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Joint ground states of two directed polymers in a random medium are investigated. Using exact min-cost flow optimization, the true two-line ground-state is compared with the single line ground state plus its first excited state with "worst-possible" initial conditions, where the two lines start next to each other. It is found that these two-line configurations are (for almost all disorder configurations) distinct implying that the true two-line ground-state is nonseparable, which means that the two-line ground state cannot be obtained by adding a second line to the first line in the one-line ground state without deforming the first line. The effective interaction energy between the two lines scales with the system size with the scaling exponents 0.39 ± 0.03 and 0.21 ± 0.02 in 2D and 3D, respectively.

In the presence of buoyancy, multiple diffusion coefficients, and porous media, the dispersion of solutes can be remarkably complex. The lattice-Boltzmann (LB) method is ideal for modeling dispersion in flow through complex geometries; yet, LB models of solute fingers or slugs can suffer from peculiar numerical conditions (e.g., denormal generation) that degrade computational performance by factors of 6 or more. Simple code optimizations recover performance and yield simulation rates up to ~3 million site updates per second on inexpensive, single-CPU systems. Two examples illustrate limits of the methods: (1) Dispersion of solute in a thin duct is often approximated with dispersion between infinite parallel plates. However, Doshi, Daiya and Gill (DDG) showed that for a smooth-walled duct, this approximation is in error by a factor of ~8. But in the presence of wall roughness (found in all real fractures), the DDG phenomenon can be diminished. (2) Double-diffusive convection drives "salt-fingering", a process for mixing of fresh-cold and warm-salty waters in many coastal regions. Fingering experiments are typically performed in Hele-Shaw cells, and can be modeled with the 2D (pseudo-3D) LB method with velocity-proportional drag forces. However, the 2D models cannot capture Taylor–Aris dispersion from the cell walls. We compare 2D and true 3D fingering models against observations from laboratory experiments.

The random deposition model must be enhanced to reflect the variety of surface roughness due to some material characteristics of the film growing by vacuum deposition or sputtering. The essence of the computer simulation in this case is to account for possible surface migration of atoms just after the deposition, in connection with the binding energy between atoms (as the mechanism provoking the diffusion) and/or diffusion energy barrier. The interplay of these two factors leads to different morphologies of the growing surfaces, from flat and smooth ones to rough and spiky ones. In this paper, we extended our earlier calculation by applying an extra diffusion barrier at the edges of terrace-like structures, known as the Ehrlich–Schwoebel barrier. It is experimentally observed that atoms avoid descending when the terrace edge is approached, and these barriers mimic this tendency. Results of our Monte Carlo computer simulations are discussed in terms of surface roughness, and compared with other model calculations and some experiments from literature. The power law of the surface roughness σ against film thickness t was confirmed. The nonzero minimum value of the growth exponent β near 0.2 was obtained which is due to the limited range of the surface diffusion and the Ehrlich–Schwoebel barrier. Observations for different diffusion ranges are also discussed. The results are also confirimed with some deterministic growth models.

In this paper we present a generalization of a simple solid-on-solid epitaxial model of thin film growth, when surface morphology anisotropy is provoked by anisotropy in the model control parameters of binding energy and/or diffusion barrier. The anisotropy is discussed in terms of the height–height correlation function. It was experimentally confirmed that the difference in diffusion barriers yields anisotropy in morphology of the surface. We obtained antisymmetric correlations in the two in-plane directions for antisymmetric binding.