No description available.

Elementary Wigner function calculations of the infinite square well and Schroedinger cat states are presented as an introduction to the quasi-probability function. An entangled cat state is calculated and the Wigner function of the state is found. Properties of the entanglement of the state and the nature of its entanglement are found to be distinguishable by this distribution. This work is mostly concerned with obtaining the Wigner function via a path integral method, following a previously published technique. The method approximates the ground state Wigner function by finding the classical path associated with each point in phase space, assuming the P-function of the Hamiltonian of the system is able to be found. The imaginary part of action determines the phase of the path integral and depends on the geometry of the path; specifically the area which it encloses. An investigation into two systems, the Morse potential and the double well potential, was performed to try and find classical paths enclosing area and thus recreating the negative features of the exact Wigner function. The minimisation of the action found the classical path for each phase space point. This was performed numerically using tools created in Excel and Mathematica. In general, it was discovered that the classical paths did not enclose any area and therefore the Wigner function approximations were everywhere positive. The majority of those paths which were found to enclose some area produce a phase which is not large enough to change the sign of the path integral.

In this thesis we study the BPS spectrum and vacuum moduli spaces of membrane matrix models derived from dimensional reduction of the BLG and ABJM M2- brane theories. We explain how these reduced models may be mapped into each other, and describe their relationship with the IKKT matrix model. We construct BPS solutions to the reduced BLG model, and interpret them as quantized Nambu- Poisson manifolds. We study the problem of topologically twisting the reduced ABJM model, and along the way construct a new twist of the IKKT matrix model. We construct a cohomological matrix model whose partition function localizes onto the BPS moduli space of the ABJM matrix model. This partition function computes an equivariant index enumerating framed BPS states with specified R-charges.

Driven cyclic or periodic motion is a recurrent feature of many of the microscopic mechanical systems that support life. For example, fields of cilia, nano-scale hair-like structures, beat together to transport fluids though mammalian tracts. Bacteria and other micro-organisms are able to swim using similar organelles known as flagella, while "molecular motors" provide traction in muscle tissue. These mechanisms rely on the hydrodynamics and statistical mechanics of driven cyclic motion at the micro-scale. The purpose of this thesis is to investigate these principles in an abstract sense, in order to better understand these aspects of biology and to provide a framework for the design of future biomimetic devices. In particular, two aspects of rotational motion at low Reynolds number, the viscous dominated regime occupied by microorganisms and micro-machines,are investigated in this work. First, hydrodynamic synchronisation at low Reynolds number is considered. A model system is created that comprises two colloidal spheres driven along circular paths. The driving forces are applied using optical tweezers, a tool that employs a highly focussed laser beam to exert known forces on micro-particles. Each sphere is driven such that it experiences a given optical force profile, but the net force, and thus the resulting rotation rate, are free to vary. A fluid-mediated interaction force also acts on the spheres, and spontaneously induces synchronisation of their rotational motion. This system is an experimental demonstration of minimal models that were previously proposed to describe the synchronous behaviour of flagella. Synchronisation is only possible under certain conditions at low Reynolds number. In the system employed here, synchronisation can occur either via small deformations of each sphere's circular path, or by modulation of the optical driving force. Synchronisation strength is found to depend on these two mechanisms as predicted by theory. Next, the effect of thermal fluctuations on a rotating system are considered. A micro-rotor that experiences a torque when optically trapped is fabricated using photo-polymerisation. This rotor is used to experimentally demonstrate a rotational Fluctuation Theorem, which describes the probability of observing a trajectory over which the surrounding medium does work on the rotor. In the macro-world, a trajectory of this kind would be said to violate the second law of thermodynamics, but is made possible at the micro-scale because the relevant forces and energies are similar in magnitude to the thermal energy of the system. The probability of observing these trajectories is shown to decrease exponentially with the time over which the rotor is studied, as predicted by the Fluctuation Theorem.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.