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Physical Review A

Bose-Einstein condensation in a three-dimensional system at constant pressure

A. N. Chaba, Raj K. Pathria

Published Sep 1, 1978• 8 citations
Publisher

Abstract

Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature ${T}_{c}$, the volume of the system changes abruptly from being proportional to the total number of particles $N$ to being proportional to ${N}^{\frac{3}{4}}$, the number of particles in the condensate rises sharply from $\ensuremath{\propto}{N}^{\frac{1}{4}}$ to $\ensuremath{\propto}N$, and the specific heat ${C}_{P}$ turns out to be proportional to ${N}^{\frac{9}{8}}$. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with ${C}_{P}$, becomes $\ensuremath{\propto}{N}^{\frac{3}{5}}$.

DOI
10.1103/physreva.18.1277

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Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number $N$ of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature ${T}_{c}$ in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of ${x}_{0}$ up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of ${T}_{c}$ for relatively small atom numbers. For instance, an increase of about $+$40% is expected with ${10}^{4}$ atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.

arXiv:1011.6477•Physical Review A•Feb 24, 2011•29 citations
Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature Tc{T}_{c}Tc​, the volume of the system changes abruptly from being proportional to the total number of particles NNN to being proportional to N34{N}^{\frac{3}{4}}N43​, the number of particles in the condensate rises sharply from \ensuremath∝N14\ensuremath{\propto}{N}^{\frac{1}{4}}\ensuremath∝N41​ to \ensuremath∝N\ensuremath{\propto}N\ensuremath∝N, and the specific heat CP{C}_{P}CP​ turns out to be proportional to N98{N}^{\frac{9}{8}}N89​. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with CP{C}_{P}CP​, becomes \ensuremath∝N35\ensuremath{\propto}{N}^{\frac{3}{5}}\ensuremath∝N53​.
Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\ensuremath{\simeq}{T}_{c}$, is approached from above, the volume of the system becomes subextensive, Vc=O(NlnN){V}_{c}=O(\frac{N}{\mathrm{ln}N})Vc​=O(lnNN​), which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume V(=Vc)V(={V}_{c})V(=Vc​), the condensate fraction gradually builds up and becomes nonnegligible as TTT approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below T0{T}_{0}T0​, it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant PPP its volume, over a thermodynamically negligible range of temperatures, would reduce to values O(N12)O({N}^{\frac{1}{2}})O(N21​) which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat CP{C}_{P}CP​, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.
An asymptotic evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) is carried out, under a variety of boundary conditions, without converting summations into integrations. The theoretical results for the Dirichlet boundary conditions (${\ensuremath{\psi}}_{S}=0$) contain all the significant features of the numerical results obtained earlier by Goble and Trainor; in particular, we reproduce the characteristic length D∗{D}^{*}D∗ at which the specific heat of the system is at an absolute maximum. It turns out that D∗{D}^{*}D∗ is directly proportional to the mean interparticle distance l‾\overline{l}l, with a constant of proportionality $c(\ensuremath{\approx}20\ensuremath{-}30)$. No such characteristic length appears if boundary conditions other than Dirichlet's are employed. The shift, and the rounding off, of the specific-heat maximum are also studied, and the distinguishing influence of the boundary conditions examined.
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.
In a recent paper we carried out a rigorous, asymptotic analysis of the low-temperature behavior of a Bose gas confined to a finite cubic geometry and subjected to periodic boundary conditions. That analysis is now extended to Dirichlet, Neumann, and antiperiodic boundary conditions. As before, the thermogeometric parameter yyy and the condensate fraction N0N\frac{{N}_{0}}{N}NN0​​ are studied as functions of temperature and are evaluated explicity for cubes with Ll‾=40and100\frac{L}{\overline{l}}=40 \mathrm{and} 100lL​=40and100, where LLL is the edge length of the enclosure and l‾\overline{l}l the mean interparticle distance in the system. The finite-size corrections to the condensate fraction under Dirichlet or Neumann boundary conditions turn out to be qualitatively different from those under periodic or antiperiodic boundary conditions. This difference becomes manifest when one tries to express these corrections in a form consistent with the standard scaling theory for finite-size effects.
Several authors have proposed criteria for the onset of Bose-Einstein condensation in finite systems. Using mathematical techniques developed by Pathria and collaborators (1972) these various proposals are systematically examined by applying them to an ideal Bose gas confined to a cuboidal enclosure under periodic boundary conditions. The 'transition points' thus obtained have been subjected to a comparative study, which indicates that the most useful general criterion is the macroscopic one based on the maximum of the specific heat C v .
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number NNN of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature Tc{T}_{c}Tc​ in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of x0{x}_{0}x0​ up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of Tc{T}_{c}Tc​ for relatively small atom numbers. For instance, an increase of about $+$40% is expected with 104{10}^{4}104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.