Theoretical Study Of Phase Transition Dynamics In Complex Systems: Developments And Applications Of Mesoscopic Statisitical Methods

With the development of life science and nanotechnology, the study of phase transition dynamics in complex systems has been one of the frontier topics in the fields of statistical physics and theoretical chemistry. The dynamical behaviors in these complex systems usually span a large space scale and a long time scale, such as nucleation and growth in a first-order phase transition, self-assembled nanostructures and dynamical patterns on soft-matter surface and bacterial cloning process. It is known that quantum mechanics and molecular dynamics are limited to treat small systems and short-term behavior, while phenomenological theory and mean-field method lack of direct relation with microscopic mechanism. Therefore, aiming to understanding microscopic mechanism, dynamical laws, and control means, developments and applications of mesoscopic statistical methods is an effective way. Especially, it is of fundamental importance to unveil the roles of fluctuations and disorder in complex systems. In this thesis, we propose a coarse-grained Monte Carlo (CGMC) method that satisfies the conditions of statisitcal consistency and apply the method to study equlibrium and nonequilibrium phase transitions on complex networks, and apply forward flux sampling to study nucleaion dynamics of Ising model on complex network and study self-assmebled vortex pattern transition in a system of self-propelled particles, and study topological- and parametric-induced corehence resonance in forced coupled two-state systems. The detailed summary is described below.CGMC simulations for phase transitions on complex networks.We propose a degree-based coarse-grained approach to simulate phase transitions and critical phenomena on complex networks, which have been successfully applied to two representative systems, i.e., equilibrium Ising model and nonequilibrium epidemic SIS model. Under the annealed network approximation, we prove that our CG approach satisfies the conditions of statistical consistency, that is, consistent equilibrium probabilities distribution and consistent nonequilibrium reactive flux. Extensive simulations show that our CG approach can not only reproduce the phase curves and fluctuations information, but also can efficiently study the size effect of critical phenomena. In addition, the conditions of statistical consistency may be used to test other CG models in the future.Nucleation on complex networksPhase transitions and critical phenomena on complex networks have received considerable attention in recent years. However, most of previous works focus on the relation between network topology and phase transition points. There is almost no mention of the dynamical or kinetic aspects of phase transitions in the literature. We apply umbrella sampling and forward flux sampling to study nucleation of the Ising model on complex networks. For scale-free networks, homogeneous nucleation starts from nodes with smaller degrees, and nucleation rate decreases exponentially with network size and critical nucleus increases with network size. For heterogeneous nucleation, different scaling law between nucleation rate and impurity numbers and degree exponent is found. For nucleation on small-world networks, we find that as rewire probability increases the size-effect of nucleation undergoes a transition. For modular networks, as the modularity worsens nucleation undergoes a transition from two-stage to one-stage process. Interestingly, the nucleation rate shows a nonmonotonic dependence on the modularity, in which a maximal nucleation rate occurs at a moderate level of modularity.Disorder-induced coherence resonanceComplex nonlinear systems are inevitably influenced by noise and disorder, e.g., internal fluctuations, external noise, structural disorder, and diversity so on. By the interaction with nonlinear systems, disorder can sometimes become a driving force of order. Examples include stochastic resonance, coherence resonance, array-enhanced synchronization, and disorder-tamed chaos. We study the effects of topological disorder and parameter disorder on a two-state system subject to a subthreshold signal. It is found that the two types of disorder can both induce coherence resonance. Especially, using a heterogeneous mean-filed theory we present an effective potential of the system, so that numerical results can be well understood.Self-assembled pattern transition for self-propelled particles (SPP)In recent years, the dynamics of SPP have received much attention, which can well describe collective motion in nature such as flocks of birds, schools of fish, bacteria colonies. SPP systems can exhibit rich nonequilibrium spatiotemporal patterns such as flocking, global vortex, droplet vortex, random walk and so on. We apply forward flux sampling to study noise-driven chiral transition of vortex for SPP. We calculate the rate and demonstrate the pathway for the transition, and find that the transition starts from peripheral particles of vortex.