Effect of the transition of networks from floppy to rigid on the diffusion coefficient.

We investigate a tracer particle moving on a two-dimensional square lattice created by network formers (NF). The positions of all network formers are randomly displaced by a small amount from the nodes of the network. Each NF can be in a "floppy" or "rigid" state, depending on the number of bonds connecting it to neighboring network formers. The NF that have more than a specified number of m bonds are in rigid state, the remaining ones are in a floppy state. The energy of the tracer particle depends on its distance from those of the four nearest NF that are in "rigid" state. The NF in floppy state do not contribute to the energy. We here demonstrate that the a priori increase in the diffusion coefficient with the concentration of the floppy states goes through a crossover point, after which the increase is much sharper.

Percolation of randomly distributed growing clusters: finite-size scaling and critical exponents for the square lattice.

We study the percolation properties of the growing clusters model on a 2D square lattice. In this model, a number of seeds placed on random locations on the lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The model exhibits a discontinuous transition for very low values of the seed concentration p and a second, nontrivial continuous phase transition for intermediate p values. Here we study in detail this continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component. Using finite size scaling and large scale Monte Carlo simulations we determine the value of the percolation threshold where the giant component first appears, and the critical exponents that characterize the transition. We find that the transition belongs to a different universality class from the standard percolation transition.

Accurate estimation of the survival probability for trapping in two dimensions.

In this work we study the mean survival probability Phi(n,c) of random walks on a two-dimensional lattice in the presence of traps of concentration c, as a function of the number of steps n. The computation of this quantity is performed indirectly by using the distribution of the number of sites visited S(n). In order to achieve an accurate description of this distribution we use a combination of numerical techniques. The method allows an accurate calculation of Phi down to very small values (of the order of 10(-100), for example), which is not possible via direct simulations. The survival probability is analyzed in terms of an asymptotic expansion, following the results of Donsker and Varadhan [Commun. Pure Appl. Math. 28, 525 (1975); 32, 721 (1979)], and by using the outcome of a scaling ansatz, as described in our earlier work.

Thermodynamics and diffusion of a lattice gas on a simple cubic lattice.

A lattice gas model with nearest neighbor attractive interactions on a simple cubic lattice is considered. The method of nonequilibrium statistical ensembles due to Zubarev is used to derive expressions for jump and chemical diffusion coefficients. For thermally activated hopping dynamics in the hydrodynamical (low frequency, long wavelength) limit, and neglecting specific memory effects, these expressions are represented in a simple form in terms of the zero concentration limit of the chemical diffusion coefficient and equilibrium characteristics, i.e., the chemical potential, and the probability for two nearest neighbor sites to be vacant. These equilibrium characteristics are calculated by means of the self-consistent diagram approximation. The equilibrium characteristics and diffusion coefficients are in a good agreement with extensive Monte Carlo simulation results.

Reaction front structure in the diffusion-limited A+B model with initially randomized reactants.

Subtle features of the reaction front formation in the A+B-->0 reaction are reported for the initially random and equal A+B reactant distribution. Three nonclassical parameters (initial linewidth, minimum, and maximum), for each interparticle gap and nearest neighbor distance distributions, are derived, as a function of time, using Monte Carlo simulations. These empirical front measures and their temporal scaling exponents are compared with the previously studied ones for the reactant interparticle distributions.

Trapping and survival probability in two dimensions.

We investigate the survival probability Phi(n,c) of particles performing a random walk on a two-dimensional lattice that contains static traps, which are randomly distributed with a concentration c, as a function of the number of steps n. Phi(n,c) is analyzed in terms of a scaling ansatz, which allows us to locate quantitatively the crossover between the Rosenstock approximation (valid only at early times) and the asymptotic Donsker-Varadhan behavior (valid only at long times). While the existence of the crossover has been postulated before, its exact location has not been known. Our scaling hypothesis is based on the mean value of the quantity S(n), the number of sites visited in an n-step walk. We make use of the idea of self-interacting random walks, and a "slithering" snake algorithm, available in the literature, and we are thus able to obtain accurate survival probability data indirectly by Monte Carlo simulation techniques. The crossover can now be determined by our method, and it is found to depend on a combination of c and n. It occurs at small Phi(n,c) values, which is typically the case for large values of n.

Influence of auto-organization and fluctuations on the kinetics of a monomer-monomer catalytic scheme.

We study analytically the kinetics of an elementary bimolecular reaction scheme of the Langmuir-Hinshelwood type taking place on a d-dimensional catalytic substrate. We propose a general approach that takes into account explicitly the influence of spatial correlations on the time evolution of the mean particle density. With this approach, we recover some known results concerning the time evolution of the mean particle density and establish others.

A heterogeneous tube model of intestinal drug absorption based on probabilistic concepts.

PURPOSE: To develop an approach based on computer simulations for the study of intestinal drug absorption. METHODS: The drug flow in the gastrointestinal tract was simulated with a biased random walk model in the heterogeneous tube model (Pharm. Res. 16, 87-91, 1999), while probability concepts were used to describe the dissolution and absorption processes. An amount of drug was placed into the input end of the tube and allowed to flow, dissolve and absorb along the tube. Various drugs with a diversity in dissolution and permeability characteristics were considered. The fraction of dose absorbed (Fabs) was monitored as a function of time measured in Monte Carlo steps (MCS). The absorption number An was calculated from the mean intestinal transit time and the absorption rate constant adhering to each of the drugs examined. RESULTS: A correspondence between the probability factor used to simulate drug absorption and the conventional absorption rate constant derived from the analysis of data was established. For freely soluble drugs, the estimates for Fabs derived from simulations using as an intestinal transit time 24500 MCS (equivalent to 4.5 h) were in accord with the corresponding data obtained from literature. For sparingly soluble drugs, a comparison of the normalized concentration profiles in the tube derived from the heterogeneous tube model and the classical macroscopic mass balance approach enabled the estimation of the dissolution probability factor for five drugs examined. The prediction of Fabs can be accomplished using estimates for the absorption and the dissolution probability factors. CONCLUSIONS: A fully computerized approach which describes the flow, dissolution and absorption of drug in the gastrointestinal tract in terms of probability concepts was developed. This approach can be used to predict Fabs for drugs with various solubility and permeability characteristics provided that probability factors for dissolution and absorption are available.

Heterogeneous tube model for the study of small intestinal transit flow.

PURPOSE: A Monte-Carlo computer simulation technique was employed to study the details of the small intestinal transit flow in the gastrointestinal (GI) tract. METHODS: A heterogeneous tube model was constructed using a numerical computer simulation technique. The model was built from first principles and included several heterogeneous characteristics of the GI tract structure. We used a random, dendritic-type internal structure representing the villi of the GI tract. The small intestinal transit flow was simulated using two diffusion models, namely, the blind ant and the myopic ant models, which are different models to account the elapse of time, and which are both based on statistical properties of random walks. For each one of the models we utilize two types of biased random walk, placing different emphasis in the motion towards the output of the tube. We monitored the flow of the drug in terms of Monte-Carlo time steps (MCS) through the tube walls and dendritic villi present. RESULTS: The frequency of the transit times was dependent on the structure of the dendritic villi and on the type of biased random walk. The small intestinal flow profile of literature data for a large number of drugs was well characterized by the heterogeneous model using, as parameters, a certain number of villi per unit length of the tube and specific characteristics for both types of the biased random walk. A correspondence between the MCS and real time units was achieved. CONCLUSIONS: The transit process of the oral dosage forms in the GI tract can be reproduced with the heterogeneous model developed. This model can be used to study GI absorption phenomena.

Gastrointestinal drug absorption: is it time to consider heterogeneity as well as homogeneity?

The current analysis of gastrointestinal absorption phenomena relies on the concept of homogeneity. However, drug dissolution, transit and uptake in the gastrointestinal tract are heterogeneous processes since they take place at interfaces of different phases under variable stirring conditions. Recent advances in physics and chemistry demonstrate that the geometry of the environment is of major importance for the treatment of heterogeneous processes. In this context, the heterogeneous character of in vivo drug dissolution, transit and uptake is discussed in terms of fractal concepts. Based on this analysis, drugs are classified in accordance with their gastrointestinal absorption characteristics into two broad categories i.e. homogeneous and heterogeneous. The former category includes drugs with satisfactory solubility and permeability which ensure the validity of the homogeneous hypothesis. Drugs with low solubility and permeability are termed heterogeneous since they traverse the entire gastrointestinal tract and therefore are more likely to exhibit heterogeneous dissolution, transit and uptake. The high variability of whole bowel transit and the unpredictability of conventional dissolution tests for heterogeneous drugs are interpreted on the basis of the fractal nature of these processes under in vivo conditions. The implications associated with the use of strict statistical criteria in bioequivalence studies for heterogeneous drugs are also pointed out.