Thermodynamics of inhomogeneous imperfect quantum gases in harmonic traps

We discuss thermodynamic properties of harmonically trapped imperfect quantum gases. The spatial inhomogeneity of these systems imposes a redefinition of the mean-field interparticle potential energy as compared to the homogeneous case. In our approach, it takes the form a/2 N-2 omega(d), where N is the number of particles, omega-the harmonic trap frequency, d-system's dimensionality, and a is a parameter characterizing the interparticle interaction. We provide arguments that this model corresponds to the limiting case of a long-ranged interparticle potential of vanishingly small amplitude. This conclusion is drawn from a computation similar to the well-known Kac scaling procedure, which is presented here in a form adapted to the case of an isotropic harmonic trap. We show that within the model, the imperfect gas of trapped repulsive bosons undergoes the Bose-Einstein condensation provided d> 1. The main result of our analysis is that in d= 1 the gas of attractive imperfect fermions with a = -a(F) < 0 is thermodynamically equivalent to the gas of repulsive bosons with a = a(B) > 0 provided the parameters a(F) and a(B) fulfill the relation a(B) + a(F) = (h) over bar. This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional (2D) uniform imperfect repulsive Bose and attractive Fermi gases.