Casimir force induced by imperfect Bose gas

We present a study of the Casimir effect in an imperfect (mean-field) Bose gas contained between two infinite parallel plane walls. The derivation of the Casimir force follows from the calculation of the excess grand canonical free energy density under periodic, Dirichlet, and Neumann boundary conditions with the use of the steepest descent method. In the one-phase region the force decays exponentially fast when distance $D$ between the walls tends to infinity. When Bose-Einstein condensation point is approached the decay length in the exponential law diverges with critical exponent $\nu_{IMP}=1$, which differs from the perfect gas case where $\nu_{P}=1/2$. In the two-phase region the Casimir force is long-range, and decays following the power law $D^{-3}$, with the same amplitude as in the perfect gas.