  The relation between the bulk correlation length and the decay length of
thermodynamic Casimir forces is investigated microscopically in two
three-dimensional systems undergoing Bose-Einstein condensation: the perfect
Bose gas and the imperfect mean-field Bose gas. For each of these systems, both
lengths diverge upon approaching the corresponding condensation point from the
one-phase side, and are proportional to each other. We determine the
proportionality factors and discuss their dependence on the boundary
conditions. The values of the corresponding critical exponents for the decay
length and the correlation length are the same, equal to 1/2 for the perfect
gas, and 1 for the imperfect gas.
