The Gross-Pitaevskii Equation and Bose-Einstein condensates

The Gross-Pitaevskii equation is discussed at the level of an advanced course on statistical physics. In the standard literature the Gross-Pitaevskii equation is usually obtained in the framework of the second quantisation formalism, which in many cases goes beyond the material covered in many advanced undergraduate courses. In this paper, we motivate the derivation of the Gross-Pitaevskii equation (GPE) in relationship to concepts from statistical physics, highlighting a number of applications from dynamics of a Bose-Einstein condensate to the excitations of the gas cloud. This paper may be helpful not only in encouraging the discussion of modern developments in a statistical mechanics course, but also can be of use in other contexts such as mathematical physics and modelling. The paper is suitable for undergraduate and graduate students, as well as general physicists.