Additional constraints in adsorption-desorption kinetics.

In this work, the adsorption-desorption kinetic in the framework of the lattice gas model is analyzed. The transition probabilities are written as an expansion of the occupation configurations. Due to that, the detail balance principle determine half of the adsorption A{i} and desorption D{i} coefficients, consequently, different functional relations between them are proposed. Introducing additional constrains, it is demonstrated that when those coefficients are linearly related through a parameter gamma , there are values of lateral interaction V , that lead to anomalous behavior in the adsorption isotherms, the sticking coefficient and the thermal programmed desorption spectra. Diagrams for the allowed values of V and gamma are also shown. Alternatively, a more reliable formulation for the adsorption desorption kinetic based on the transition state theory is introduced. In such way the equilibrium and non equilibrium observables do not present anomalous or inconsistent behavior.

Analysis of the convergence of the 1t and Wang-Landau algorithms in the calculation of multidimensional integrals.

In this Brief Report, the convergence of the 1t and Wang-Landau algorithms in the calculation of multidimensional numerical integrals is analyzed. Both simulation methods are applied to a wide variety of integrals without restrictions in one, two, and higher dimensions. The efficiency and accuracy of both algorithms are determined by the dynamical behavior of the errors between the exact and the calculated values of the integral. It is observed that the time dependence of the error calculated with the 1t algorithm varies as N;{-12} [with N the number of Monte Carlo (MC) trials], in quantitative agreement with the simple sampling Monte Carlo method. In contrast, the error calculated with the Wang-Landau algorithm saturates in time, evidencing the nonconvergence of this method. The sources of error for both methods are also determined.

Fast algorithm to calculate density of states.

An algorithm to calculate the density of states, based on the well-known Wang-Landau method, is introduced. Independent random walks are performed in different restricted ranges of energy, and the resultant density of states is modified by a function of time, F(t) proportional to t-1, for large time. As a consequence, the calculated density of state, gm(E,t) , approaches asymptotically the exact value g(ex)(E) as proportional to t-1/2, avoiding the saturation of the error. It is also shown that the growth of the interface of the energy histogram belongs to the random deposition universality class.

Nonconvergence of the Wang-Landau algorithms with multiple random walkers.

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/sqrt[m]. However, if the number of walkers increases above a certain critical value m>m_{x}, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value m_{x}, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.

Hard versus soft dynamics for adsorption-desorption kinetics: Exact results in one-dimension.

The adsorption-desorption kinetics is discussed in the framework of the kinetic lattice-gas model. The master equation formalism has been introduced to describe the evolution of the system, where the transition probabilities are written as an expansion of the occupation configurations of all neighboring sites. Since the detailed balance principle determines half of the coefficients that arise from the expansion, it is necessary to introduce ad hoc, a dynamic scheme to get the rest of them. Three schemes of the so-called hard dynamics, in which the probability of transition from single site cannot be factored into a part which depends only on the interaction energy and one that only depends on the field energy, and five schemes of the so-called soft dynamics, in which this factorization is possible, were introduced for this purpose. It is observed that for the hard dynamic schemes, the equilibrium and nonequilibrium observables, such as adsorption isotherms, sticking coefficients, and thermal desorption spectra, have a normal or physical sustainable behavior. While for the soft dynamics schemes, with the exception of the transition state theory, the equilibrium and nonequilibrium observables have several problems. Some of them can be regarded as abnormal behavior.

Wang-Landau algorithm: a theoretical analysis of the saturation of the error.

In this work we present a theoretical analysis of the convergence of the Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced years ago to calculate the density of states in statistical models. We study the dynamical behavior of the error in the calculation of the density of states. We conclude that the source of the saturation of the error is due to the decreasing variations of the refinement parameter. To overcome this limitation, we present an analytical treatment in which the refinement parameter is scaled down as a power law instead of exponentially. An extension of the analysis to the N-fold way variation of the method is also discussed.