Scaling and pattern formation in condensed matter systems

In this dissertation, I present analytical and numerical work regarding the scaling behavior of three physical systems.;I begin by analyzing the scaling behavior of Griffiths ferromagnets near the Griffiths-paramagnetic transition point. By deriving the asymptotic behavior of the magnetization of the system using an ansatz for the Yang-Lee zero density, I find that the scaling behavior of Griffiths ferromagnet is dominated by an essential singularity in the external magnetic field. Excellent agreement is found by comparing this prediction to the experimental data on La 0.7Ca0.3MnO3, from which I also extract the critical exponents.;Next I report on a mathematical framework to describe landscape formation due to carbonate precipitation near geothermal hot springs. I derive analytically the shape and stability of the spherically symmetric domes. The solution agrees with field observations and simulation results. In addition, I apply a similar conceptual framework to study the formation and stability of stalactites in limestone caves. The shape of stalactites is calculated and the solution is found to be unconditionally stable, as opposite to the unstable dome solutions. By studying the linear stability of a uniform sheet of fluid flowing down a constant slope, moreover, I show that our theory gives results that are consistent with the scale-free terraced landscapes observed.;Finally, I study multiscale patterns in polycrystalline materials, with the phase field crystal (PFC) model. I first show that the complex amplitudes representation incorporates the correct form of nonlinear elasticity. I then analyze the plastic properties of the model by applying a shearing force. Dislocation avalanches, which resembling the scaling behavior in driven ferromagnetic, are observed. Critical exponents are extracted from power laws extending over 5 decades. I extend the PFC model to accommodate actual atomic configurations and vacancies. I use the extended PFC model to simulate a liquid and reproduce the correct form of the two-point correlation functions. Finally, I extend the PFC model to describing binary systems. The resulting theory describes both atomic hopping events on microscopic scales and diffusion on macroscopic scales. It also reproduces the activated Arrhenius form of the diffusion coefficient.