Bose-Einstein condensation (BEC) of a noninteracting Bose gas of N particles in a two-dimensional box with Dirichlet boundary conditions is studied. Confirming previous work, we find that BEC occurs at finite N at low temperatures T without the occurrence of a phase transition. The conventionally-defined transition temperature T(E) for an infinite three-dimensional (3D) system is shown to correspond in a 2D system with finite N to a crossover temperature between a slow and rapid increase in the fractional boson occupation N(0)/N of the ground state with decreasing T. We further show that T(E)∼1/logN at fixed area per boson, so in the thermodynamic limit there is no significant BEC in 2D at finite T. Thus, paradoxically, BEC only occurs in 2D at finite N with no phase transition associated with it. Calculations of thermodynamic properties versus T and area A are presented, including Helmholtz free energy, entropy S, pressure p, ratio of p to the energy density U/A, heat capacity at constant volume (area) C(V) and at constant pressure C(p), isothermal compressibility κ(T) and thermal expansion coefficient α(p), obtained using both the grand-canonical ensemble (GCE) and canonical ensemble (CE) formalisms. The GCE formalism gives acceptable predictions for S, p, p/(U/A), κ(T) and α(p) at large N, T and A but fails for smaller values of these three parameters for which BEC becomes significant, whereas the CE formalism gives accurate results for all thermodynamic properties of finite systems even at low T and/or A where BEC occurs.
Bose-Einstein condensation in an ideal (i.e. interactionless) boson gas can be studied analytically, at university-level statistical and solid state physics, in any positive dimensionality ( d >0) for identical bosons with any positive-exponent ( s >0) energy-momentum (i.e. dispersion) relation. Explicit formulae with arbitrary d / s are discussed for: the critical temperature (non-zero only if d / s >1); the condensate fraction; the internal energy; and the constant-volume specific heat (found to possess a jump discontinuity only if d / s >2). Classical results are recovered at sufficiently high temperatures. Applications to `ordinary' Bose-Einstein condensation, as well as to photons, phonons, ferro- and antiferromagnetic magnons, and (very specially) to Cooper pairs in superconductivity, are mentioned.
Abstract The Bose–Einstein condensation (BEC) in three-dimensional (3D) anisotropic lattices is studied. We present theoretical results for the critical temperature for BEC, chemical potential, condensate fraction and relevant thermodynamic quantities like: internal energy, entropy, specific heat and compressibility as a function of anisotropy parameter being the ratio of the nearest-neighbor in-plane (
t∥ t ‖ ) and out-of-plane (
t⊥ t ⊥ ) hopping amplitudes. In particular, considered scenarios include weakly coupled two-dimensional (2D) planes (
t⊥/t∥≪1 t ⊥ / t ‖ ≪ 1 , relevant for layered structures) as well as a rod-like geometry of interacting one-dimensional (1D) chains (
t∥/t⊥≪1 t ‖ / t ⊥ ≪ 1 ). The impact of the dimensional crossover as the system is tuned away from a set of disconnected 2D layers, or traverses from a set of separate 1D chains to a regime where a fully isotropic 3D structure emerges is elucidated. Both numerical and analytic approaches are employed, (the latter in a form of series expansions involving
t∥,t⊥ t ‖ , t ⊥ amplitudes) for internal energy, entropy, specific heat and isothermal compressibility. The theoretical outcome of the present study may be of interest to a number of scenarios in solid-state physics, where the relevant quasi-particles are bosonic-like, as well as might be applicable to the physics of cold bosons loaded in artificially engineered 3D optical lattices.
We investigate Bose-Einstein condensation for ultracold bosonic atoms in two-dimensional systems. The functional renormalization group for the average action allows us to follow the effective interactions from molecular scales (microphysics) to the characteristic extension of the probe
l (macrophysics). In two dimensions the scale dependence of the dimensionless interaction strength $\ensuremath{\lambda}$ is logarithmic. Furthermore, for large
l the frequency dependence of the inverse propagator becomes quadratic. We find an upper bound for $\ensuremath{\lambda}$, and for large $\ensuremath{\lambda}$ substantial deviations from the Bogoliubov results for the condensate depletion, the dispersion relation and the sound velocity. The melting of the condensate above the critical temperature
Tc is associated to a phase transition of the Kosterlitz-Thouless type. The critical temperature in units of the density
Tc∕n vanishes for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ logarithmically.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.