We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a 1D or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. Paying particular attention to 1D, we validate our variational treatment by comparing numerical simulations of a trapped oblate system in 3D with the 1D theory, and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two-components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.

As a test of the Kibble-Zurek mechanism (KZM) of defect formation, we simulate the Bose-Einstein condensation transition in a toroidally confined Bose gas using the stochastic projected Gross-Pitaevskii equation (SPGPE), with and without the energy-damping reservoir interaction. Energy-damping alters the scaling of the winding number distribution with the quench time -a departure from the universal KZM theory that relies on equilibrium critical exponents. Numerical values are obtained for the correlation-length critical exponent ν and the dynamical critical exponent z for each variant of reservoir interaction theory. The energy-damping reservoir interactions cause significant modification of the dynamical critical exponent of the phase transition, whilst preserving the essential KZM critical scaling behavior. Comparison of numerical and analytical two-point correlation functions further illustrates the effect of energy damping on the correlation length during freeze out.

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