Figure 5.7 – Variance comparison of MALA and GHMALA on the quartic Gaussian dis-tribution
ofE[f(X)] build with GHMALA (wheref is given by Equation (5.31)). We can observe that the decrease in variance between MALA and GHMALA for small h is around 280.
Nevertheless, for larger h, the explicit integration is not accurate enough, which leads to an increase in the asymptotic variance for larger time steps. Eventually, the smallest asymptotic variance of the time average estimator of GHMALA is around 50 times lower than the smallest asymptotic variance for MALA.
5.6 Conclusion
We presented a class of unbiased algorithm that enables us to benefit from the variance reduction of the nonreversible Langevin equations (5.1) with respect to the reversible dynamics (5.10). More precisely, we presented two variations of these algorithms. The first one (GMALA) can be viewed as a lifting method, and more specifically as a generalized Metropolis Hastings methods on a lifted state space. The second one (GHMALA), similar to the first one, can be viewed as a Generalized Hybrid Monte-Carlo method.
Numerical experiments show that variance reductions (compared with classical MALA) of several orders of magnitude can be achieved for potentials concentrated on a lower dimensional submanifold. We also expect these algorithms to perform better in the case of entropic barriers. The main difficulty is, in the case of GMALA, to use a proposal that allows to achieve a sufficiently high average acceptance ratio (to compete with MALA).
For example this can be done by using a mid-point discretization. Even though this scheme is implicit, the computation of the Metropolis-Hastings acceptance probability does not require the computation of the Hessian of logπ. In the case of GHMALA, numerical experiments show that the choice of a suitable integrator for the conservative dynamics
may lead to large improvements and computational cost reduction compared with the mid-point method.
Chapitre 6
Numerical analysis of metastable dynamics in rotating BEC
6.1 Introduction and motivations
This chapter aims at providing a numerical method to analyse the metastable be-haviour of a slowly rotating Bose-Einstein condensate at finite temperature, modelled by the Stochastic Projected Gross-Pitaevskii Equation.
6.1.1 The physical setting
It has been experimentally observed that rotating Bose-Einstein condensates take the form of vortex lattices, with a number of vortices that depends on the rotating speed of the thermal cloud [3,30,34,122]. We are interested in the situation where several vortex lattices configurations, characterised by different numbers of vortices, are locally stable for a same fixed rotation speed. We aim at investigating metastable dynamics between these vortex configurations in this case. We make precise this notion of metastability in Section 6.1.2.
In the following, we consider the case of a rotating dilute Bose gas trapped into an isotropic harmonic trap centred at the origin O. Moreover, we suppose that the thermal cloud rotates around theO−→z axis. We also assume that the intensity of the trap in the zdirection is strong, so that the condensate is flattened enough to be described by a two dimensional model. We refer to the Section 2.4.2 in Chapter 2 for precisions about this simplification. Thus, we limit the present study to the two dimensional case. Because of the isotropy of the system, its energy is invariant by rotations around theO−→z axis. Moreover, the energy is also invariant by any uniform change of phase. In the following we will say that two states of the system areequivalent is they are equal up to a rotation around the O−→z axis and a uniform shift of the phase.
In order to study this kind of dynamics in the isotropic case, we will model the system 179
by the Stochastic Projected Gross-Pitaevskii Equation (SPGPE). See [30, 80, 81] for a derivation of this model, and [25] for a review of similar techniques and models. This model aims at describing the dynamics of (possibly rotating) Bose-Einstein condensate at finite temperature. It has mainly been built to provide a tool to analyse the growth dynamics of a condensate during the phase transition and especially the spontaneous symmetry breaking and the nucleation of vortices during this process [111,166,167]. Yet, when the Gross-Pitaevskii energy at zero temperature (defined later in (6.5)) has several different local minima which are not equivalent, the solution of the SPGPE exhibits a metastable behaviour. From a mathematical point of view, the vortices correspond to singularity points of the phase field of the wave function, but not of the complex field itself. Thus the density of the condensate vanishes at these points. They create holes inside the condensate and thus they can be observed during experiments. Note that it is also possible to observe some phase differences in practice. This kind of methods can provide experimental vortex localisation methods.
To our knowledge, there is no experimental evidence of such a metastable behaviour.
We recall that during experimentations the condensate is generally destroyed, during an expansion procedure, to make the vortices observable. Thus, most of the experimenta-tions only give access to the law of an observable at a given time, but not to trajecto-rial observables. In this chapter we do not pretend to highlight a metastable behaviour of the considered system. Indeed, we only propose a numerical method to analyse the metastable dynamics of the solutions of the SPGPE with rotating thermal cloud, with-out judging if this model is valid and relevant to predict such a physical phenomenon.
Nevertheless we note that in [119] the authors try to form condensates, tuning the speed of rotation to prescribe the number of vortices. These configurations are shown in Figure 6.1. They observe that a prescribed rotation velocity may lead to different numbers of vortices in a condensate. They cannot identify if these fluctuations come from a lack of experimental reproducibility, or if they are intrinsic to the system considered. Our work proposes an explanation, by means of a metastable behaviour, for this observation. More recently in [171] the authors study experimentally Bose-Fermi superfluid mixtures. The number of counted vortices fluctuates quite a lot with respect to the rotation frequency.
These phenomena could be caused by metastable dynamics between vortex configurations.
Metastable behaviours should also occur in rotating fermionic superfluids. Zwierlein and co-authors report observations of vortex lattices in such a superfluid in [173]. Numerical experiments in the context of rotating BEC at finite temperature have been carried out in [100]. The model is not based on the SPGPE, but on the Zaremba, Nikuni, and Griffin (ZNG) approach. The authors study the dynamics of an off-centred vortex in a harmoni-cally trapped pancake-shaped condensate. The vortex is known to decay by spiraling out toward the edge of the condensate. They quantify the dependence of this decay on tem-perature, atomic collisions, and thermal cloud rotation. This experiment has been done using a numerical scheme proposed in [101].
6.1. Introduction and motivations 181
Figure 6.1 – Transverse absorption images of a Bose-Einstein condensate stirred with a laser beam for various rotation frequencies (from [119]).
6.1.2 Basics about metastability
We usually say that an ergodic dynamics ismetastable when it can be split up between a slow and a fast time scale and that the fast dynamics consists of fluctuations inside a metastable region inside which the dynamics stays trapped for large times (compared with the fast time scale). Let us focus on a typical kind of metastable dynamics given by the reversible overdamped Langevin equation,
dXtε=−∇E(Xtε) dt+√
2εdWt, (6.1)
where the energy landscapeE presents some potential or someentropic barriers. A poten-tial barrier appears when the energy E has several local minima separated by a saddle-point of higher energy than both local minima. It means that any continuous trajectory that links two local minima needs to cross some level sets of higher energy than this saddle point. The scarcity of a change of metastable region comes from the fact that crossing the barrier requires some unlikely moves of the stochastic fluctuations that enable to provide enough energy to the system to cross the potential barrier. An entropic barrier appears when several regions of the energy are separated by a narrow continuous path of constant energy. In this case, the rare event comes from the fact that the system must find the good path to cross metastable regions. We refer to Figure 1.1 [115] for an illustration for these two kinds of energy landscape. In our application case, the metastability of the condensate dynamics is caused by a potential barrier.
In the simplifying case of a dynamics given by (6.1), we can formally define a metastable state as a state such that the typical exit time from it is much larger than the local
equilibration time inside the state. One convenient way to give a rigorous sense to this idea is to use the notion of quasi-stationary distribution which provides a way to define a metastable state with respect to the first eigenvalues of the generator of the diffusion (6.1). We refer to [46,64,115] for further preferences.
The interesting long time behaviour of a metastable dynamics relies on the exit events of the metastable regions, defined by the exit times and possibly the exit points. Since the system remains trapped a long time inside the metastable region, it is natural to hope that the exit event can be modeled as a jump Markov process as explained in [64]. These exit times highly depend on the kind of barrier as stated above. In the small temperature limit (ε → 0), they grow linearly with respect to ε−1 in the case of an entropic barrier, and exponentially with respect toε−1 in the case of a potential barrier. This last result is known as the Arrhenius law, and can be formulated and proven with the theory of large deviations [62, 78] under appropriate assumptions on E. In the case of potential barriers for the reversible overdamped Langevin dynamics (6.1), some works have been carried out to estimate the prefactors of the average exit time of a metastable state. See for instance [93]. This kind of results are known as Eyring-Kramers formulas. Suppose that that E describes a double-well energy in an arbitrary space dimension. Let x− and x+ be the two minima of each well. Let x∗ be the only saddle point. Then, we denote by D a neighbourhood of x+ of size δ, chosen small enough, and by τD the hitting time of the setD. Then, the Eyring-Kramers formula is given by,
Ex−(τD) = 2π λ1(x∗)
s
det(∇2E(x∗))
det(∇2E(x−)) e(E(x−)−E(x∗))/ε h
1 +O
ε1/2|logε|3/2i
, (6.2) whereλ1(x∗) is the unique negative eigenvalue of∇2E(x∗), andEx−(τD) is the expectation ofτD for a Markov process given by (6.1) and starting fromx−.
The SPGPE model is given by a nonreversible overdamped Langevin equation, of the form,
dXtε=−(ε+i)∇E(Xtε) dt+√
2εdWt. (6.3)
This dynamics exhibits a metastable behaviour similar to the one of (6.1). The additional term corresponds to a purely Hamiltonian dynamics that conserves the energy E, and thus only the reversible overdamped Langevin component of this dynamics allows to cross saddle points. We refer to [29] for a generalization of the Eyring-Kramers formula to the nonreversible case.
In practice, the direct computation of the law of the exit times is a difficult problem.
The Eyring-Kramers formula is only asymptotic, and requires to look for the order one saddle points of the energy (see Section 2.1 of [61] for a presentation of such methods).
Efficient Monte Carlo methods have been developed to compute these exit times, without using the Eyring-Kramers formula. They consist in sampling the ensemble of paths joining
6.2. The Stochastic Projected Gross-Pitaevskii Equation 183