|dc.description.abstract||A grand canonical C-field theory has been previously developed for modelling finite temperature Bose gases, the stochastic projected Gross-Pitaesvkii equation (SPGPE) . Previous investigations have shown quantitative agreement between experiments and the SPGPE, with a fitted growth rate [2,3] or no fitted parameters at all . These and other works [5,6] have used the number-damping SPGPE, a sub-theory of the SPGPE neglecting a scattering process between the coherent and incoherent regions that conserves the particle number of each region, known as energy-damping. The systems in these works were in quasi-equilibirium and as such a growth process, also known as number-damping, is thought to be dominant. Evidence suggests that energy-damping is significant when the system is far from equilibrium ; we may also postulate systems where energy-damping is the only allowed process. In this thesis we use the full SPGPE including the energy-damping reservoir interaction in systems where this process plays an important role in the dissipative evolution.
We model quenches of chemical potential across the Bose-Einstein condensation transition in a one dimensional Bose gas confined to a toroid. We use two different models; the full SPGPE and the number-damping SPGPE. The purpose of this is to test the results of our simulations against the predictions of the Kibble-Zurek mechanism (KZM), a theory of defect formation in second order phase transitions. We find that both models give results consistent with KZM, in that various measurable quantities obey a power law with respect to the quench time. The power law exponents are determined by critical exponents, which depend on the universality class of the phase transition. We find the number-damping SPGPE results are consistent with the critical exponents predicted by mean field theory. We are unable to find a universality class with critical exponents consistent with the results of the full theory, and in particular the dynamical critical exponent differs from that predicted by mean field theory.
We also use the SPGPE to simulate the motion of a bright soliton in a one dimensional attractive Bose gas confined to a toroid and in contact with a thermal cloud of a second component. The bright soliton is an analytical solution of the one-dimensional Gross-Pitaevskii equation for an attractive Bose-Einstein condensate, which can propagate in space without changing its functional form. We derive a stochastic differential equation for the soliton velocity, finding that the energy-damping reservoir interaction manifests as an Ornstein-Uhlenbeck process for velocity decay, affording a complete analytic solution for the damping and diffusion rates of the bright soliton. The results of simulating the bright soliton using the SPGPE are compared against the analytic solutions of the velocity stochastic differential equation, including the mean, variance, two-time correlations, and power spectra of the velocity. We find that the numerical and analytical solutions show excellent agreement for all these quantities, validating our procedure for obtaining the velocity equation of motion.||