J. T. Lewis

An operational approach to quantum probability

In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.

The exact calculation of long-range forces between atoms by perturbation theory

A general sum rule is described which has many variants; it permits, for example, the exact calculation of the long-range forces between a proton and a hydrogen atom using conventional perturbation theory. The method is exemplified by the calculation of the second-order term and a precise assessment of the significance of the continuum states is made. The basic identity for a given function g is (r|gl8)/(Er — Es)≡(r |f|8 ), where the function f satisfies a certain differential equation (13). The limitation of the method is the difficulty which may arise in solving the equation. The sum rule technique appears to have applications to many fields.

Entropy of ATM traffic streams: a tool for estimating QoS parameters

For the purposes of estimating quality-of-service parameters, it is enough to know the large deviation rate-function of an ATM traffic stream; modeling procedures can be bypassed if we can estimate the rate-function directly, exploiting the analogy between the rate-function and thermodynamic entropy. We show that this proposal is soundly based on statistical sampling theory. Experiments on the Fairisle ATM network at the University of Cambridge have established that it is feasible to collect the required data in real time. >

Quantum measurement and decoherence

Distribution functions defined in accord with the quantum theory of measurement are combined with results obtained from the quantum Langevin equation to discuss decoherence in quantum Brownian motion. Closed form expressions for wave packet spreading and the attenuation of coherence of a pair of wave packets are obtained. The results are exact within the context of linear passive dissipation. It is shown that, contrary to widely accepted current belief, decoherence can occur at high temperature in the absence of dissipation. Expressions for the decoherence time with and without dissipation are obtained that differ from those appearing in earlier discussions.

The large deviation principle and some models of an interacting boson gas

This is a study of the equilibrium thermodynamics of the Huang-Yang-Luttinger model of a boson gas with a hard-sphere repulsion using large deviation methods; we contrast its properties with those of the mean field model. We prove the existence of the grand canonical pressure in the thermodynamic limit and derive two alternative expressions for the pressure as a function of the chemical potential. We prove the existence of condensate for values of the chemical potential above a critical value and verify a prediction of Thouless that there is a jump in the density of condensate at the critical value. We show also that, at fixed mean density, the density of condensate is an increasing function of the strength of the repulsive interaction. In an appendix, we give proofs of the large deviation results used in the body of the paper.

Condensation in the imperfect boson gas

We prove that Bose-Einstein condensation persists in the imperfect boson gas; it is not destroyed by the mean field interaction.

Invited Practical connection admission control for ATM networks based on on-line measurements

We present algorithms for connections admission control (CAC) in an ATM network based upon the use of on-line measurement and estimation. (The algorithms described in this paper are the subject of a UK patent application.) Our approach uses the theory of large deviations, which is a valuable tool for reasoning about rare events in stochastic systems. We have shown previously that the large deviations rate function, or entropy, of ATM traffic can be estimated from activity measurements. This can be used to determine the effective bandwidth of the traffic. In this paper we present practical measurement-based CAC algorithms which are computationally efficient, and can be used with traffic whose statistical characteristics are unknown.

The equivalence of ensembles for lattice systems: Some examples and a counterexample

We describe the problem of the equivalence of ensembles at the level of states for classical lattice systems. We discuss circumstances where the vanishing of the specific information gain of a sequence of microcanonical measures with respect to a sequence of grand canonical measures implies the equivalence of ensembles. We give a simple derivation of a criterion for the vanishing of the specific information gain in terms of thermodynamic functions. The proof uses ideas from the theory of large deviations but is self-contained. We show how the criterion works in a simple model of a paramagnet and in the Ising model of a ferromagnet in any dimension but fails in the case of the Curie-Weiss mean-field model.

Independent Oscillator Model of a Heat Bath: Exact Diagonalization of the Hamiltonian

The problem of a quantum oscillator coupled to an independent-oscillator model of a heat bath is discussed. The transformation to normal coordinates is explicitly constructed using the method of Ullersma. With this transformation an alternative derivation of an exact formula for the oscillator free energy is constructed. The various contributions to the oscillator energy are calculated, with the aim of further understanding this formula. Finally, the limitations of linear coupling models, such as that used by Ullersma, are discussed in the form of some critical remarks.

On generalized condensation in the free boson gas

We investigate the free boson gas with Dirichlet boundary conditions in d-dimensional Euclidean space. For d > 2 we find three types of condensation when the mean density ρ exceeds a critical value ρc, depending on how the bulk limit is taken.