The Casimir effect is an interaction between the boundaries of a finite system when fluctuations in that system correlate on length scales comparable to the system size. In particular, the critical Casimir effect is that which arises from the long-ranged thermal fluctuation of the order parameter in a system near criticality. Recent experiments on the Casimir force in binary liquids near critical points and 4He near the superfluid transition have redoubled theoretical interest in the topic. It is an unfortunate fact that exact models of the experimental systems are mathematically intractable in general. However, there is often insight to be gained by studying approximations and toy models, or doing numerical computations. In this work, we present a brief motivation and overview of the field, followed by explications of the O(2) model with twisted boundary conditions and the O(n->infinity) model with free boundary conditions. New results, both analytical and numerical, are presented.

The notion of phase transitions as a characterization of a change in physical properties

pervades modern physics. Such abrupt and fundamental changes in the behavior of physical systems are evident in condensed matter system and also occur in nuclear and subatomic settings. While this concept is less prevalent in the field of biology, recent advances have pointed to its relevance in a number of settings. Recent studies have modeled both the cell cycle and cancer as phase transition in physical systems. In this dissertation we construct simplified models for two biological systems. As described by those models, both systems exhibit phase transitions.

The first model is inspired by the shape transition in the nuclei of neutrophils during differentiation. During differentiation the nucleus transitions from spherical to a shape often described as “beads on a string.” As a simplified model of this system, we investigate the spherical-to-wrinkled transition in an elastic core bounded to a fluid shell system. We find that this model exhibits a first-order phase transition, and the shape that minimizes the energy of the system scales as (µr 3 /κ).

The second system studied is motivated by the dynamics of globular proteins. These proteins may undergoes conformational changes with large displacements relative to their size. Transitions between conformational states are not possible if the dynamics are governed strictly by linear elasticity. We construct a model consisting of an predominantly elastic region near the energetic minimum of the system and a non-linear softening of the system at a critical displacement. We find that this simple model displays very rich dynamics include a sharp dynamical phase transition and driving-force-dependent symmetry breaking.

Electrostatic interactions are shown to exert a significant effect on the buckling instability of a rod. In particular, the threshold value of the compressional force needed to induce buckling is found to be independent of rod length for long charged rods. In the case of rods of intermediate length, the critical buckling force crosses over from the classic inverse-square length dependence to asymptotic length-independent form with increasing rod length. It is suggested that this effect leads to the possibility of electromechanical stiffening of nanotubes, which would allow relatively long segments of them to be used as atomic force probes. (C) 2004 American Institute of Physics.