We analyze the behavior of the anomalous density as function of the radial distance at different temperatures in a variational framework. We show that the temperature dependence of the anomalous density agrees with the Hartree-Fock-Bogoliubov (HFB) calculations. Comparisons between the normal and the anomalous fractions at low temperature show that the latter remains higher and consequently the neglect of the anomalous density may destabilize the condensate. These results are compatible with those of Yukalov. Surprisingly, the study of the anomalous density in terms of the interaction parameter shows that the dip in the central density is destroyed for sufficiently weak interactions. We explain this effect.

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We use the time-dependent variational principle of Balian and Vénéroni to derive a set of equations governing the dynamics of a trapped Bose gas at finite temperature. We show that this dynamics generalizes the Gross-Pitaevskii equations in that it introduces a consistent dynamical coupling between the evolution of the condensate density, the thermal cloud and the "anomalous" density.

The time-dependent variational principle proposed by Balian and Vénéroni is used to provide the best approximation to the generating functional for multi-time Green's functions of a set of (bosonic) observables Q µ . By suitably restricting the trial spaces, the computation of the two-time Green's function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature Φ 4 -theory in curved space and coupled to gravity.