We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that $0<\ensuremath{\int}d\mathbf{r}V(r)=a<\ensuremath{\infty}.$ Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\ensuremath{\rho}(\ensuremath{\mu})=F(\ensuremath{\mu}\ensuremath{-}a\ensuremath{\rho}(\ensuremath{\mu}))$ between the density $\ensuremath{\rho}$ and the chemical potential $\ensuremath{\mu},$ valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling ${V}_{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\gamma}}^{3}V(\ensuremath{\gamma}r),$ we prove that in the mean-field limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}0,$ only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that $0<\ensuremath{\int}d\mathbf{r}V(r)=a<\ensuremath{\infty}.$ Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\ensuremath{\rho}(\ensuremath{\mu})=F(\ensuremath{\mu}\ensuremath{-}a\ensuremath{\rho}(\ensuremath{\mu}))$ between the density $\ensuremath{\rho}$ and the chemical potential $\ensuremath{\mu},$ valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling ${V}_{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\gamma}}^{3}V(\ensuremath{\gamma}r),$ we prove that in the mean-field limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}0,$ only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.

arXiv:cond-mat/0304392Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJul 14, 200314 citations

Condensation Phenomena in a Clustering Bose-Einstein Gas

William Band

It has been shown that Mayer's theory of imperfect gases can be handled approximately as a problem in association in which the actual gas is replaced by a mixture of ideal gases consisting of clusters of all sizes and between which dissociative equilibrium exists. Kahn and Uhlenbeck have shown how Mayer's theory can be generalized to apply to quantum degenerate gases. The purpose of the present paper is to examine the clustering approximation to such a theory of the quantum degenerate Bose-Einstein gas. The symmetrized wave functions for the clustering assembly are set up and the partition function evaluated for all ways of clustering. Equations of state are deduced that include condensation into the liquid phase as a clustering avalanche at the saturation pressure. It is found that clustering has the effect of lowering the lambda-temperature due to statistical degeneracy right down to the saturated vapor line, so that the lambda-transition is entirely inhibited from the gas phase.It has been shown that Mayer's theory of imperfect gases can be handled approximately as a problem in association in which the actual gas is replaced by a mixture of ideal gases consisting of clusters of all sizes and between which dissociative equilibrium exists. Kahn and Uhlenbeck have shown how Mayer's theory can be generalized to apply to quantum degenerate gases. The purpose of the present paper is to examine the clustering approximation to such a theory of the quantum degenerate Bose-Einstein gas. The symmetrized wave functions for the clustering assembly are set up and the partition function evaluated for all ways of clustering. Equations of state are deduced that include condensation into the liquid phase as a clustering avalanche at the saturation pressure. It is found that clustering has the effect of lowering the lambda-temperature due to statistical degeneracy right down to the saturated vapor line, so that the lambda-transition is entirely inhibited from the gas phase.

Physical ReviewSep 1, 19506 citations

ON THEORY OF NON-IDEAL BOSE GAS

Yu. V. Sanochkin

Soviet physics. DokladyApr 11, 19590 citations

Effective Dynamics of a Heavy Particle in an Ideal Bose Gas in the Thermodynamic Limit

Dirk-André Deckert, Jürg Martin Fröhlich, Peter Pickl +1 more

We study a system of a heavy quantum particle, from now on called tracer particle, coupled to an ideal gas of $N$ light Bose particles with a ratio between the masses of the tracer particle and of a gas particle being proportional to the gas particle density. While all particles obey the non-relativistic dispersion relation, the tracer particle is also driven by an external potential and couples to the gas particles through a pair potential. We compare the corresponding quantum dynamics to effective dynamics, given by a Newtonian equation for the tracer particle and a corresponding effective equation for the Bose gas. We quantify the closeness of these two dynamics depending on the gas particle density and the volume of the support of their wave function. The provided estimates are such that we get convergence in any limit taking the density and the volume to infinity. In particular, they allow to interchange the thermodynamic with the mean field limit.We study a system of a heavy quantum particle, from now on called tracer particle, coupled to an ideal gas of $N$ light Bose particles with a ratio between the masses of the tracer particle and of a gas particle being proportional to the gas particle density. While all particles obey the non-relativistic dispersion relation, the tracer particle is also driven by an external potential and couples to the gas particles through a pair potential. We compare the corresponding quantum dynamics to effective dynamics, given by a Newtonian equation for the tracer particle and a corresponding effective equation for the Bose gas. We quantify the closeness of these two dynamics depending on the gas particle density and the volume of the support of their wave function. The provided estimates are such that we get convergence in any limit taking the density and the volume to infinity. In particular, they allow to interchange the thermodynamic with the mean field limit.

Sep 30, 20120 citations

Thermodynamics of a Weakly Interacting Bose–Einstein Gas

T. Haugset, H. Haugerud, F. Ravndal

arXiv:cond-mat/9706088Annals of PhysicsJun 1, 199831 citations

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential

V(r)

such that $0<\ensuremath{\int}d\mathbf{r}V(r)=a<\ensuremath{\infty}.$ Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\ensuremath{\rho}(\ensuremath{\mu})=F(\ensuremath{\mu}\ensuremath{-}a\ensuremath{\rho}(\ensuremath{\mu}))$ between the density $\ensuremath{\rho}$ and the chemical potential $\ensuremath{\mu},$ valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling ${V}_{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\gamma}}^{3}V(\ensuremath{\gamma}r),$ we prove that in the mean-field limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}0,$ only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.

We study a system of a heavy quantum particle, from now on called tracer particle, coupled to an ideal gas of

N

light Bose particles with a ratio between the masses of the tracer particle and of a gas particle being proportional to the gas particle density. While all particles obey the non-relativistic dispersion relation, the tracer particle is also driven by an external potential and couples to the gas particles through a pair potential. We compare the corresponding quantum dynamics to effective dynamics, given by a Newtonian equation for the tracer particle and a corresponding effective equation for the Bose gas. We quantify the closeness of these two dynamics depending on the gas particle density and the volume of the support of their wave function. The provided estimates are such that we get convergence in any limit taking the density and the volume to infinity. In particular, they allow to interchange the thermodynamic with the mean field limit.