Bose–Einstein Condensation of an Ideal Gas in a Solid Torus

ABSTRACT

We study Bose–Einstein condensation of an ideal gas in a solid torus (major radius , minor radius , Dirichlet walls) for – particles and aspect ratios . The low‐energy spectrum is dominated by a soft ladder of states winding around the ring, and this ladder controls the finite‐size thermodynamics. We derive a closed‐form expansion for the ladder energies through eighth order in the thickness. The finite‐ thermodynamics is evaluated exactly in the grand‐canonical and canonical ensembles, and the heat capacity is smooth in both. Measured from the saturation temperature, at which the excited states can hold all particles, the heat‐capacity maximum lies slightly below it in the ball and in the cube but well above it in the torus, by a relative amount decaying as . This dimensional crossover is traced to the soft ring direction by a control computation on a simply connected elongated box. The torus belongs, for every aspect ratio, to the anomalous universality class of condensate statistics; at the saturation temperature its chemical potential lies below the ground‐state energy by an ‐independent amount, leaving a condensate fraction that decays as . A scan over aspect ratios finds no resonant geometry: curvature suppresses spectral shell structure. Quantitative connections to ring‐shaped traps are given.