Finite-size Scaling In The Canonical Ensemble And Some Properties Of The Percolation Model

Many-body interacting systems play an important role in physical sciences. They possess lots of interesting properties, including:state of an equilibrium system can be described by a few parameters; there exist universal features which tell that systems with quite different microscopic parameters behave the same; when tuning certain parame-ters of a system, it may undergo phase transitions, which relate to changing of symmetry or topology properties, and rich critical phenomena exist near phase transition points. Due to the large number of degrees of freedom and nonlinear interactions between the components, it is very challenging to explore properties of a many-body system. Only a few systems can be solved exactly, such as the two dimensional Ising model and some one dimensional quantum lattice models. For other cases we perform approximate calculations (such as mean-field calculation), conduct experiments, or make computer simulations. Numerical simulations have been putting forward tremendously the study of many-body interacting systems since the invention of electronic computers. Promi-nent types of simulation algorithms include Monte Carlo, molecular dynamics, transfer-matrix, exact diagonalization and numerical renormalization methods etc. Combining with coarse-grained models, renormalization theory or scaling analysis etc., they have led to enormous useful information. However, there are still important physics prob-lems remaining to be answered, such as to what extent certain properties are universal, how to gain information of a very large system from simulation results for small sys-tems, and what kind of coarse-grained models we should use to capture the relevant physics.Focusing on the study of the above problems, this thesis presents our research on several topics in the field of critical phenomena. Chapter 1 introduces the theories, models, and numerical methods used in the research.Chapter 2 and 3 contain our study on the finite-size scaling in the canonical en-semble. In chapter 2, by scaling analysis, we first derive the Fisher renormalization for finite-size systems in the canonical ensemble, which relates to the dependence of parti-cle density on the system size in the grand-canonical ensemble. Then we consider the fluctuation-suppression effects, and derive the change of the scaling of the wrapping probability. In the grand-canonical ensemble, the critical value of wrapping probabili-ties are universal. We derive that, for systems with 2yt-d>0 (yt being the exponent of the thermal scaling field, d being the spatial dimension), the critical value of wrapping probabilities are different from their values in the grand-canonical ensemble, however, they are still universal. Other dimensionless observables, such the Binder ratio, show similar behavior. Scaling analysis also tells us that there exist new finite-size corrections in the canonical ensemble, such as a term with exponent -|2yt-d|. For both ensembles, we simulated the pure and dilute Potts model by Monte Carlo methods, which verifies these analytical results. Chapter 3 presents our study on percolation in the canonical ensemble. For the percolation model, the density of particles (bonds or occupied sites) does not depend on the system size, thus different scaling behaviors between canonical and grand-canonical ensembles come solely from the suppression of particle number fluctuation. We derive a new correction exponent 2yt-d in the canonical ensemble (the percolation model has 2yt-d< 0 for d≥ 2), which agrees with results in Chapter 2. More interestingly, we find that quantities which are universal in the grand-canonical ensemble may not be universal, such as the excess cluster number. These analyses have also been verified by Monte Carlo simulations.Though percolation is simple by definition, it contains rich physical contents. Chap-ter 4 and 5 show our further exploration on the percolation model. In Chapter 4, we present our study on short-range correlations in percolation. Based on exact results for the Potts model in two dimensions, we derive exact values of the nearest-neighbor con-nectivity for the percolation model in the square, triangular and honeycomb lattices. Re-sults from Monte Carlo simulations agree with these values. In the simulations, we also observed the fluctuation of neighboring connectivities, and found logarithmic behaviors in the finite-size scaling. We relate the fluctuation to four-point correlation functions studied in the literature, and derive the observed logarithmic behaviors from a logarith-mic factor in the four-point correlations obtained by conformal field theory analysis in the literature. Chapter 5 reports our study on hole-size distribution in percolation clus-ters. We find from the simulation results that a hyperscaling relation T’= 1+ d’/d’F is satisfied, where r’is the Fisher exponent which describes the hole-size distribution in a large percolation cluster, d’is the fractal dimension of the percolation cluster, and d’F is the dimension of the holes. Recently, a variant percolation model was proposed to explain the distribution of clusters formed in quasi two-dimensional active gels, which exhibit critical properties. The variant model can capture qualitatively the critical ex-ponent observed in experiment, but not quantitatively. Via Monte Carlo simulations, we discover that the hole-size distribution of the largest cluster in ordinary percolation can quantitatively explain the critical exponent observed in the experiment, while the clusters observed in the variant percolation model is equivalent to holes of the largest backbone cluster in ordinary percolation. We also find that for site percolation on the triangular lattice, site or bond percolation on the square lattice, under proper definition, the size of the largest hole in the largest percolation cluster is half of the system size in the limit L →∞.