St. Petersburg Paradox and Failure Probability

The St. Petersburg paradox provides a simple paradigm for systems that show sensitivity to rare events. Here, we demonstrate a physical realization of this paradox using tensile fracture, experimentally verifying for six decades of spatial and temporal data and two different materials that the fracture force depends logarithmically on the length of the fiber. The St. Petersburg model may be useful in a variety fields where failure and reliability are critical

Onsager's missing steps retraced

Onsager's paper on phase transition and phase coexistence in anisotropic colloidal systems is a landmark in the theory of lyotropic liquid crystals. However, an uncompromising scrutiny of Onsager's original derivation reveals that it would be rigorously valid only for ludicrous values of the system's number density (of the order of the reciprocal of the number of particles) Based on Penrose's tree identity and an appropriate variant of the mean-field approach for purely repulsive, hard-core interactions, our theory shows that Onsager's theory is indeed valid for a reasonable range of densities

Dimensional Analysis and the Time Required to Urinate

According to the recently discovered 'Law of Urination', mammals, ranging in size from mice to elephants, take, on the average, 21s to urinate. We attempt to gain insights into the physical processes responsible for this uniformity using simple dimensional analysis. We assume that the biological apparatus for urination in mammals simply scales with linear size, and consider the scenarios where the driving force is gravity or elasticity, and where the response is dominated by inertia or viscosity. We ask how the time required for urination depends on the length scale, and find that for the time to be independent of body size, the dominant driving force must be elasticity, and the dominant response viscosity. Our note demonstrates that dimensional analysis can indeed readily give insights into complex physical and biological processes

The Freedericksz Transition in a Spatially Varying Magnetic Field

Much is known about the Freedericksz transition induced by uniform electric and magnetic fields in nematic liquid crystals. In this work, we are interested in the effects of a spatially varying field on the transition. Specifically, we study the director configuration in a homeotropic nematic cell in a spatially varying magnetic field with cylindrical symmetry. The experiment is conducted with a ring magnet which provides a radial magnetic field with magnitude monotonically decreasing to zero at the center. The nematic cell is positioned in the central plane of the ring, with the cell normal parallel to the ring normal. Interference patterns of the nematic cell between crossed polarizers were observed. The director configuration in the nematic cell is modeled with Frank–Oseen theory, and the computed interference pattern from the simulated director field are compared with experiment. We conclude that if the magnetic field strength varies with position in the plane of the cell, there is no Freedericksz transition

Impedance Matching in an Elastic Actuator

We optimize the performance of an elastic actuator consisting of an active core in a host which performs mechanical work on a load. The system, initially with localized elastic energy in the active component, relaxes and distributes energy to the rest of the system. Using the linearized Mooney-Rivlin hyperelastic model in a cylindrical geometry and assuming the system to be overdamped, we show that the value of the Young's modulus of the impedance matching host which maximizes the energy transfer from the active component to the load is the geometric mean of Young's moduli of the active component and the elastic load. This is similar to the classic results for impedance matching for maximizing the transmittance of light propagating through dielectric media.Comment: 6 pages, 2 figure

Contributions of Repulsive and Attractive Interactions to Nematic Order

Both repulsive and attractive molecular interactions can be used to explain the onset of nematic order. The object of this paper is to combine these two nematogenic molecular interactions in a unified theory. This attempt is not unprecedented; what is perhaps new is the focus on the understanding of nematics in the high density limit. There, the orientational probability distribution is shown to exhibit a unique feature: it has compact support on configuration space. As attractive interactions are turned on, the behavior changes, and at a critical attractive interaction strength, thermotropic behavior of the Maier-Saupe type is attained.Comment: 14 pages, 4 figure

Surface anchoring energy of cholesteric liquid crystals

In this paper, we propose a suitable surface energy expression for cholesteric liquid crystals. We show that there exists a symmetry allowed term for chiral nematics that doesn’t appear in the traditional Rapini-Papoular surface energy form. We discuss some consequences of this new surface anchoring term

Curvature-driven Foam Coarsening on a Sphere: A Computer Simulation

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state