Interacting Brownian motions and the Gross-Pitaevskii formula

For a long time these predictions were considered as a curiosity of non-interacting gases and had no practical impact. After the observation of superfluidity in liquid ⁴He below the $\lambda$ temperature (2.17 K) was made, London (1938) suggested that, despite the strong interatomic interactions, BEC indeed occurs in this system and is responsible for the superfluidity properties. This suggestion has stood the test of time and is the basis of our modern understanding of the properties of the superfluid phase.

The first self-consistent theory of super-fluids was developed by Landau (1941) in terms of the spectrum of elementary excitations of the fluid. In 1947 Bogoliubov developed the first microscopic theory of interacting Bose gases, based on the concept of Bose-Einstein condensation. This initiated several theoretical studies; a recent account on the state of the art can be found in Adams, S. and Bru, J.-B. (2004a, b) and on its contribution to superfluidity theory in Adams, S. and Bru, J.-B. (2004c). After Landau, L.D. and Lifshitz, E.M. (1951) had appeared, Penrose (1951) and Onsager, L. and Penrose, O. (1956) introduced the concept of the non-diagonal long-range order and discussed its relationship with BEC. An important development in the field took place with the prediction of quantised vortices by Onsager (1949) and Feynman (1955). The experimental studies on dilute atomic gases were developed much later, starting from the 1970s, benefiting from the new techniques developed in atomic physics based on magnetic and optical trapping, and advanced cooling mechanisms.

In 1995, the first experimental realisations of BEC were achieved in a system that is as different as possible from ⁴He, namely, in dilute atomic alkali gases trapped by magnetic fields. These realisations are due to Anderson, M.H. et al. (1995), Bradley, C.C. et al. (1995), Davis, K.B. et al. (1995), after appropriate cooling methods had been developed. For this remarkable achievement, the Nobel prize in physics 2001 was awarded to E.A. Cornell, W. Ketterle and C.E. Wieman. Over the last few years these systems have been the subject of an explosion of research, both experimental and theoretical. A comprehensive account on Bose-Einstein condensation is the recent monograph Pitaevskii, L. and Stringari, S. (2003).

Perhaps the most fascinating aspect of BEC is best illustrated by the cover of Science magazine of December 22, 1995, in which the Bose condensate is declared as the “molecule of the year”. The Bose condensate is pictured as a platoon of soldiers marching in lookstep: every atom in the condensate must behave in exactly the same way. One of the most striking consequences is that effects, which are so small that they are practically invisible at the level of a single atom, are spectacularly amplified.

Motivated by the experimental success, in a series of papers Lieb, E.H. et al. (2000a), Lieb, E.H. et al. (2000b), Lieb, E.H. et al. (2001), Lieb, E.H. and Seiringer, R. (2002) obtained a mathematical foundation of Bose-Einstein condensation at zero temperature. The mathematical formulation of the $N$-particle Boson system is in terms of an $N$-particle Hamilton operator, ${\mathcal{H}}_{N}$, whose ground states describe the Bosons under the influence of a trap potential and a pair potential, see Section 2. Lieb et al. rigorously proved that the ground state energy per particle of ${\mathcal{H}}_{N}$ (after proper rescaling of the pair potential) converges towards the energy of the well-known Gross-Pitaevskii functional. The ground state is approximated by the $N$-fold product of the Gross-Pitaevskii minimiser mulitplied by a correlated term involving the solution of the associated scattering equation. Moreover, they also showed the convergence of the reduced density matrix, which implies the Bose-Einstein condensation. As had been generally predicted, the scattering length of the pair interaction potential plays a key role in this description.

These rigorous results are only for zero temperature, whereas the experiments show BEC at very low, but positive temperature. The mathematical understanding of BEC at positive temperature is rather incomplete yet. Its analysis represents an important challenging and ambitious research area in the field of many-particle systems. Thermodynamic equilibrium states are described by traces of ${\rm e}^{-\beta{\mathcal{H}}_{N}}$, where $\beta\in(0,\infty)$ is the inverse temperature and ${\mathcal{H}}_{N}$ is the $N$-particle Hamilton operator. Via the Feynman-Kac formula (see e.g. Feynman (1953) and Ginibre (1970)), these traces are expressed as exponential expectations of $N$ interacting Brownian motions with time horizon $[0,\beta]$. This opens up the possibility to use probabilistic approaches for the study of these traces, in particular stochastic analysis and the theory of large deviations.

In this review we present our probabilistic approaches to dilute systems of interacting many-particle systems at positive temperature using the Gross-Pitaevskii approximation. Using the the theory of large deviations, we characterise the large-$N$ and the large-$\beta$ behaviour of various exponential expectations of $N$ interacting Brownian motions with time horizon $[0,\beta]$ in terms of variants of the Gross-Pitaevskii variational formula. In particular we introduce and analyse a new model, which we call the Hartree model, whose ground states are the ground product states of the Hamilton operator ${\mathcal{H}}_{N}$. Their large-$N$ behaviour is characterised in terms of the Gross-Pitaevskii formula, with the scattering length replaced by the integral of the pair interaction potential. This nice assertion is complemented by an analogous result for positive temperature. Our programme started with Adams, S. et al. (2006a, b), which we summarise here. Further aspects are considered in Adams, S. and Dorlas, T. (2007a), Adams, S. and König, W. (2007), Adams, S. and Dorlas, T. (2007b) and Adams (2007a). Under current development are Adams (2007c), Adams (2007d), Adams, S. et al. (2007) and Adams, S. et al. (2007) in which non-dilute systems are studied.

We give a brief introduction to the physics of dilute quantum gases and their mathematical treatment at zero temperature in Section 2. In particular we introduce the Gross-Pitaevskii formula and the scattering length and describe the results by Lieb et al. and our results of the ground product state. Our probabilistic models are introduced in Section 3. Section 4 is devoted to our large deviations results and the variational analysis.

We introduce the modelling of the Gross-Pitaevskii theory which will be the starting point for our probabilistic models in Section 3. Let us comment briefly on some issues of the 1995 experiments as these are the motivation for the renewed interest in the Gross-Pitaevskii theory and its analytical proof by Lieb et al.

The experimental systems are collections of individual neutral alkali-gas atoms (e.g., ⁶Li, ${}^{4}0$K, ${}^{8}7$Rb, ${}^{2}3$Na, ⁷Li and ${}^{8}5$Rb, ⁸⁷Rb,¹³³Cs,¹⁷⁴Yb, ⁸⁵Rb₂, and ⁶Li₂)), with total number $N$ ranging from a few hundreds up to $\sim 10^{10}$, confined by magnetic and/or optical means to a relatively small region of space. Their densities range from $\sim 10^{11}\mbox{cm}^{-1}$ to $\sim 5\times 10^{15}\mbox{cm}^{-1}$, and their temperatures are typically in the range of a few tenths of nK up to $\sim 5\mu$K.

In a typical system, we are faced with several length scales. One of them is the two-body interaction energy $\hbar^{2}/m{\alpha}^{2}$, where $m$ is the reduced mass of the two particles, $\hbar$ is Heisenberg’s constant and ${\alpha}$ is the scattering length (see Section 2.1 below), expressing the strength of the interatomic interaction. A second one is the mean interparticle spacing $r_{\rm int}$, and a third one is the oscillator frequency $a_{\rm osc}$ of the confining trap potential. Note that the first scale does not depend on the trap geometry, whereas the oscillator frequency $a_{\rm osc}$, the mean interparticle spacing, the transition temperature $T_{\rm c}$ and the mean-field energy $U_{0}$ (to be specified later) do depend on the shape of the confining potential. Introduce the “healing length” $\xi=(2mnU_{0}\hbar)^{-1/2}$ and the de Broglie wavelength $\lambda_{\rm DB}$. Note that $a_{\rm osc}$ is the zero-point spread of the ground-state wave function of a free particle in the trap. The relations between these scales are as follows.

Typical values are ${\alpha}\sim 50$ Å, $r_{\rm int}\sim 2000$ Å, $\xi\sim 4000$ Å, $a_{\rm osc}\sim 1\mu$. If one compares these numbers with those of liquid helium, one sees that the dilute gas condition ${\alpha}\ll r_{\rm int}$, which is characteristic for the BEC of alkali gases, is very far from satisfied for liquid helium. As a consequence, liquid helium is a much more strongly interacting system than BEC gases, by many orders of magnitude.

We now turn to a mathematical modelling and introduce the potentials and the scattering length in Section 2.1 and the Gross-Pitaevskii theory in Section 2.2.

Much thermodynamic information about the Boson system is contained in the traces of the Boltzmann factor ${\rm e}^{-\beta{\mathcal{H}}_{N}}$ for $\beta>0$, like the free energy, or the pressure. Since the 1960ies, interacting Brownian motions are generally used for probabilistic representations for these traces. The parameter $\beta$, which is interpreted as the inverse temperature of the system, is then the length of the time interval of the Brownian motions.

However, the traces do not contain much information about the ground state. Since the pioneering work of Donsker and Varadhan in the early 1970ies it is basically known that the ground states are intimately linked with the Brownian occupation measures. This link is rigorously established via the theory of large deviations for diverging time, which corresponds to vanishing temperature.

We look at the distribution of the $N$ Brownian motions under the transformed path measure

Here $\mathbb{E}$ denotes the Brownian expectation for deterministic start at the origin and time horizon $[0,\beta]$. We call $\widehat{}\mathbb{P}_{N,\beta}$ the canonical ensemble model, since it is derived, via a Feynman-Kac formula, from the trace-class operator of the canonical ensemble, ${\rm e}^{-\beta{\mathcal{H}}_{N}}$. That is, the trace is given as

Here $\mathbb{E}_{x_{i},x_{i}}^{\beta}$ denotes the expectation with respect to a Brownian bridge starting in $x_{i}$ and terminating in $x_{i}$ after time $\beta$.

However, a system of $N$ Bosons is described by a trace of the projection to symmetric wave functions, i.e., wave functions that are invariant under permutations of the single particle indices. Hence the trace for a system of Bosons is given as

where $\mathfrak{S}_{N}$ is the group of permutation of $N$ elements. These symmetrised systems are the subject of the review Adams (2007b) in these proceedings. Recent results can be found in Adams, S. and Dorlas, T. (2007a), Adams, S. and König, W. (2007) and Adams (2007c), Adams (2007d).

The path measure $\mathbb{P}_{N,\beta}$ is a model for $N$ Brownian motions in a trap $W$ with the presence of a repellent pair interaction. We can conceive the $N$-tuple of the motions, $B_{t}=(B_{t}^{{\scriptscriptstyle{({1}})}},\dots,B_{t}^{{\scriptscriptstyle{({N}})}}),$ as one Brownian motion in $\mathbb{R}^{dN}$. Introduce the normalised occupation measure of the $dN$-dimensional motion,

which is a random element of the set ${\mathcal{M}}_{1}(\mathbb{R}^{dN})$ of probability measures on $\mathbb{R}^{dN}$. It measures the time spent by the tuple of $N$ Brownian motions in a given region. Note that there is only one time scale involved for all the motions, i.e., the Brownian particles interact with each other at common time units. We can write the Hamiltonians in terms of the occupation measure as

Note that the energy functional $\langle h,{\mathcal{H}}_{N}h\rangle$ may be rewritten $\langle h,{\mathcal{H}}_{N}h\rangle=I_{N}(\mu)+\langle\mathfrak{W},\mu\rangle+\langle\mathfrak{v},\mu\rangle$ for the probability measure $\mu({\rm d}x)=h^{2}(x)\,{\rm d}x$.

We introduce the normalised occupation measure of the $i$-th motion,

The tuple of the $N$ occupation measures, $(\mu_{\beta}^{{\scriptscriptstyle{({1}})}},\dots,\mu_{\beta}^{{\scriptscriptstyle{({N}})}})$, plays a particular role in this model. We can write the Hamiltonians as

where we recall the operator $V$ with kernel $v\circ|\cdot|$, and $\mu_{\beta}^{\otimes}=\mu_{\beta}^{{\scriptscriptstyle{({1}})}}\otimes\dots\otimes\mu_{\beta}^{{\scriptscriptstyle{({N}})}}$ is the product measure.

We present our main large deviations results for both the canonical ensemble and the Hartree model. In Section 4.1 the zero temperature (i.e., large-$\beta$) limit is considered, and in Section 4.2 the large-$N$ limit, both at zero temperature and positive temperature.