Phase transition in an ideal bose gas in an external field

Evgeny Levich, Victor Yakhot

B. Ya. Balagurov, Валентин Григорьевич Вакс

L. Goldstein

The peculiar condensation in phase space of ideal Bose-Einstein gases is discussed when the internal energy states of the atoms or molecules is taken into account and when the gas of atoms without internal structure is placed in an external conservative field of force. The inclusion of the internal energy states decreases while the application of the external field increases the condensation temperature from its value in absence of either of these causes. The transition phenomenon, apart from the preceding change in the condensation temperature, proceeds in the same way as in the case of the free gas atoms without internal structure; in particular, the order of the transition remains invariable, as the continuity of the heat capacity at constant volume is maintained at the transition point.

D. N. Zubarev, Iu. A. Tserkovnikov

Martin J. Cooper, Melville S. Green

The condensation of an ideal Bose gas, in which the phase symmetry of the boson states has been removed by a linear coupling of the creation and annihilation operators ${\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ to a fictitious external field $C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, is considered as a model of a second-order phase transition. In the vicinity of the transition, the spontaneous-order parameter (thermal expectation value of $\ensuremath{\psi}$) is related to the conjugate field by a series of terms of which the first represents a scaled equation of state that exhibits a power-law behavior with nonclassical exponents. The expectation value for the ordered product of boson operators ${〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1) {\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\cdots}{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(M) \ensuremath{\psi}({1}^{\ensuremath{'}})\ensuremath{\cdots}\ensuremath{\psi}({N}^{\ensuremath{'}})〉}_{\ensuremath{\beta}}$ is determined and its behavior across the coexistence curve discussed. The existence of a conjectured critical eigenvector is demonstrated and a related asymptotic property of the ordered product determined.

B. Ya. Balagurov, Валентин Григорьевич Вакс