atoms $N=\sum_{\bf p}(a^{\dagger}_{\bf p}a_{\bf p})$. The corresponding Gibbs statistical operator for the above truncated Hamiltonian reads $$\label{eq:3.10}
w(\Psi)\simeq\exp\left[\Omega-\beta\left({\cal H}^{(0)}(\Psi)+{\cal H}^{(2)}(\Psi)\right)\right],$$ where $$\begin{gathered}
{\cal H}^{(0)}(\Psi)=H^{(0)}(\Psi)-\mu{\cal V}\left(\Psi^{*}\Psi\right), \label{eq:3.11} \\
{\cal H}^{(2)}(\Psi)=H^{(2)}(\Psi)-\mu\sum_{{\bf p}\neq 0}\left(a^{\dagger}_{\bf p}a_{\bf p}\right). \label{eq:3.12}\end{gathered}$$ The grand thermodynamic potential $\Omega$ as a function of reciprocal temperature $\beta={1/T}$, chemical potential $\mu$, and variational parameter $\Psi_{\alpha}$ is found from the normalization condition ${\rm Tr}\,w=1$, $$\label{eq:3.13}
\Omega=\beta{\cal H}^{(0)}(\Psi)-\ln{\rm Tr}\left[ \exp(-\beta{\cal H}^{(2)}(\Psi))\right],$$ where the trace is taken in the space of occupation numbers of bosons with ${\bf p}\neq 0$. In the standard Bogoliubov approach the relation between the condensate wave function and chemical potential is determined by the $c$-number part of the thermodynamic potential assuming that it represents the leading term, $$\label{eq:3.14}
\omega^{(0)}={U(0)\over 2}(\Psi^{*}\Psi)^{2}+{J(0)\over 2}(\Psi^{*}\lambda^{a}\Psi)^{2}-h(\Psi^{*}\lambda^{2}\Psi)-\mu(\Psi^{*}\Psi),$$ where we introduced the thermodynamic potential density $\omega=\Omega/\beta{\cal V}$. The later, up to a sign, coincides with pressure $P$, $\omega=-P$ and it is employed when studying macroscopic dynamics of superfluid systems, both classical and relativistic [@JPhysA; @TMF]. The variation of Eq. (\[eq:3.14\]) over $\Psi^{*}_{\alpha}$ yields $$\label{eq:3.15}
\mu\Psi_{\alpha}-U(0)(\Psi^{*}\Psi)\Psi_{\alpha}-J(0)(\Psi^{*}\lambda^{a}\Psi)\lambda^{a}_{\alpha\beta} \Psi_{\beta}+h\lambda^{2}_{\alpha\beta}\Psi_{\beta}=0$$ (we do not write the complex conjugate equation). This equation ensures the minimum of the thermodynamic potential and gives a relation between the chemical potential and condensate wave function. The contribution of 