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 2N2 ωd, where N is the number of particles, ω—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 = −aF < 0 is thermodynamically equivalent to the gas of repulsive bosons with a = aB > 0 provided the parameters aF and aB fulfill the relation aB + aF = . This result supplements similar recent conclusion about thermodynamic equivalence of two-dimensional (2D) uniform imperfect repulsive Bose and attractive Fermi gases.