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    Towards exact numerical solutions of quantum many-body problems in ultracold Bose gases : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2022) Yang, Mingrui
    The main objective of this thesis is to examine how the full configuration interaction quantum Monte Carlo (FCIQMC) method can be best utilized for studying ultracold Bose gases. FCIQMC is a stochastic approach for finding the ground state of a quantum many-body Hamiltonian. It is based on the dynamical evolution of a walker population in Hilbert space, which samples the ground state configuration vector over many iterations. The method has been previously applied to studies of the electronic structure of molecules, solids and certain spin models, as well as recently to ultracold Fermi gases. Whereas in this work we are interested in using the method to examine ultracold bosonic atoms. In this thesis, we cover methodological developments and applications of the FCIQMC method. Two main themes are covered in this thesis: methodological developments and applications of the full configuration interaction quantum Monte Carlo method. Firstly, we present a modification of the original protocol in FCIQMC for walker population control of Booth et al. [J. Chem. Phys. 131, 054106 (2009)] in order to achieve equilibration at a pre-defined average walker number and to avoid walker number overshoots. Next, we investigate a systematic statistical bias found in FCIQMC, known as the population control bias, that originates from controlling a walker population with a fluctuating shift parameter and can become large in bosonic systems. We use an exactly solvable stochastic differential equation to model the bias. Lastly, we showcase an application of FCIQMC in studying the properties of the lowest-energy momentum eigenstates, known as yrast states, of Bose gases coupled with a mobile impurity in one spatial dimension. Based on the results of our computations, we identify different dynamical regimes: the polaron and depleton regimes and transitions between them.
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    Melting temperatures of the noble gases from ab-initio Monte Carlo simulations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2019) Smits, Odile R.
    This thesis describes simulations to determine the melting temperatures of the noble gases based on first-principles ab-initio methods. The melting temperatures of bulk krypton, xenon, radon and oganesson are determined using parallel-tempering Monte Carlo with the interaction potential approximated by two- and three-body contributions. The employed interaction potentials are obtained from relativistic coupled cluster theory including spin-orbit coupling and are the most accurate ab-initio potentials to this date. These potentials are fitted to computationally efficient functions utilized to calculate the interaction energy during the Monte Carlo melting simulation. Two different techniques of obtaining the melting temperature are presented. First, the melting temperature is studied by simulating finite clusters in a canonical ensemble. The melting temperature is then deducted from extrapolation of the finite cluster results to the bulk. Second, the melting temperature is determined by direct sampling of the bulk using cells with periodic boundary conditions in the isobaric-isothermal ensemble. Upon correction for superheating, an excellent agreement to the melting temperatures obtained from cluster simulations is obtained. The numerically determined melting temperatures of krypton and xenon are in close agreement with available experimental data. That is, for krypton a melting temperature of 109.5 K and 111.7 K is obtained for cluster and periodic simulations respectively, which is approximately 5 Kelvin lower than the corresponding experimental value of 115.78 K. The melting point of xenon is determined to be 156.1 K and 161.6 K respectively, which compares to the experimental value of 161.40 K. The long debated value of the radon melting temperature of 202 K is confirmed by our simulations (200 K for both techniques). And finally, the melting point of oganesson is determined to be 330 K and therefore surprisingly high compared to the other rare gases. This implies that oganesson is a solid at room temperature. Furthermore, an analytical formula to compute the temperature of the solid-liquid phase transition based on the analytically expressed bulk modulus and interaction potential is presented, and the superheating correction factor is evaluated.
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    A Monte Carlo study of the effect of sample bias on the multinomial logit coefficients : a thesis presented in partial fulfilment of the requirements of the degree of the Master of Business Studies at Massey University
    (Massey University, 1996) Bell, Grant K
    This thesis reports the findings of a Monte Carlo simulation into the effect of sample bias on the parameters of the multinomial logit (MNL) choice model. At issue is the generalisability of parameter estimates obtained from biased samples to the balance of the population. An actual data set of 164 respondents was used to estimate an aggregate model. Using these parameters as the true coefficients of choice behaviour, an unbiased sampling distribution of the MNL parameters was derived by repeatedly fitting aggregate models to artificially generated sets of individual responses. Subsequently, the biased sampling distribution was derived by selectively eliminating those individuals at the tails of the sample distribution based on their correlation with one of the independent variables. The expected values of the biased and unbiased sampling distributions were compared to assess the sensitivity of the model to sample bias. The research found the biased coefficients changed by significantly more than the proportion of individuals removed. However, this sensitivity was predictable as the percentage change in the value of the coefficients was related to the size of the coefficient. It was also found that the coefficients of the unbiased variables were not significantly influenced by bias on another variable. The ratio between the unbiased variables was also maintained. It was concluded that although sensitive to bias, the estimates produced by the MNL model could be modified to reflect the different effect of the bias on the coefficients. Additionally, there was no evidence to suggest that the MNL estimates were not reflecting the effects of interest when calibrated on unbiased samples.