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Physical Review A

Comment on the Specific Heat of a Bose Gas Confined to a Thin-Film Geometry

Stanley Greenspoon, Raj K. Pathria

Published Nov 1, 1973• 23 citations
Publisher

Abstract

Our previous evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry is extended to second order in ($\frac{\overline{l}}{D}$), where $\overline{l}$ is the mean interparticle distance. Explicit expressions, involving a dimensionless parameter $y$, are obtained both for the specific heat of the system and for the temperature at which the specific heat is maximum. Occurrence of the parameter $y$, which is a measure of the chemical potential of the system in terms of the spacing of the single-particle energy levels, enables us to cast our expression for the specific heat of the system into a form consistent with the scaling theory of Fisher and Barber. Our results for the specific-heat maximum, as a function of ($\frac{D}{\overline{l}}$), are a considerable improvement over the first-order ones reported earlier, and have come remarkably close to the ones obtained numerically by Goble and Trainor.

DOI
10.1103/physreva.8.2657

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Effect of boundary conditions on finite Bose-Einstein assemblies

Z. R. Hasan, D. F. Goble

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Physical Review A•Aug 1, 1974•10 citations
Our previous evaluation of the specific heat of an ideal Bose gas confined to a thin-film geometry is extended to second order in (l‾D\frac{\overline{l}}{D}Dl​), where l‾\overline{l}l is the mean interparticle distance. Explicit expressions, involving a dimensionless parameter yyy, are obtained both for the specific heat of the system and for the temperature at which the specific heat is maximum. Occurrence of the parameter yyy, which is a measure of the chemical potential of the system in terms of the spacing of the single-particle energy levels, enables us to cast our expression for the specific heat of the system into a form consistent with the scaling theory of Fisher and Barber. Our results for the specific-heat maximum, as a function of (Dl‾\frac{D}{\overline{l}}lD​), are a considerable improvement over the first-order ones reported earlier, and have come remarkably close to the ones obtained numerically by Goble and Trainor.
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.
The specific heat at constant density of an ideal-Bose-fluid film of thickness $L=l{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, but infinite lateral extent, is calculated analytically to order ${l}^{\ensuremath{-}2}$. Both hard-wall and periodic boundary conditions are considered. Good agreement with the numerical calculations of Goble and Trainor for the total specific heat under hard-wall conditions is obtained down to $l\ensuremath{\simeq}10$. In the critical region, the large-lll behavior accords with the scaling theory of finite-size effects. The appropriate scaling functions and the surface specific heat are explicitly calculated.
The recent observation by Hasan and Goble of a discrepancy between their numerical results on the specific heat of an ideal Bose gas confined to a thin-film geometry and the earlier analytic calculations of Pathria employing Neumann boundary conditions is shown to be unjustified. In those cases where a true comparison is possible, e.g., under Dirichlet or periodic boundary conditions, the observed deviations turn out to be insignificant if one uses the more recent second-order calculations of Greenspoon and Pathria rather than the original first-order ones of Pathria. For comparison under Neumann boundary conditions, fresh numerical calculations are required. We also present here second-order analytic calculations of the specific heat of this system (i) under mixed boundary conditions (Dirichlet on one wall, Neumann on the other) and (ii) under antiperiodic boundary conditions, and compare them with the relevant numerical calculations of Hasan and Goble.
An asymptotic evaluation of the specific heat of an ideal relativistic Bose gas confined to a cuboidal enclosure (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$) is carried out, under periodic boundary conditions, taking into account the possibility of particle-antiparticle pair production in the system. Finite-size corrections to the standard bulk behavior are calculated explicitly in the regions t>0t>0t>0 and t<0t<0t<0, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, such that $|t|\ensuremath{\ll}1$ and $|{L}_{i}t|\ensuremath{\gg}1$. While for t>0t>0t>0 finite-size corrections turn out to be exponential for all geometries, for t<0t<0t<0 this is true only in the case of a film; for other geometries, such as a cuboid or a rectangular channel, these corrections conform to a power law instead. Finally, we consider the situation in the core region, where ∣Lit∣=O(1)|{L}_{i}t|=O(1)∣Li​t∣=O(1), and examine the location t∗{t}^{*}t∗ and the height ${c}_{\ensuremath{\rho}}^{*}$ of the specific-heat maximum; finite-size corrections in this region turn out to be $O({L}_{<}^{\ensuremath{-}1})$, where L<{L}_{<}L<​ denotes the shortest side of the enclosure.
It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$ with L1{L}_{1}L1​, ${L}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\infty}>{L}_{3}\ensuremath{\gg}{\ensuremath{\lambda}}_{T}=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}$) is found to go from order $\frac{{\ensuremath{\lambda}}_{T}{T}_{\mathrm{cB}}}{(T\ensuremath{-}{T}_{\mathrm{cB}})}$ to order ${L}_{3}\mathrm{exp}{\frac{1}{2}\sqrt{\ensuremath{\pi}}{\ensuremath{\zeta}}_{\frac{3}{2}}(\frac{{L}_{3}}{{\ensuremath{\lambda}}_{T}})[{(\frac{{T}_{\mathrm{cB}}}{T})}^{\frac{3}{2}}\ensuremath{-}1]}$ as the temperature is lowered through the bulk critical temperature ${T}_{\mathrm{cB}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\zeta}}_{\ensuremath{\chi}} \mathrm{is}\mathrm{the}\mathrm{Riemann} \ensuremath{\zeta} \mathrm{function}.)$ The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.
The specific heat of an ideal semi-infinite ($\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}D$) Bose gas is calculated numerically using a number of boundary conditions. The results provide a suitable comparison with the analytical calculations of Fisher and Barber and of Pathria. The agreement is very good except for the Neumann boundary condition where the numerical result shows deviation from that of Pathria. However, the earlier assumption of the existence of a "correlation length" appears to have little physical basis. For the hard-wall case, the anomaly in the specific heat is probably due to constraint on particle density. The distinctive effect of such a boundary condition (${\ensuremath{\Psi}}_{s}=0$) is explored more thoroughly in maximal assemblies ($L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}D$) where its effect on specific heat shows up prominently.