Scaling analysis of nonequilibrium processes in random media

Abstract

Statistical physics has the simplest form in the thermodynamic limit. However, especially in the uncharted territory of nonequilibrium phenomena and disordered systems, finding the thermodynamic limit is often made difficult by the absence of general formalism and strong finite-size effects. In order to overcome such difficulties, a systematic scaling analysis must be performed to filter out irrelevant properties. This thesis discusses a few examples of nonequilibrium processes in random media in which such scaling analysis provides useful information about the thermodynamic limit. Chapter 2 discusses the graphicaliy problem of scale-free networks, which is related to the geometric constraint on key structural parameters of many complex systems, namely the degree exponent $\gamma$ and the upper cutoff exponent $\alpha$. Using the graphicality criterion proved by Erd\H{o}s and Gallai, the realizable $\alpha$ is shown to be lower than $1/\gamma$ for $\gamma < 2$, whereas any upper cutoff is possible for $\gamma > 2$. This result is also numerically verified by random and deterministic samplings of degree sequences. Chapter 3 discusses the metastable states of zero-temperature Glauber dynamics on Erd\H{o}s--R\`{e}nyi networks, which concerns the fate of curvature-driven coarsening process in the quenched random media. Previous studies have been inconclusive about how close the system can approach the ground state after the quench from infinite to zero temperature. Simulations at various system sizes reveal that there exist two different self-averaging subgroups of metastable states, one approaching close to the ground state and the other staying far away. Scaling analysis suggests that the latter type might be asymptotically dominant, which implies the existence of large stable domain walls in the quenched random media. Chapter 4 discusses the totally asymmetric simple exclusion process (TASEP) on directed random regular networks, which is a simple model of activ...