Phase Transitions in Quantum Microcanonical Equilibrium

2013-02-17T03:16:44Z (GMT) by Daniel Hook
<p>Thesis: Submitted for the degree of Doctor of Philosophy of the University of London (Imperial College), 2007.</p> <p>Supervisor: Dorje C Brody</p> <p>Collaborators: Carl M Bender, Dorje C Brody, Lane P Hughston</p> <p>Examiners: Professor Des Johnston, Herriot-Watt University (external); Dr Ian RC Buckley, Kings College London<br><br></p> <p>A mathematically consistent formulation of quantum statistical mechanics is developed for finite-dimensional quantum systems. A quantum microcanon- ical postulate is proposed as a basis for a new equilibrium state of a closed system. Equilibrium is characterised by a uniform distribution on a level sur- face of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, how- ever, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures, without the consideration of thermodynamic limits. Several examples are investigated in detail to work out their physical characteristics in equilibrium. The energy variance and Fisher information are calculated for systems in microcanonical equilibrium: the new equilibrium ensemble has the physically desirable property that it represents the minimum energy uncertainty state. For a system that has nondegenerate equally spaced en- ergy eigenvalues we have introduced a scaling scheme to consider the limit as the number of energy levels approaches infinity. In this limit, the density of states is shown to converge to a delta-function centred at the intermediate value, (E_max + E_min)/2, of the energy. Determining this limit requires an elaborate asymptotic study of an infinite sum whose terms alternate in sign.</p>