Moment ratios and dynamic critical behavior of a reactive system with several absorbing configurations

Critical properties of the Ziff, Gulari, and Barshad (ZGB) model with inert sites

We study the model proposed by Ziff, Gulari, and Barshad to mimic the oxidation of carbon monoxide (CO) in the presence of fixed impurities distributed over the catalytic surface. Our focus is on the continuous phase transition between the active phase, where occurs the production of carbon dioxide (CO2), and the inactive phase, where all the non inert sites become filled with oxygen molecules. We employ Monte Carlo simulations to calculate the different ratios between moments of the order parameter at the critical point, as well as, we determine the critical exponents β and ν⊥ as a function of the concentration of impurities. We show that the presence of impurities over the catalytic surface changes the critical behavior of the system. The critical exponents depend on the concentration of impurities and the model does not belong to the directed percolation universality class.

A network model of correlated growth of tissue stiffening in pulmonary fibrosis

During the progression of pulmonary fibrosis, initially isolated regions of high stiffness form and grow in the lung tissue due to collagen deposition by fibroblast cells. We have previously shown that ongoing collagen deposition may not lead to significant increases in the bulk modulus of the lung until these local remodeled regions have become sufficiently numerous and extensive to percolate in a continuous path across the entire tissue [Bates et al. 2007 Am. J. Respir. Crit. Care Med. 176 617]. This model, however, did not include the possibility of spatially correlated deposition of collagen. In the present study, we investigate whether spatial correlations influence the bulk modulus in a two-dimensional elastic network model of lung tissue. Random collagen deposition at a single site is modeled by increasing the elastic constant of the spring at that site by a factor of 100. By contrast, correlated collagen deposition is represented by stiffening the springs encountered along a random walk starting from some initial spring, the rationale being that excess collagen deposition is more likely in the vicinity of an already stiff region. A combination of random and correlated deposition is modeled by performing random walks of length N from randomly selected initial sites, the balance between the two processes being determined by N. We found that the dependence of bulk modulus, B(N, c), on both N and the fraction of stiff springs, c, can be described by a strikingly simple set of empirical equations. For c < 0.3, B(N, c) exhibits exponential growth from its initial value according to B(N, c) ≈ B₀exp(2c)[1 + c^β ln(N^aI)], where β = 0.994 ± 0.024 and a_I = 0.54 ± 0.026. For intermediate concentrations of stiffening, 0.3 ≤ c ≤ 0.8, another exponential rule describes the bulk modulus as B(N, c) = 4B₀exp[a_II (c - c_c )], where a_II and c_c are parameters that depend on N. For c > 0.8, B(N, c) is linear in c and independent of N, such that B(N, c) = 100B₀ - 100a_III (1 - c)B₀, where a_III = 2.857. For small concentrations, the physiologically most relevant regime, the forces in the network springs are distributed according to a power law. When c = 0.3, the exponent of this power law increases from -4.5, when N = 1, and saturates to about -2, as N increases above 40. These results suggest that the spatial correlation of collagen deposition in the fibrotic lung has a strong effect on the rate of lung function decline and on the mechanical environment in which the cells responsible for remodeling find themselves.

Absorbing states in a catalysis model with anti-Arrhenius behavior

We study a model of heterogeneous catalysis with competitive reactions between two monomers A and B. We assume that reactions are dependent on temperature and follow an anti-Arrhenius mechanism. In this model, a monomer A can react with a nearest neighbor monomer A or B, however, reactions between monomers of type B are not allowed. We assume attractive interactions between nearest neighbor monomers as well as between monomers and the catalyst. Through mean-field calculations, at the level of site and pair approximations, and extensive Monte Carlo simulations, we determine the phase diagram of the model in the plane y(A) versus temperature, where y(A) is the probability that a monomer A reaches the catalyst. The model exhibits absorbing and active phases separated by lines of continuous phase transitions. We calculate the static, dynamic, and spreading exponents of the model, and despite the absorbing state be represented by many different microscopic configurations, the model belongs to the directed percolation universality class in two dimensions. Both reaction mechanisms, Arrhenius and anti-Arrhenius, give the same set of critical exponents and do not change the nature of the universality class of the catalytic models.