The thermodynamic behavior of the one-dimensional Bose-gas–attractive-δ impurity system is studied in this paper. The system is shown to undergo the Bose–Einstein condensation and the cause of the phase transition is attributed to the bound state introduced by the impurity in the free-particle energy spectrum. The condensed phase is composed of particles captured by the impurity, forming a drop of particles well localized in space. This gives to the Bose–Einstein condensation in this system the appearance of the ordinary vapor–liquid phase transition. The expression for the pressure is obtained and is plotted as a function of the temperature and the system length. The shapes of the isothermal curves reinforce the interpretation of a vapor–liquid transition. The evaluation of the heat capacity at constant length shows the existence of a finite discontinuity at the transition temperature.
Through exact numerical solutions we show Bose-Einstein condensation (BEC) for the one-dimensional (1D) bosons with repulsive short-range interactions at zero temperature by taking a particular large size limit. Following the Penrose-Onsager criterion of BEC, we define condensate fraction by the fraction of the largest eigenvalue of the one-particle reduced density matrix. % We show the finite-size scaling behavior such that condensate fraction is given by a scaling function of one-variable: interaction parameter multiplied by a power of particle number. Condensate fraction is nonzero and constant for any large value of particle number or system size, if the interaction parameter is proportional to the negative power of particle number. %Here the interaction parameter is defined by the coupling constant of the delta-function potentials devided by the density. %With the scaling behavior we derive various themodynamic limits where condensate fraction is constant for any large system size; for instance, it is the case even in the system of a finite particle number.
We obtain the thermodynamic properties for a noninteracting Bose gas constrained on multilayers modeled by a periodic Kronig-Penney delta potential in one direction and allowed to be free in the other two directions. We report Bose-Einstein condensation (BEC) critical temperatures, chemical potential, internal energy, specific heat, and entropy for different values of a dimensionless impenetrability $P\ensuremath{\geqslant}0$ between layers. The BEC critical temperature
Tc coincides with the ideal gas BEC critical temperature
T0 when
P=0 and rapidly goes to zero as
P increases to infinity for any finite interlayer separation. The specific heat
CV as a function of absolute temperature
T for finite
P and plane separation
a exhibits one minimum and one or two maxima in addition to the BEC, for temperatures larger than that of BEC
Tc. This highlights the effects due to particle confinement. We then discuss a distinctive dimensional crossover of the system through the specific heat behavior driven by the magnitude of
P. For
T<Tc the crossover is revealed by a change in slope of
logCV(T) and when
T>Tc, it is exhibited by a broad minimum in
CV(T).