The thesis is devoted to the study of the influence of local fluctuations in the reactant concentration and reactant transport properties on the change in scaling laws in reaction-diffusion processes, as well as the influence of heterogeneous properties of the catalyst on the equilibrium behavior of the formed adsorbates. Achieving the set goal, several problems were solved.
First, the change in scaling at the crossover between diffusion-limited and reaction-limited cooperative regimes in reaction-diffusion systems was investigated. Such a crossover is analyzed by comparing the universal behavior of a single-species coagulation-diffusion process with stochastic reset at large times on a one-dimensional chain and a Bethe lattice. For this case, the well-studied empty-interval method for one-dimensional systems was extended using the ben-Avraham-Glasser approximation. Logarithmic corrections to the scaling of the particle-density were found, as well as the crossover scaling functions and the effective critical exponents were calculated.
A two-species system of mobile trap-particles that can mutually annihilate or coagulate, while target-particles can be adsorbed by traps was also considered. Anomalous diffusion of particles of both species is modeled by long-range jumps of Lévy flights type. The system is considered in the diffusion-limited regime, which is achieved for the space dimension below the upper critical one. The field-theoretic renormalization group method was used to study the scaling behavior of such a two-species system. Universal exponents for the density and the density-density correlation function of target-particles in the one-loop approximation were calculated. When replacing the space dimension with an effective one (that depend on the control parameter of the Lévy distribution), the obtained exponents can be related to similar expressions for the case of the ordinary diffusion. In the one-dimensional case, the obtained analytical results for the density decay exponent of target-particles are confirmed by the numerical simulations.
To study the influence of catalyst heterogeneous properties on the equilibrium behavior of the adsorbates formed in the process of catalytically activated reactions, the problem was considered in one-dimensional and effectively high-dimensional cases. At the same time, the structure of the catalytic substrate, which is often not precisely defined geometrically, was considered as a set of mobile or localized randomly distributed catalytic sites or bonds between sites.
In the first problem, the equilibrium properties of the adsorbate formed in the process of catalytically activated reactions A + B → ⊘ in a two-species system on chains with catalytic elements (bonds between sites -- model I and sites -- model II), placed randomly according to the annealed disorder scenario (in equilibrium with the system) and quenched one (localized) are investigated. In the case of one-dimensional chains, exact solutions were found for both types of catalytic elements and for both types of disorder. In the case of annealed disorder for models I and II, exact expressions for the disorder-averaged grand canonical partition function and, therefore, for the adsorbate pressure and its thermodynamic derivatives were obtained. In the case of quenched disorder, the problem of averaging the logarithm of a grand canonical partition function is solved by two approaches.
In the second problem, the equilibrium properties of the adsorbate in such a two-species system on Bethe and Hushimi pseudolattices are investigated. In this problem, we extend model I, where the catalytic bonds are placed randomly according to the annealed disorder scenario and consider that the same species may interact with each other when they meet at neighboring sites. For two types of pseudolattices, for the symmetric case with equal chemical potentials of both species and the same interaction of the same species, a full phase diagram of the two-component adsorbate was obtained. It is shown that the phase diagram is quite complex and consists of several phases. Moreover, due to its bipartite nature, two additional phases with structural ordering exist on the Bethe lattice, whereas on the Hushimi lattice, such phases are absent due to stronger frustration effects.
