Statistical physics of topological emulsions and expanding populations

This thesis studies how microscopic interactions lead to large scale phenomena in two very different systems: two-dimensional liquid crystals and expanding populations.;First, we explore the interactions among circular droplets embedded in a two-dimensional liquid crystal. The interactions arise due to anchoring boundary conditions on the surface of the inclusions and the elastic deformations of the orientational order parameter in the continuous phase. We analytically compute the texture around a single droplet and the far-field droplet-droplet pair potential. The near-field pair potential is computed numerically. We find that droplets attract at long separations and repel at short separations, which results in a well-defined preferred distance between the droplets and stabilization of the emulsion. Self-organization, barriers to coalescence, and the effects of thermal fluctuations are also discussed.;Second, we study the role of randomness in the number of offspring on the evolutionary dynamics of expanding populations. Several equally fit genetic variants (alleles) are considered. We find that spatial expansion combined with demographic fluctuations leads to a substantial loss of genetic diversity and spatial segregation of the alleles. The effects of these processes on recurring mutations and selective sweeps are studied as well.;Third, the competition between two alleles of different fitness is investigated. We find that the essential features of this competition can be captured by a non-linear reaction-diffusion equation. During a range expansion the fitter allele forms growing sectors that eventually engulf the less fit allele. The applications to measuring relative fitness in microbiological experiments are discussed.;Finally, we analyze how a combination of strong stochasticity and weak competition affects the spreading of beneficial mutations in stationary, non-expanding, populations.