion coefficient inside the capsule. Based on these relations, a phenomenological expression for the characteristic release in terms of size $L$ and erosion rate $\kappa$ is proposed, which can be useful for predicting the crossover erosion rate ${\kappa }_c$. We applied this relation to the experimental literature data for the release of acetaminophen immersed in a wax matrix and found them to be consistent with our numerical results.

We present a method to derive an analytical expression for the roughness of an eroded surface whose dynamics are ruled by cellular automaton. Starting from the automaton, we obtain the time evolution of the height average and height variance (roughness). We apply this method to the etching model in $1+1$ dimensions, and then we obtain the roughness exponent. Using this in conjunction with the Galilean invariance we obtain the other exponents, which perfectly match the numerical results obtained from simulations. These exponents are exact, and they are the same as those exhibited by the Kardar-Parisi-Zhang (KPZ) model for this dimension. Therefore, our results provide proof for the conjecture that the etching and KPZ models belong to the same universality class. Moreover, the method is general, and it can be applied to other cellular automata models.

We use molecular dynamics to determine the force needed to rupture a chain molecule being stretched at constant loading rate and temperature. When all energy bonds of the molecule are identical, we find that the force $F$ depends on the pulling rate $r$ and temperature $T$ according to $F\sim \mathrm{const}-T^{1∕3}{\mid \ln (r∕T)\mid }^{1∕3}$. When a single weak bond is introduced, this result is modified to $F\sim \mathrm{const}-T^{2∕3}{\mid \ln (r∕T)\mid }^{2∕3}$. This scaling, which is model independent, can be used with experiment to quantitatively extract relevant microscopic parameters.