We introduce a time-dependent projected Gross-Pitaevskii equation to describe a partially condensed homogeneous Bose gas, and find that this equation will evolve randomized initial wave functions to equilibrium. We compare our numerical data to the predictions of a gapless, second order theory of Bose-Einstein condensation [S. A. Morgan, J. Phys. B 33, 3847 (2000)], and find that we can determine a temperature when the theory is valid. As the Gross-Pitaevskii equation is nonperturbative, we expect that it can describe the correct thermal behavior of a Bose gas as long as all relevant modes are highly occupied. Our method could be applied to other boson fields. DOI: 10.1103/PhysRevLett.87.160402 PACS numbers: 03.75.Fi, 05.30.Jp, 11.10.Wx The achievement of Bose-Einstein condensation (BEC) in a dilute gas offers the possibility of studying the dynamics of a quantum field at finite temperatures in the laboratory [1,2]. However, direct numerical simulation of the full equations of motion for such systems is well beyond the capability of today's computers. Even equilibrium calculations in the region of a phase transition require nonperturbative methods, meaning that fully quantal treatments are unfeasible.At finite temperature, when there are an appreciable number of noncondensed particles, the fully quantal second order theory of Morgan [3] (and other equivalent treatments [4,5]) should be sufficient for an accurate description of many properties of the dilute Bose gas in equilibrium and away from the region of critical fluctuations. Dynamical treatments are much harder, and in general require significant approximations. For example, calculations have been performed for small systems [6], with a restricted number of modes [7], and for the dynamics of condensate formation where the ground state is assumed to grow adiabatically [8].The Gross-Pitaevskii equation (GPE) has been used to predict the properties of condensates near T 0, when there are very few noncondensate atoms present. Both statically and dynamically it has shown excellent agreement with experiment [9][10][11]. It has been argued, however, that the GPE can be used to describe the dynamics of a BEC at finite temperature [12][13][14]. In the limit where the modes of the system are highly occupied ͑N k ¿ 1͒, the classical fluctuations of the field overwhelm the quantum fluctuations, and these modes may therefore be represented by a coherent wave function. This is analogous to the situation in laser physics, where the highly occupied laser modes can be well described by classical equations.Using this argument, Damle et al. have performed calculations of the approach to equilibrium of a near ideal superfluid [15], and similar approximations to other quantum field equations have been successful elsewhere [16]. also use the GPE to represent the classical modes of a Bose-condensed system. The main advantage of this method is that realistic calculations, while still a major computational issue, are feasible -methods for solving the GPE are well developed. Al...
