Superfluidity in the Stochastic Limit

The theory of superfluidity was developed by Landau and Bogolyubov (cf. [10, 11, 14]), for an introduction see [12]. An analogue of the approach of [10, 11] was applied to superconductivity theory (cf. [13]). The essence of the superfluidity phenomenon is that the Bose condensate becomes superfluid and friction between condensate and normal phase disappears. Bogolyubov in [10, 11] found that this can be explained as an effect of stabilization of the condensate by interaction between particles. Since the main point of superfluidity is the absence of quantum dissipation, it is natural to expect that the stochastic limit of quantum theory, which is the mathematical theory of quantum dissipation and transport (cf. [8]), should play a role in the description of this phenomenon. In the present paper, using as a starting point the approach of [10, 11], we introduce a Hamiltonian that describes the interaction between the Bose condensate and the normal phase and investigate this Hamiltonian using the stochastic limit approach. The name stochastic limit is due to the property that in this limit, the quantum fields become quantum white noises and the Schrödinger equation becomes a white noise Hamiltonian equation. The main advantage of this new technique is that this limit preserves all the relevant physical information in the scale of interest and at the same time essentially simplifies the mathematical description so to allow to extract this information and put it to use.