Critical phenomena and phase transition in long-range systems
In this dissertation, I study critical phenomena and phase transitions in systems with long-range interactions, in particular, the ferromagnetic Ising model with quenched site dilution and the asset exchange model with growth. In the site-diluted Ising model, I focus on the effects of quenched disorder on both critical phenomena and nucleation. For critical phenomena, I generalize the Harris criterion for the mean-field critical point and the spinodal, and find that they are not affected by dilution, whereas pseudospinodals are smeared out. For nucleation, I find that dilution reduces the lifetime of the metastable state. I also investigate the structure of nucleating droplets in both nearest-neighbor and long-range Ising models. In both cases, nucleating droplets are more likely to occur in spatially more dilute regions. I also modify the asset exchange model to include different types of economic growth, such as constant growth and geometric growth. For constant growth, one agent eventually gets almost all the wealth regardless of the growth rate. For geometric growth, the wealth distribution depends on the way that the growth is distributed among agents, which is represented by the parameter 𝛾. For the evenly distributed growth, 𝛾=0, and as 𝛾 increases, the growth in the total wealth is distributed preferentially to richer agents. For 𝛾=1, the wealth of every agent grows at a rate that is linearly proportional to his/her wealth. I find a phase transition at 𝛾=1. For 𝛾<1, there is an rescaled steady state wealth distribution and the system is effectively ergodic. In this state, the wealth at all ranks grows exponentially in time and inequality stays constant. For 𝛾>1, one agent eventually obtains almost all the wealth, and the system is not ergodic. For 𝛾=1$, the dynamics of the poor agents' wealth is similar to that of a geometric random walk. In addition, I elucidate the effects of unfair trading, inhomogeneity in agents, modified growth which only depends on richest $1% agents' wealth, and a finite range of wealth exchange.