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Abstract In this paper, a theoretical approach to calculate site percolation thresholds on two-dimensional lattices is proposed. The method, based on exact counting of configurations on finite cells, arises as a generalization of the analytical approximation introduced by Rosowsky (2000). The resulting methodology was applied to calculate the percolation thresholds corresponding to four systems: monomers on honeycomb lattices (p c = 0. 71278), dimers on square lattices (p c = 0. 5713), dimers on honeycomb lattices (p c = 0. 6653) and dimers on triangular lattices (p c = 0. 4783). The obtained results are in good agreement with previous values calculated by very accurate simulations: 0.69704, 0.5649, 0.6902 and 0.4872. The technique can be easily extended to deal with three-dimensional lattices. Highlights • The percolation problem of monomers and dimers on 2D lattices is studied. • A theoretical approach based on counting configurations on finite cells is applied. • Percolation thresholds are reported. • The obtained results are in good agreement with previous calculations in the literature. [ABSTRACT FROM AUTHOR]

The phase behaviour of aligned rigid rods of length k ( k -mers) adsorbed on two-dimensional square lattices has been studied by Monte Carlo (MC) simulations and histogram reweighting technique. The k -mers, containing k identical units (each one occupying a lattice site) were deposited along one of the directions of the lattice. In addition, attractive lateral interactions were considered. The methodology was applied, particularly, to the study of the critical point of the condensation transition occurring in the system. The process was monitored by following the fourth order Binder cumulant as a function of temperature for different lattice sizes. The results, obtained for k ranging from 2 to 7, show that: (i) the transition coverage exhibits a decreasing behaviour when it is plotted as a function of the k -mer size and (ii) the transition temperature, T c , exhibits a power law dependence on k , T c ∼ k 0 , 4 , shifting to higher values as k increases. Comparisons with an analytical model based on a generalization of the Bragg–Williams approximation (BWA) were performed in order to support the simulation technique. A significant qualitative agreement was obtained between BWA and MC results. [ABSTRACT FROM AUTHOR]

A simple method to estimate percolation thresholds p c in lattice models is presented. The technique is based on the calculation of the probabilities R L X (p) for finding a percolating cluster of type X [ X could be horizontal (H), vertical (V), average (A), intersection (I), or union (U)] on a finite lattice of side length L at concentration p of occupied sites. The functions R L X (p) are obtained by numerical simulations. Then, (1) effective percolation thresholds p c X , r (L) are estimated from the condition R L X (p) = r (where r is a parameter ranging between 0 and 1); and (2) the percolation threshold in the thermodynamic limit is determined by extrapolating the effective percolation thresholds to infinite L. The methodology presented herein contains and extends the classical scheme developed by Yonezawa, Sakamoto and Hori, offering a more complete and versatile theoretical/simulation framework to calculate percolation thresholds. The approach is validated by successfully comparing with well-known results obtained for the problem of random site percolation on square lattices. Taking advantage of the behavior of the curves of p c X , r (L) for different values of the parameter r , more general scaling relations are proposed for the effective percolation thresholds. Finally, the technique is applied to model experimental data of clogging transitions in a two-dimensional silo with a vibrated base. • A new procedure to calculate percolation thresholds in lattice systems is presented. • Multiple effective thresholds are obtained from the percolation probability functions. • The method is applied to the problem of random site percolation on square lattices. • The obtained results are in good agreement with previous calculations in literature. • The clogging transition in a 2D silo is discussed in terms of percolation theory. [ABSTRACT FROM AUTHOR]

Thermodynamics of straight k -mers gas on the square lattice is addressed from its relation to state exclusion statistics through a new analytical approximation along with a detailed MC analysis. Multiple state exclusion arising as consequence of the spatial correlation between particle states is described through exclusion spectrum functions and related to the thermodynamic potentials. An effective lattice analytical approximation is introduced through configurational mapping and applied to dimers (k = 2) and trimers (k = 3). Simple closed analytical forms for the coverage dependence of multiple state exclusion probabilities and chemical potential are obtained. Remarkable agreement with fast relaxation MC simulations carried out in this work is already found for the lowest degree of approximation. • A new approach to model multisite occupancy adsorption is presented. • The formalism is based on statistics of multiple excluded states. • The theory is applied to a lattice-gas model of straight rigid k-mers on square lattices. • An analytical expression for the adsorption isotherm is obtained for dimers and trimers. • Theoretical results are compared with Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

It is known that the Maximum relative Entropy (MrE) method can be used to both update and approximate probability distributions functions in statistical inference problems. In this manuscript, we apply the MrE method to infer magnetic properties of ferromagnetic materials. In addition to comparing our approach to more traditional methodologies based upon the Ising model and Mean Field Theory, we also test the effectiveness of the MrE method on conventionally unexplored ferromagnetic materials with defects . [ABSTRACT FROM AUTHOR]

Random sequential adsorption of straight rigid rods of length k ( k -mers) on a simple cubic lattice has been studied by numerical simulations and finite-size scaling analysis. The k -mers were irreversibly and isotropically deposited into the lattice. The calculations were performed by using a new theoretical scheme, whose accuracy was verified by comparison with rigorous analytical data. The results, obtained for k ranging from 2 to 64, revealed that (i) the jamming coverage for dimers ( k = 2 ) is θ j = 0.918388 ( 16 ) . Our result corrects the previously reported value of θ j = 0.799 ( 2 ) (Tarasevich and Cherkasova, 2007); (ii) θ j exhibits a decreasing function when it is plotted in terms of the k -mer size, being θ j ( ∞ ) = 0.4045 ( 19 ) the value of the limit coverage for large k ’s; and (iii) the ratio between percolation threshold and jamming coverage shows a non-universal behavior, monotonically decreasing to zero with increasing k . [ABSTRACT FROM AUTHOR]

In this paper, the adsorption of interacting binary mixtures on square lattices has been studied. By using Monte Carlo simulation and finite-size scaling analysis, the connection between the surface ordered phases and the percolating properties of the adsorbed phase has been investigated. A rich phase diagram separating a percolating from a non-percolating region has been determined. The main features of the phase diagram have been discussed in terms of simple considerations related to the interactions present in the problem. [ABSTRACT FROM AUTHOR]

The site-bond percolation problem in two-dimensional (3,12 2) lattices has been studied by means of numerical simulation and analytical calculations. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as S ∩ B and S ∪ B) were considered. In S ∩ B (S ∪ B), two points are connected if a sequence of occupied sites AND (OR) bonds joins them. The analytical method is based on the approximation introduced by Tsallis (2004), which allows to calculate the S ∩ B and S ∪ B percolation functions from the percolation functions corresponding to pure site and pure bond percolation problems. Theoretical and numerical data (supplemented by analysis using finite-size scaling theory) were used to determine, for the first time, the complete phase diagram of the system (phase boundary between the percolating and nonpercolating regions). Comparisons between results obtained using the Tsallis's scheme and simulation data were performed in order to test the reaches and limitations of the approach developed here. [ABSTRACT FROM AUTHOR]

In present work we discuss the problem of introducing stochasticity into 3D atomistic kinetic mean-field simulations. As this is a new approach for simulating the time evolution of material systems, it should be positioned in the field of available methods. Compared to the most used stochastic techniques: while atomistic kinetic Monte Carlo (KMC) methods generate microstates of the system they simulate, stochastic kinetic mean-field (SKMF) seems to generate mesostates of the system during a finite time-window. Previously, strong interrelation have been found for the dispersion of composition fluctuations in ideal solutions between SKMF states and averaged KMC states. In present work we compare the statistical nature of fluctuations in the two approaches in case of equilibrium solid solutions with non-zero mixing energies. At the investigated high temperature cases we show how at a certain noise amplitude in SKMF the composition fluctuations can be related to results received by averaging a certain number of independent KMC states. Furthermore, the correlation between the neighboring sites, emerging because of the chemical interactions, shows the same statistical behavior in both methods as well. [ABSTRACT FROM AUTHOR]