Quantum kinetic theory: modelling and numerics for Bose-Einstein

Quantum kinetic theory: modelling andnumerics for Bose-Einstein condensation

Weizhu Bao1, Lorenzo Pareschi2 and Peter A.Markowich3

1 Department of Computational Science, National University of Singapore,Singapore 117543 [email protected]

2 Department of Mathematics, University of Ferrara, Via Machiavelli 35, 4110Ferrara Italy [email protected]

3 Department of Mathematics, University of Vienna, Boltzmanngasse 9, 1090Vienna, Austria [email protected]

Summary. We review some modelling and numerical aspects in quantum kinetictheory for a gas of interacting bosons and we try to explain what makes Bose-Einsteincondensation in a dilute gas mathematically interesting and numerically challenging.Particular care is devoted to the development of efficient numerical schemes for thequantum Boltzmann equation that preserve the main physical features of the contin-uous problem, namely conservation of mass and energy, the entropy inequality andgeneralized Bose-Einstein distributions as steady states. These properties are essen-tial in order to develop numerical methods that are able to capture the challengingphenomenon of bosons condensation. We also show that the resulting schemes canbe evaluated with the use of fast algorithms. In order to study the evolution of thecondensate wave function the Gross-Pitaevskii equation is presented together withsome schemes for its efficient numerical solution.

1 Introduction

The quantum dynamics of many body systems is often modelled by a non-linear Boltzmann equation which exhibits a gas-particle-like collision behav-ior. The application of quantum assumptions to molecular encounters leadsto some divergences from the classical kinetic theory[15] and despite theirformal analogies the Boltzmann equation for classical and quantum kinetictheory present very different features.

The interest in the quantum framework of the Boltzmann equation hasincreased dramatically in the recent years. Although the quantum Boltzmannequation, or QBE, for a single specie of particles is valid for a gas of fermions aswell as for a gas of bosons, blow up of the solution in finite time may occur onlyin the latter case. As a consequence the QBE for a gas of bosons represents themost challenging case both mathematically and numerically. In particular this

2 Weizhu Bao, Lorenzo Pareschi and Peter A.Markowich

equation has been successfully used for computing non-equilibrium situationswhere Bose-Einstein condensate occurs.

Bose-Einstein condensation, or BEC, has a long history dating from theearly 1920s (see [11],[19],[20]). The notion of Bose statistics dates back to a1924 paper in which Bose used a statistical argument to derive the black-bodyphoton spectrum. Bose was unable to publish his work, so he sent it to Einsteinwho translated it into German and got it published. Einstein then extendedthe idea of Bose statistics to the case of noninteracting atoms. The result wasBose-Einstein statistics. Einstein noticed a peculiar feature of the distributionof the atoms over the quantized energy levels predicted by these statistics. Atvery low but finite temperature a large fraction of the atoms would go into thelowest energy quantum state. This phenomenon is now called Bose-Einsteincondensation.

Although it was a source of debate for decades, it is now recognized that theremarkable properties of superconductivity and superfluidity in both helium3 and helium 4 are related to BEC. The appeal of superconductivity and ofsuperfluidity, along with that of laser light, the third common system in whichmacroscopic quantum behavior is evident, provided the main motivation tothe study of BEC in a gas. Experimentally this has been achieved thanks tostrong advancements in trapping and cooling techniques for neutral atomsleading to the 2001 Nobel prize in physics by Cornell, Ketterle and Wiemann[16, 33].

A different description is provided by the time dependent Gross-Pitaevskiiequation, or GPE, which can be viewed as an equation for the condensate wavefunction (order parameter for the Bose condensate)[28, 29]. This equation is asimplified description in that it includes no quantum fluctuations, or thermalor irreversible effects, but it may well be valid for a large number of condensateparticles. Both of these equations contain essential aspects of the problem.However, in practice the process of creating a Bose-Einstein condensate in atrap by means of evaporative cooling starts in a regime covered by the QBEand finishes in a regime where the GPE is thought to be valid.

In the first part of this paper we briefly recall the main properties of theQBE and derive accurate numerical discretizations, which maintain the basicanalytical and physical features of the continuous problem, namely, mass andenergy conservation, entropy growth and equilibrium distributions. Althoughour treatment is valid for both fermions and bosons we shall mainly concen-trate on a gas of bosons since we are interested in methods capable to describethe formation of condensate. To this aim we consider the homogeneous Boltz-mann equation for a quantum gas and derive first and second order accuratequadrature formulas for the collision operator in some relevant physical cases.Due to their ’direct’ derivation from the continuous operator these schemespossess all the desired physical properties at a discrete level. In addition weshow that with a suitable choice of the interacting kernel the computationscan be performed with fast algorithms. Next we extend our method to themore general case of time dependent trap potentials [31].

Quantum kinetic theory: modelling and numerics for BEC 3

For the sake of completeness we mention the recent works [14, 36, 41,42, 43] in which fast methods for Boltzmann equations were derived usingdifferent techniques like multipole methods, multigrid methods and spectralmethods. For a mathematical analysis of the quantum Boltzmann equation inthe space homogeneous isotropic case we refer to [38, 39, 22, 23, 24]. We remarkthat already the issue of giving mathematical sense to the collision operatoris highly nontrivial (particularly if positive measure solutions are allowed,as required by a careful analysis of the equilibrium states). Derivations ofthe QBE from the evolution of interacting quantum particles are found in[10, 21, 26]. Finally we mention here a recent paper by Buet and Cordier[13] where the asymptotic Kompaneets equation limit of the QBE has beenstudied numerically.

In the second part, we study the evolution of the condensate wave func-tion, by means of the Gross-Pitaevskii equation [30, 44] together with someschemes for its efficient numerical solution. Here we briefly review the time-splitting spectral method introduced in [5, 3, 2] and its main properties. Amathematical model for coupling QBE and GPE is also introduced [51].

The rest of the work is organized as follows. In the next Section we shallintroduce the QBE for bosons and its main physical properties. In Section 3we discuss the details of our numerical schemes and some numerical examplesare performed. The results confirm the capability of the methods to capturethe concentration behavior of bosons. In Section 4 we extend the method tothe case of time dependent trapping potentials. The GPE equation is thendiscussed in Section 5 together with its numerical solution. Finally in Section6 some future directions of research and possible strategies for numericallycoupling QBE and GPE are outlined.

2 The quantum Boltzmann equation

We consider a gas of interacting bosons, which are trapped by a confiningpotential V = V (x) with min V (x) = 0. We denote the total energy of aboson with momentum p = (p1, p2, p3) and position x = (x, y, z) (after anappropriate non-dimensionalization) by

ε(x,p) =|p|22

+ V (x). (1)

Let F = F (x,p, t) ≥ 0 be the phase-space density of bosons.

2.1 The QBE in energy space

Assuming a boson distribution which only depends on the total energy ε wewrite

F (x,p, t) = f

( |p|22

+ V (x), t)

, (2)


where f = f(ε, t) ≥ 0 is the boson density in energy space.Following [45, 46, 25, 26, 27] we write a Boltzmann-type equation (referred

to as boson Boltzmann equation in the sequel) in energy space

ρ(ε)∂f

∂t= Q(f)(ε), t > 0, (3)

with the collision integral

Q(f)(ε) = C

∫

R3+

δ(ε + ε∗ − ε′ − ε′∗)S(ε, ε∗, ε′, ε′∗)[f′f ′∗(1 + f)(1 + f∗)

(4)− ff∗(1 + f ′)(1 + f ′∗)] dε∗dε′dε′∗,

where C = m/(π2~3) and S ≥ 0 is a given function. Here ~ is the Planckconstant and m denotes the mass of a particle. In the sequel we assume theequation has been normalized so that C = 1.

We denoted the density of states by

ρ(ε) =∫

R6δ

(ε−

( |p|22

+ V (x)))

dp dx, (5)

andf ′ = f(ε′, t), f ′∗ = f(ε′∗, t), f = f(ε, t), f∗ = f(ε∗, t). (6)

As usual ε and ε∗ are the pre-collisional energies of two interacting bosonsand ε′ and ε′∗ are the post-collisional ones.

The positive measure

δ(ε + ε∗ − ε′ − ε′∗)S(ε, ε∗, ε′, ε′∗) (7)

denotes the energy transition rate, i.e. Sdε′ dε′∗ is the transition probabilityper unit volume and per unit time that two bosons with incoming energies ε,ε∗ are scattered with outgoing energies ε′, ε′∗.

We recall that the phase-space density F = F (x,p, t) satisfies the momentum-position space Boltzmann equation

∂F

∂t+ p · ∇xF −∇xV (x) · ∇pF = Q(F ), (8)

with the scattering integral

Q(F )(x,p) =Q(F )

(|p|2/2 + V (x))

ρ (|p|2/2 + V (x)). (9)

In the homogeneous case V (x) ≡ 0, F independent of x, we set

ρ(ε) =∫

R3δ

(ε− |p|2

2

)dp (10)


and computeρ(ε) = 4π

√2ε (11)

then equation (8) is formally identical to the Boson Boltzmann equation con-sidered in [23],[24]

∂F

∂t=

∫

R9δ(p + p∗ − p′ − p′∗)δ(ε + ε∗ − ε′ − ε′∗)W (p,p∗,p′,p′∗)

(12)[F ′F ′∗(1 + F )(1 + F∗)− FF∗(1 + F ′)(1 + F ′∗)] dp∗dp

′dp′∗,

with ε(p) = |p|2/2 and W , S are related by∫

S2×S2×S2δ(p + p∗ − p′ − p′∗)W (p,p∗,p′,p′∗) dσ∗dσ′dσ′∗

=S

(|p|2/2, |p∗|2/2, |p′|2/2, |p′∗|2/2)

ρ(|p|2/2) |p∗| |p′| |p′∗|.

Here we denoted p∗ = |p∗|σ∗, p′ = |p′|σ′, p = |p|σ, and p′∗ = |p′∗|σ′∗. Inparticular for W ≡ 1 we have

S(ε, ε∗, ε′, ε′∗) = σ ρ(εmin), (13)

where (see [23])εmin = min(ε, ε∗, ε′, ε′∗), (14)

and σ is the total scattering cross section. We assume σ to be constant which istrue for sufficiently small temperatures (where σ ≈ a2

s and as is the wave scat-tering length[49]). Even in the non-homogeneous case V (x) 6= 0 the equation(3) is formally identical to the isotropic version of the homogeneous bosonicBoltzmann equation (12) (after the introduction of |p|2/2 as new independentvariable). However, the density of states is computed by formula (5) in thenon homogeneous case instead of (10) in the space homogeneous case.

The above arguments remain valid in the case of a gas of fermions, thatis quantum particles with half integer spin for which, by the Pauli exclusionprinciple, we have at most one particles on each orbital (electrons, protons,neutrons, ...) in contrast to bosons that have integer spin and for which thenumber of particles in a given state is arbitrary (carrier particles, mesons, ...).For a gas of fermions the QBE has the same structure except sign changes inthe collision operator

Q(f)(ε) =∫

R3+

δ(ε + ε∗ − ε′ − ε′∗)S(ε, ε∗, ε′, ε′∗)[f′f ′∗(1− f)(1− f∗)

(15)− ff∗(1− f ′)(1− f ′∗)] dε∗dε′dε′∗.


2.2 Physical properties

A simple calculation gives the weak form of the collision operator. Let φ = φ(ε)be a test function. Then, at least formally∫ ∞

0

Q(f)φdε =12

∫

R4+

δ(ε + ε∗ − ε′ − ε′∗)S(ε, ε∗, ε′, ε′∗)[f′f ′∗(1 + f)(1 + f∗)

(16)− ff∗(1 + f ′)(1 + f ′∗)][φ + φ∗ − φ′ − φ′∗]dεdε∗dε′dε′∗.

Here we used the micro-reversibility property, i.e. the fact that each collision isreversible and that each pair of interacting bosons represents a closed physicalsystem. Mathematically this amounts to the requirement [23]

S(ε, ε∗, ε′, ε′∗) = S(ε∗, ε, ε′, ε′∗) = S(ε′, ε′∗, ε, ε∗). (17)

The symmetry properties (17) immediately imply the analogous propertiesfor the energy transition rate (7) and the weak form (16) follows from thevariable substitution in the integral using these symmetries.

As a consequence we have the following collision invariants

1.φ(ε) ≡ 1 ⇒

∫ ∞

0

Q(f)(ε) dε = 0, (18)

2.φ(ε) ≡ ε ⇒

∫ ∞

0

Q(f)(ε)ε dε = 0. (19)

Consider now the IVP (3) supplemented by the initial condition

f(ε, t = 0) = f0(ε) ≥ 0, ε > 0. (20)

Then (18) implies mass conservation∫ ∞

0

ρ(ε)f(ε, t) dε =∫ ∞

0

ρ(ε)f0(ε) dε, ∀ t > 0, (21)

and (19) energy conservation∫ ∞

0

ρ(ε)f(ε, t)ε dε =∫ ∞

0

ρ(ε)f0(ε)ε dε, ∀ t > 0. (22)

The H-theorem for (3) is derived by setting φ(ε) = ln(1 + f(ε)) − ln f(ε)in (16).

We calculate∫ ∞

0

Q(f)(ε)(ln(1 + f(ε))− ln f(ε))dε

(23)=

12

∫

R4+

δ(ε + ε∗ − ε′ − ε′∗)S(ε, ε∗, ε′, ε′∗)e(f)dεdε∗dε′dε′∗ := D[f ],


wheree(f) = z(ff∗(1 + f ′)(1 + f ′∗), f

′f ′∗(1 + f)(1 + f∗)) (24)

andz(x, y) = (x− y)(ln x− ln y). (25)

Since the integrand of the entropy dissipation D[f ] is non-negative, we deducethe following H-theorem, obtained by multiplying (3) by φ(ε) = ln(1+f(ε))−ln f(ε)

d

dtS[f ] = D[f ], (26)

which implies that the entropy

S[f ] :=∫ ∞

0

ρ(ε)((1 + f) ln(1 + f)− f ln f)dε, (27)

is increasing along trajectories of (3). We remark that trivially the third phys-ical conservation law, namely momentum conservation, also holds. Clearly thephase-space density F of (2) satisfies

∫

R3pF (x,p, t)dx ≡ 0, ∀t ≥ 0. (28)

2.3 Steady states

We now turn to the issue of steady states of the QBE.The main qualitative characteristics of f are described by these two prop-

erties: conservations and increasing entropy. It is therefore natural to expectthat as t tends to ∞ the function f converges to a function f∞ which realizesthe maximum of the entropy S[f ] under the moments constraint (19)-(18).Clearly if f∞ solves the entropy maximization problem with constraints (19)-(18), there exist Lagrange multipliers α, β ∈ R such that

∫ρ(ε)(ln(1 + f∞(ε))− ln f∞(ε))φ(ε) dε =

∫ρ(ε)(αε + β)φ(ε) dε, ∀φ

(29)which implies

ln(1 + f∞(ε))− ln f∞(ε) = αε + β

and thereforef∞(ε) =

1eαε+β − 1

, α > 0, β ∈ R. (30)

The function f∞ is called a Bose-Einstein distribution. Again if we considera gas of of fermions the only difference results in the sign and reads

f∞(ε) =1

eαε+β + 1, α > 0, β ∈ R (31)

called a Fermi-Dirac distribution (see [23] for more detalis).


The problem of equilibrium distributions for bosons has a very long his-tory, going back to Bose and Einstein in the twenties of the last century (see[11],[19],[20]), who noticed that the class of Bose-Einstein distributions (30)is not sufficient to assume all possible values of equilibrium mass

N∞ =∫ ∞

0

ρ(ε)f∞(ε)dε, (32)

and equilibrium energy

E∞ =∫ ∞

0

ρ(ε)εf∞(ε)dε, (33)

such that Dirac distribution have to be included in the set of equilibriumstates. In [23] it was shown that for every pair (N∞, E∞) ∈ R2

+ there existα ≥ 0, β ∈ R such that the generalized Bose-Einstein distribution defined by

ρ(ε)f∞(ε) =ρ(ε)

eαε+β+ − 1+ |β−|δ(ε), (34)

is an equilibrium state of (3) satisfying (32)-(33). Here we denoted β+ =max(β, 0) and β− = −max(−β, 0). The value Nc = |β−| represents the mass-fraction of particles which are condensed in equilibrium, i.e. in their quantummechanical ground state with ε = 0. The parameters α and β+ can be relatedto the chemical potential µ and the temperature T of the gas by

α =1

kT, β =

−µ

kT,

where k is the Boltzmann constant.In the following sections we shall use

S(ε, ε∗, ε′, ε′∗) = ρ(εmin). (35)

Notice that the condensation is fully localized in phase space, i.e. it mayonly occur at p = 0 (vanishing momentum) and at those points in positionspace, where the potential assumes its minimum value 0. The reason for this isthe form (2) of the phase space distribution and a semiclassical limit processwhich leads to the Boson Boltzmann equation (3).

3 Numerical methods

We consider the IVP for the quantum Boltzmann equation

ρ(ε)∂f

∂t= Q(f)(ε), t > 0, (36)

f(ε, t = 0) = f0(ε) ≥ 0. (37)


Here the independent variable ε > 0 represents the kinetic energy, ρ = ρ(ε) ≥0 is the (given) density of states and the boson collision operator now reads

Q(f)(ε) =∫

R3+

δ(ε + ε∗ − ε′ − ε′∗)ρ(εmin)[f ′f ′∗(1 + f)(1 + f∗)

(38)− ff∗(1 + f ′)(1 + f ′∗)] dε∗dε′dε′∗.

Obviously the equation (36) maintains a minimum principle such that solutionof (36), (37) satisfy f(ε, t) ≥ 0 for ε ≥ 0, t > 0 if f0(ε) ≥ 0 for ε > 0. Althoughour treatment will be restricted to the case of of bosons it can be appliedstraightforwardly to the case of fermions.

3.1 Discretization and main properties

The starting point in the development of a numerical scheme for (38) is thedefinition of a bounded domain approximation of the collision operator Q.

Let f be defined for ε ∈ [0, R] and denote

QR(f)(ε) =∫

[0,R]3δ(ε + ε∗ − ε′ − ε′∗)ρ(εmin)[f ′f ′∗(1 + f)(1 + f∗)

(39)− ff∗(1 + f ′)(1 + f ′∗)]ψ(ε ≤ R) dε∗dε′dε′∗

where ψ(I) is the indicator function of the set I. Then, at least formally∫ ∞

0

QR(f)φdε =12

∫

[0,R]4δ(ε + ε∗ − ε′ − ε′∗)ρ(εmin)[f ′f ′∗(1 + f)(1 + f∗)

(40)− ff∗(1 + f ′)(1 + f ′∗)][φ + φ∗ − φ′ − φ′∗]dεdε∗dε′dε′∗

for any test function φ = φ(ε). The proof follows the lines of the correspondingweak form of Q discussed in Section 2. It is easy to check that the weak for-mulation (40) of QR implies mass and energy conservation as well as entropyinequality over the bounded domain.

Let us now introduce a set of equally spaced discrete energy grid pointsε1 ≤ ε2 ≤ . . . ≤ εN in [0, R]. We will restrict to product quadrature rules withequal weights w = R/N such that

∫ R

0

f(ε) dε ≈ w

N∑

i=1

f(εi).

A general quadrature formula for (39) is given by


QR(f)(εi) ≈ QR(f)(εi) = w3N∑

j,k,l=1

δklij ρ(εmin)[fkfl(1 + fi)(1 + fj)

(41)− fifj(1 + fk)(1 + fl)]ψ(εi ≤ R),

where now fi = f(εi) and εmin = minεi, εj , εk, εl. The quantities δklij are

suitable discretization of the δ-function on the grid.In order to maintain the conservation properties on the discrete level it is

of paramount importance that the discretized δ-function will reduce the pointsin the sum to a discrete index set which satisfies the relation i + j = k + l.

We now consider the set of ODEs which originates from the energy dis-cretization of the IVP for QR

ρ(εi)dfi

dt= QR(f)(εi), t > 0, (42)

fi(t = 0) = f0,R(εi) ≥ 0. (43)

and state [40]

Proposition 1. If we define

δklij =

1/w i + j = k + l0 otherwise (44)

the solutions of the IVP (42), (43) satisfy the following discrete conservationproperties and entropy principle

w

N∑

i=1

ρ(εi)dfi

dtφ(εi) = 0, φ(ε) = 1, φ(ε) = ε, (45)

w

N∑

i=1

ρ(εi)dh(fi)

dt≥ 0, h(fi) = (1 + fi) log(1 + fi)− fi log fi. (46)

Due to the definition of δklij we have the quadrature formula

QR(f)(εi) = w2N∑

j,l=11≤k=i+j−l≤N

ρ(εmin)[fkfl(1 + fi)(1 + fj)

(47)− fifj(1 + fk)(1 + fl)].

It is easy to check by direct verification [40] that these schemes admitsdiscrete Bose-Einstein equilibrium states of the form

f∞(εi) =1

eαεi+β − 1, α > 0, β ∈ R. (48)

More delicate is the question of ’generalized’ discrete Bose-Einstein equilib-rium which will be discussed later on.


3.2 First and second order methods

Let us rewrite for ε ∈ [0, R] the collision integral (39) as

QR(f)(ε) =∫ R

0

∫ D(ε,ε′)

S(ε,ε′)ρ(εmin)F (ε, ε′, ε′∗) dε′∗dε′, (49)

where F (ε, ε′, ε′∗) = [f ′f ′∗(1 + f)(1 + f∗) − ff∗(1 + f ′)(1 + f ′∗)], with ε∗ =ε′ + ε′∗ − ε, and S(ε, ε′) = maxε− ε′, 0, D(ε, ε′) = minε− ε′ + R, R. Theintegration domain for a fixed value of ε in the (ε′, ε′∗) plane is shown in figure1.

0 0

ε’

ε *’

I

IV II

III

R

Rε

ε

R+ε

R+ε

Fig. 1. The computational domain (dark gray region) in the (ε′, ε′∗) plane for a fixedε

We need the following [40]

Lemma 1. We have

ρ(εmin) =

ρ(ε∗) (ε′, ε′∗) ∈ Iρ(ε) (ε′, ε′∗) ∈ IIρ(ε′∗) (ε′, ε′∗) ∈ IIIρ(ε′) (ε′, ε′∗) ∈ IV

(50)

where the regions I, II, III, IV represent a partition of the computationaldomain and are shown in figure 1.

Using the previous lemma the integral (49) over the four regions can bedecomposed as

QR(f)(ε) = I1(ε) + I2(ε) + I3(ε) + I4(ε), (51)

where for example we have


I1(ε) =∫ ε

0

∫ ε

ε−ε′ρ(ε′ + ε′∗ − ε)F (ε, ε′, ε′∗) dε′∗dε′, (52)

and similarly for the other regions.In the same way the quadrature formula (47) can be decomposed as

QR(f)(εi) = I1(εi) + I2(εi) + I3(εi) + I4(εi), (53)

where now

I1(εi) = w2i∑

k=1

i∑

l=i−k+1

ρ(εk + εl − εi)F (εi, εk, εl). (54)

From the point of view of accuracy we can state [40]

Theorem 1 (Consistency). Let the function f and ρ be Cm([0, R]), m = 1or m = 2, then the quadrature formula (47) satisfies

|QR(f)(εi)− QR(f)(εi)| ≤ R2Cm(∆ε)mMm, ∆ε = R/N, (55)

where Mm is a constant that depends on f and ρ and their derivatives up tothe order m. Moreover if εi = (i− 1)∆ε, i = 1, . . . , N (rectangular rule) thenm = 1 and Cm = 1/2, whereas if εi = (i − 1/2)∆ε, i = 1, . . . , N (midpointrule) m = 2 and Cm = 1/24.

3.3 Fast algorithms

Finally we will analyze the problem of the computational cost of the quadra-ture formula (47). A straightforward analysis shows that the evaluation ofthe double sum in (47) at the point εi requires (2(i− 1)(N − i + 1) + N2)/2operations. The overall cost for all N points is then approximatively 2N3/3.However using transform techniques and the decomposition (53) this O(N3)cost can be reduced to O(N2 log2 N).

In order to do this let us set h = k + l = i + j in (47) and rewrite

QR(εi) = w22N∑

h=2

N∑

k=1

ρ(εmin)[fkfh−k(1 + fi)(1 + fh−i)

(56)− fifh−i(1 + fk)(1 + fh−k)]Ψ [1,N ]

h−i Ψ[1,N ]h−k ,

where we have set

Ψ[s,d]i =

1 s ≤ i ≤ d0 otherwise (57)

In (56) we assume that the function fi is extended to i = 1, . . . , 2N by paddingzeros for i > N .


The sum (56) can be split into sum over the four regions which characterizeρ(εmin). We shall give the details of the fast algorithm only for region I, theother regions can be treated similarly. We have

I1(εi) = w22N∑

h=2

i∑

k=1

ρ(εh−i)[fkfh−k(1 + fi)(1 + fh−i)

(58)− fifh−i(1 + fk)(1 + fh−k)]Ψ [1,i]

h−i Ψ[1,i]h−k,

or equivalently

I1(εi) = w22N∑

h=2

ρ(εh−i)(1 + fi)(1 + fh−i)Ψ[1,i]h−i S

1h(i)

− w22N∑

h=2

ρ(εh−i)fifh−iΨ[1,i]h−i S

2h(i),

where we have set

S1h(i) =

i∑

k=1

fkfh−kΨ[1,i]h−k, S2

h(i) =i∑

k=1

(1 + fk)(1 + fh−k)Ψ [1,i]h−k. (59)

Now the two sums S1h(i) and S2

h(i) are discrete convolutions and can be eval-uated for all h and i using the FFT algorithm in O(N2 log2 N) operations[40].

Remark 1. In the case of constant ρ it is easy to show that expression (56)reduces to a double convolution sum which can be evaluated using the FFTin only O(N log2 N) operations instead of O(N2 log2 N).

3.4 Numerical examples

In this section we present some numerical examples, we refer the reader to [40]for further numerical tests. We shall refer to the first and second order fastschemes developed in the previous section by QBF1 and QBF2 respectively.The time integration is performed with standard first and second order explicitRunge-Kutta schemes after dividing equation (42) by ρ(εi) and thus rewritingthe semidiscrete schemes as

∂fi

∂t= w2

N∑j,l=1

1≤k=i+j−l≤N

ρ(εmin)ρ(εi)

[fkfl(1 + fi)(1 + fj)

(60)− fifj(1 + fk)(1 + fl)].

In all our numerical tests the density of states is given by


ρ(ε) =ε2

2, (61)

which corresponds to a harmonic potential V (x).Note that 0 ≤ ρ(εmin)/ρ(εi) ≤ 1 for εi 6= 0 and that as εi → 0 we have

ρ(εmin)/ρ(εi) → 1. Furthermore since ρ(0) = 0 the values of the distributionfunction at εi = 0 does not affect the discrete conservation of mass and energy.

The schemes were implemented using the fast algorithm described in Sec-tion 3.3.

Bose-Einstein equilibrium

The initial datum is a Gaussian profile centered at R/2

f = exp(−4(ε−R/2)2), (62)

with R = 10. We compute the large time behavior of the schemes for N =40. The stationary solution at t = 10 is given in Figure 2 for both schemestogether with the numerically computed entropy growth. As observed themethods converge to the same stationary state given by a ’regular’ discreteBose-Einstein distribution.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

ε

f ∞(ε

)

0 1 2 3 4 5 6 7 8 9 1024

26

28

30

32

34

36

t

S[f]

Fig. 2. Stationary discrete Bose-Einstein equilibrium and entropy growth for schemeQBF1 () and QBF2 (×) computed with N = 40 points.

The trend to equilibrium in time for the two schemes is reported in Figures3. Note that although the two schemes agree very well there is a remarkableresolution difference in proximity of the point ε = 0 due to the staggered gridsof the schemes.

Condensation

In this test we consider the process of condensation of bosons. It is a funda-mental results of quantum statistics of bosons that above a certain critical


02

46

810 0

2

4

6

8

100

1

2

3

4

5

6

7

t

ε

f(ε,

t)

02

46

810 0

2

4

6

8

100

1

2

3

4

5

6

7

t

ε

f(ε,

t)

Fig. 3. Trend to equilibrium in time for scheme QBF1 (left) and QBF2 (right)computed with N = 40 points.

density particles enter the ground state, that is a Bose-Einstein condensateforms (see [26],[27],[25],[45],[46]) and the equilibrium distribution f∞ is of theform (34) with β− 6= 0.

10−10

10−8

10−6

10−4

10−2

100

10−15

10−10

10−5

100

105

1010

1015

1020

ε

f(ε,

t)

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

21

t

max

(f)

Fig. 4. Distribution of bosons at different times in logarithmic scale (left) and itsmaximum value (right) during condensation with scheme QBF1 with N = 40 points.

Note that for the second order method, unlike the first order one, due tothe midpoint quadrature, ε = 0 is not a gridpoint. Thus we expect to get abetter resolution of the singularity in ε = 0 with the first order scheme.

We choose the initial distribution in the energy interval [0, R] with R = 10to be[45],[46]

f(ε) =2f0

πarctan(eΓ (1−ε/ε0)), (63)


with Γ = 5 and ε0 = R/8. At values of f0 larger than a critical f∗0 theformation of a condensate occurs (see [45],[46]). We choose f0 = 1, whichturns on to be supercritical and integrate the Boson Boltzmann equation upto T = 15. The results we obtain with the two schemes for N = 40 areconsiderably different near the point ε = 0. The results for scheme QBF1 arereported in Figures 4.

A magnified view of the numerical solutions obtained with N = 40 andN = 80 points shows that away from the singularity the two schemes are ingood agreement (see Figure 5).

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ε

f(ε)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ε

f(ε)

Fig. 5. Magnified view of the distribution of bosons at the final computation timeduring condensation with scheme QBF1 () and scheme QBF2 (×) with N = 40(left) and N = 80 (right) points.

4 Time dependent trapping potentials

We now study the formation of a Bose-Einstein condensate from an uncon-densed harmonically trapped gas of bosons by changing the shape of the trap-ping potential (as a function of time) [31]. In particular we shall extend theabove method to the case of a time-dependent trapping potential, harmonicat time t=0, with a dip at the bottom x = 0 which grows rapidly with time

The potential is characterized by a superposition of two harmonic trappotentials with frequencies ω1 and ω2 and reads

V (x, t) =12mω2

1x2 +

(12mω2

2x2 − U0(t)

)θ(R(t)− |x|) (64)

where θ is the Heavyside step function, R is the radius of the narrow dip inthe center of the trap and is given by

R(t) =√

2U0(t)/m(ω21 − ω2

2),


with

U0(t) = U0

(1− 2

1 + exp(t/ts)

), (65)

where U0 > 0 is a given parameter (−U0 is the maximal depth of the dip) andts is a reference scaling time.

In the energy dependent case we have the following QBE in conservativeform

∂

∂t(ρ(t, ε)f(t, ε)) +

∂

∂ε(G(t, ε)f(t, ε)) = Q(f)(t, ε) (66)

supplemented with the initial condition f(0, ε) = f0(ε). Equation (66) is con-sidered in the time dependent energy-interval ε ∈ Ω(t) = [ε0(t),∞[ withε0(t) = −U0(t). The density of states ρ(t, ε) and the function G(t, ε) are givenby

ρ(t, ε) =2π(2m)3/2

(2π~)3

∫

V (x,t)<ε

√ε− V (x, t) dx, (67)

G(t, ε) =2π(2m)3/2

(2π~)3

∫

V (x,t)<ε

∂V (x, t)∂t

√ε− V (x, t) dx. (68)

We assume κ = (ω1/ω2)3 ¿ 1 and find the approximate expressions [31]

ρ(t, ε) =(ε + U0(t))2

2(~ω2)3θ(ε + U0(t)) +

ε2

2(~ω1)3(69)

G(t, ε) = − ˙U0(t)(ε + U0(t))2

2(~ω2)3θ(ε + U0(t)). (70)

We recall that

Q(f)(t, ε) = Cσ

∫

Ω(t)3dε1 dε2 dε3 δ(ε1 + ε2 − ε3 − ε)ρ(t, εmin)

(71)[f1f2(1 + f3)(1 + f)− f3f(1 + f1)(1 + f2)]

with εmin = min(ε1, ε2, ε3, ε), fi = f(t, εi) and C = m/(π2~3).The time dependent change of the trapping potential creates a narrow dip

in the center of the harmonic trap. Inside of this dip the density of particles aswell as the phase-space density can increase dramatically leading eventuallyto BEC. As shown in [31] this simple model gives good agreement with theexperimental data found in [47].

4.1 The numerical method

In order to solve numerically equation (66) it is convenient to introduce thechange of variables ε = ε + U0(t) and use the notations

f(t, ε) = f(t, ε− U0(t)), ρ(t, ε) = ρ(t, ε− U0(t)), G(t, ε) = G(t, ε− U0(t)).


From (66) since

∂

∂t(ρ(t, ε)f(t, ε)) =

∂

∂t(ρ(t, ε)f(t, ε)) + U0(t)

∂

∂ε(ρ(t, ε)f(t, ε)),

we get

∂

∂t(ρ(t, ε)f(t, ε)) +

∂

∂ε((ρ(t, ε)U0(t) + G(t, ε))f(t, ε)) = Q(f)(t, ε) (72)

where now ε ∈ R+ and

Q(f)(t, ε) = C

∫

(R+)3dε1 dε2 dε3 δ(ε1 + ε2 − ε3 − ε)ρ(t, εmin)

(73)[f1f2(1 + f3)(1 + f)− f3f(1 + f1)(1 + f2)].

Problem (72) is considered with the initial condition f(0, ε) = f0(ε) = f0(ε).The numerical solution to (72) is carried out on a finite energy range

ε ∈ [0, R] with R > 0. Clearly the choice of R should be done carefully so thatthe error on the density function f due to the finite energy range is negligible.

The discretization of the collision operator Q(f) in (73) is done with themethods presented before.

Note that from (14) and (15)

ρ(t, ε)U0(t) + G(t, ε) = U0(t)(ε− U0(t))2

2(~ω1)3θ(ε− U0(t)) ≥ 0.

In particular from (7) we have that (72) can be written as

ρ(t, ε)∂

∂tf(t, ε) + (ρ(t, ε)U0(t) + G(t, ε))

∂

∂εf(t, ε) = Q(f), (74)

which describes a transport of the density function f with a nonnegativecharacteristic speed

z(t, ε) =

ρ(t, ε)U0(t) + G(t, ε)ρ(t, ε)

ε > U0(t)

0 ε ≤ U0(t),

(75)

since limε→0 G(t, ε)/ρ(t, ε) = −U0(t). Thus no boundary condition is neededon the computational domain [0, R]. The advection part in (74) has beensolved with a third order method along characteristics using equation (20).

For the time discretization we used a second order Strang[48] splittingmethod. The collision part in (74) is solved with a second order explicit Runge-Kutta scheme. Thus the overall accuracy of our schemes will be second orderboth in time and energy variables. Moreover it can be shown that the resultingschemes preserves the nonnegativity of the solution under a suitable CFLcondition that depends on z(t, ε) and the function f .


4.2 Numerical results

We have computed the numerical solution of equation (72) for several testproblems. The physical quantities have been all expressed in units of ~ω1.The initial data is an equilibrium Bose-Einstein distribution with a giventemperature kT0/~ω1 = T , a given chemical potential µ0/~ω1 = µ, a fixedtrap frequency ratio of ω2/ω1 = κ. The constant tc = tmfp/10 where tmfp =6.8 × 10−4 is the mean free collision time at the center of the trap. All thenumerical solutions have been obtained with N = 100 grid points and Cσ = 1.

We investigate the behavior of the Bose gas when creating a dip of depth−U0(t) from an initial value U0(t0) = 0 to a final value U0(tf ) = U0. Wedenote by −Uc the critical depth at which a Bose-Einstein condensate forms.The plots refers to the steady state solutions as a function of U0(t). Thedensity and energy in the time dependent energy-interval are computed as

N(t) =∫ R

0

ρ(ε, t)f(ε, t)dε, (76)

E(t) =∫ R

0

ρ(ε, t)εf(ε, t)dε. (77)

To compute the values of T and µ for a given density N and energy Mwe invert numerically the corresponding relations (76)-(77) where we assumef = f∞. The condensate fraction is computed whenever µ ≈ −U0(t).

Test 1: (U0 < Uc)

The data for this test are: T = 475, µ = −1465, U0/~ = 1500 and R = 6000.We consider the time evolution of the solution in the time interval t ∈ [0, 0.004]for κ = 0.1, 0.01, 0.001.

0 500 1000 15001

1.05

1.1

1.15

1.2

1.25

1.3

U0/h

Tf/T

0

κ=0.1

κ=0.01

κ=0.001

Fig. 6. Test 1. Temperature increase versus U0/~ in QBE for various κ.


0 500 1000 1500−3000

−2500

−2000

−1500

−1000

−500

0

U0/h

µ f

κ=0.1

κ=0.01

κ=0.001

Fig. 7. Test 1. Chemical potential decrease versus U0/~ in QBE for various κ. Thedashed line corresponds to −U0(t)/~. For κ = 0.001 the value U0/~ = −1500 isalmost critical.

In Figure 6 and 7 we report the temperature and the chemical potentialprofiles versus U0(t)/~ for various κ.

Test 2: (U0 > Uc 6= 0)

The data for this test are: T = 1000, µ = −768, U0/~ = 850, κ = 0.1, 0.01 andR = 10000. The solution has been computed in the time interval t ∈ [0, 0.001].For κ = 0.01 condensation is observed when U0(t)/~ larger then 790.

0 200 400 600 8001

1.005

1.01

1.015

1.02

1.025

U0/h

Tf/T

0 κ=0.1

κ=0.01

Fig. 8. Test 2. Temperature increase versus U0/~ in QBE (continuous line) and thecollisionless equation (dotted line).

In Figure 8, 9 and 10 we report the temperature, the chemical potentialand the condensate profiles versus U0.


0 200 400 600 800−1000

−900

−800

−700

−600

−500

−400

−300

−200

−100

0

U0/h

µ f

κ=0.01

κ=0.1

Fig. 9. Test 2. Chemical potential decrease versus U0/~ in QBE (continuousline) and the collisionless equation (dotted line). The dashed line corresponds to−U0(t)/~.

0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83−2

0

2

4

6

8

10

12

14x 10

−3

U0/Tf

Nc/N

κ=0.1

κ=0.01

Fig. 10. Test 2. Fraction of condensate versus U0/kTf in QBE. For U0(t)/~ ≥ 790condensation is observed.

Test 3: (U0 > Uc = 0)

The data for this test are: T = Tc, µ = 0, U0/~ = 400, κ = 0.2, 0.01 andR = 1500. Numerically we have used a small tolerance τ = 0.0025U0/~ asinitial value for µ to avoid the singularity at ε = 0.

The solution has been computed for t ∈ [0, 0.0005].In Figure 11 and 12 we report the temperature and the condensate profiles

versus U0/kTf .

5 The Gross-Pitaevskii equation

At temperatures T much smaller than the critical temperature Tc [34], aBEC is well described by the macroscopic wave function ψ = ψ(x, t) whose


0 100 200 3001

1.1

1.2

1.3

1.4

1.5

U0/h

Tf/T

0

κ=0.2

κ=0.01

Fig. 11. Test 3. Temperature increase versus U0/~ in QBE. Dotted lines refer tocollisionless result.

0.2 0.6 1 1.4

0.1

0.2

0.3

U0/kT

f

Nc/N

κ=0.01

κ=0.2

Fig. 12. Test 3. Fraction of condensate versus U0/kTf in QBE. Dotted lines referto collisionless result.

evolution is governed by a self-consistent, mean field nonlinear Schrodingerequation (NLSE) known as the Gross-Pitaevskii equation (GPE) [30, 44]

i~∂ψ(x, t)

∂t= − ~

2

2m∇2ψ(x, t) + V (x)ψ(x, t) + NU0|ψ(x, t)|2ψ(x, t), (78)

where m is the atomic mass, ~ is the Planck constant, N is the number ofatoms in the condensate, V (x) is an external trapping potential. When a har-monic trap potential is considered, V (x) = m

2

(ω2

xx2 + ω2yy2 + ω2

zz2)

with ωx,ωy and ωz being the trap frequencies in x, y and z-direction, respectively. Forthe following we assume (w.r.o.g.) ωx ≤ ωy ≤ ωz. U0 = 4π~2as/m describesthe interaction between atoms in the condensate with as the s-wave scatteringlength (positive for repulsive interaction and negative for attractive interac-tion). It is necessary to ensure that the wave function is properly normalized.Specifically, we require


∫

R3|ψ(x, t)|2 dx = 1. (79)

5.1 Dimensionless GPE

In order to scale the Eq. (78) under the normalization (79), we introduce

t = ωxt, x =xa0

, ψ(x, t) = a3/20 ψ(x, t), with a0 =

√~/ωxm, (80)

where a0 is the length of the harmonic oscillator ground state. In fact, here wechoose 1/ωx and a0 as the dimensionless time and length units, respectively.Plugging (80) into (78), multiplying by 1/mω2

xa1/20 , and then removing all ˜,

we get the following dimensionless GPE under the normalization (79) in threedimension

i∂ψ(x, t)

∂t= −1

2∇2ψ(x, t) + V (x)ψ(x, t) + β |ψ(x, t)|2ψ(x, t), (81)

where

V (x) =12

(x2 + γ2

yy2 + γ2zz2

), γy =

ωy

ωx, γz =

ωz

ωx, β =

U0N

a30~ωx

=4πasN

a0.

There are two extreme regimes of the interaction parameter β: (1) β =o(1), the Eq. (81) describes a weakly interacting condensation; (2) β À 1,it corresponds to a strongly interacting condensation or to the semiclassicalregime. There are two extreme regimes between the trap frequencies: (1) γy ≈1 and γz À 1, it is a disk-shaped condensation; (2) γy À 1 and γz À 1, it isa cigar-shaped condensation.

5.2 Reduction to lower dimension

In the following two cases, the 3d GPE (81) can approximately be reduced to2d or even 1d [35, 5, 2]. In the case (disk-shaped condensation)

ωx ≈ ωy, ωz À ωx, ⇐⇒ γy ≈ 1, γz À 1,

the 3d GPE (81) can be reduced to 2d GPE with x = (x, y) by assuming thatthe time evolution does not cause excitations along the z-axis since they havea large energy of approximately ~ωz compared to excitations along the x andy-axis with energies of about ~ωx. Thus we may assume that the condensa-tion wave function along the z-axis is always well described by the harmonicoscillator ground state wave function and set

ψ = ψ2(x, y, t)ψho(z) with ψho(z) = (γz/π)1/4 e−γzz2/2. (82)


Plugging (82) into (81), then multiplying by ψ∗ho(z) (where f∗ denotes theconjugate of a function f), integrating with respect to z over (−∞,∞), weget

i∂ψ2(x, t)

∂t= −1

2∇2ψ2 +

12

(x2 + γ2

yy2 + C)ψ2 + β2|ψ2|2ψ2, (83)

where

β2 = β

∫ ∞

−∞ψ4

ho(z) dz = β

√γz

2π, C =

∫ ∞

−∞

(γ2

zz2|ψho(z)|2 +∣∣∣∣dψho

dz

∣∣∣∣2)

dz.

Since this GPE is time-transverse invariant, we can replace ψ2 → ψ e−i Ct2

which drops the constant C in the trap potential and obtain the 2d GPE, i.e.

i∂ψ(x, t)

∂t= −1

2∇2ψ +

12

(x2 + γ2

yy2)ψ + β2|ψ|2ψ. (84)

The observables are not affected by this.In the case (cigar-shaped condensation) [35, 5, 2]

ωy À ωx, ωz À ωx, ⇐⇒ γy À 1, γz À 1,

the 3d GPE (81) can be reduced to 1d GPE with x = x. Similarly to the 2dcase, we derive the 1d GPE [35, 5, 2]

i∂ψ(x, t)

∂t= −1

2ψxx(x, t) +

x2

2ψ(x, t) + β1|ψ(x, t)|2ψ(x, t), (85)

where β1 = β√

γyγz/2π.In fact, the 3d GPE (81), 2d GPE (84) and 1d GPE (85) can be written

in a unified way

i∂ψ(x, t)

∂t= −1

2∇2ψ + Vd(x)ψ + βd |ψ|2ψ, x ∈ Rd, (86)

where

βd = β

√γyγz/2π,√γz/2π,

1,Vd(x) =

x2/2, d = 1,(x2 + γ2

yy2)/2, d = 2,(

x2 + γ2yy2 + γ2

zz2)/2, d = 3.

(87)

The normalization condition to (86) is∫

Rd

|ψ(x, t)|2 dx = 1. (88)

Two important invariants of (86) are the normalization of the wave func-tion

N(ψ) =∫

Rd

|ψ(x, t)|2 dx = 1, t ≥ 0 (89)

and the energy

Eβ(ψ) =∫

Rd

[12|∇ψ(x, t)|2 + Vd(x)|ψ(x, t)|2 +

βd

2|ψ(x, t)|4

]dx, t ≥ 0. (90)


5.3 Ground state solution

To find a stationary solution of (86), we write

ψ(x, t) = e−iµt φ(x), (91)

where µ is the chemical potential of the condensation and φ a function inde-pendent of time. Inserting into (86) gives the following equation for φ(x)

µ φ(x) = −12∇2φ(x) + Vd(x)φ(x) + βd |φ(x)|2φ(x), x ∈ Rd, (92)

under the normalization condition∫

Rd

|φ(x)|2 dx = 1. (93)

This is a nonlinear eigenvalue problem under a constraint and any eigenvalueµ can be computed from its corresponding eigenfunction φ by

µ = µβ(φ) =∫

Rd

[12|∇φ(x)|2 + Vd(x)|φ(x)|2 + βd |φ(x)|4

]dx

= Eβ(φ) +βd

2

∫

Rd

|φ(x)|4 dx. (94)

The Bose-Einstein condensation ground state solution φg(x) is found byminimizing the energy functional Eβ(φ) under the constraint (93), i.e. to com-pute the ground state φg, we solve the minimization problem(V) Find (µg, φg ∈ V ) such that

Eβ(φg) = minφ∈V

Eβ(φ), µg = µβ(φg) = Eβ(φg)+βd

2

∫

Rd

|φg(x)|4 dx, (95)

where the set V is defined as

V = φ | Eβ(φ) < ∞, ‖φ(x)‖ = 1 , ‖φ‖2 = ‖φ‖2L2 =∫

Rd

|φ(x)|2 dx.

In non-rotating BEC, the minimization problem (95) has a unique real valuednonnegative ground state solution φg(x) > 0 for x ∈ Rd [37].

Various algorithms for computing the minimizer of the minimization prob-lem (95) have been studied in the literature. For instance, second order in timediscretization scheme that preserves the normalization and energy diminishingproperties were presented in [1]. Perhaps one of the more popular techniquefor dealing with the normalization constraint (93) is through the followingconstruction: choose a time sequence 0 = t0 < t1 < t2 < · · · < tn < · · · with∆tn = tn+1 − tn > 0 and k = maxn≥0 ∆tn. To adapt an algorithm for thesolution of the usual gradient flow to the minimization problem under a con-straint, it is natural to consider the following splitting (or projection) scheme


which was widely used in physical literatures [12, 1] for computing the groundstate solution of BEC:

φt = −12

δEβ(φ)δφ

=12∇2φ− Vd(x)φ− βd |φ|2φ,

x ∈ Rd, tn < t < tn+1, n ≥ 0, (96)

φ(x, tn+1)4= φ(x, t+n+1) =

φ(x, t−n+1)‖φ(·, t−n+1)‖

, x ∈ Rd, n ≥ 0, (97)

φ(x, 0) = φ0(x), x ∈ Rd with ‖φ0‖ = 1; (98)

where φ(x, t±n ) = limt→t±n φ(x, t). In fact, the gradient flow (96) can be viewedas applying the steepest decent method to the energy functional Eβ(φ) with-out constraint and (97) then projects the solution back to the unit sphere inorder to satisfying the constraint (93). From the numerical point of view, thegradient flow (96) can be solved via traditional techniques and the normal-ization of the gradient flow is simply achieved by a projection at the end ofeach time step. For βd = 0, as observed in [4], the gradient flow with discretenormalization (GFDN) (96)-(98) preserves the energy diminishing property:

Theorem 2. Suppose Vd(x) ≥ 0 for all x ∈ Rd. For βd = 0, the GFDN(96)-(98) is energy diminishing for any time step k > 0 and initial data φ0,i.e.

E0(φ(·, tn+1)) ≤ E0(φ(·, tn)) ≤ · · · ≤ E0(φ(·, 0)) = E0(φ0), n ≥ 0. (99)

In fact, the normalized step (97) is equivalent to solve the following ODEexactly

φt(x, t) = µφ(t, k)φ(x, t), x ∈ Rd, tn < t < tn+1, n ≥ 0, (100)φ(x, t+n ) = φ(x, t−n+1), x ∈ Rd; (101)

where

µφ(t, k) ≡ µφ(tn+1,∆tn) = − 12 ∆tn

ln ‖φ(·, t−n+1)‖2, tn ≤ t ≤ tn+1.

Thus the GFDN (96)-(98) can be viewed as a first-order splitting method forthe gradient flow with discontinuous coefficients:

φt =12∇2φ− V (x)φ− β |φ|2φ + µφ(t, k)φ, x ∈ Rd, t ≥ 0, (102)

φ(x, 0) = φ0(x), x ∈ Rd with ‖φ0‖ = 1. (103)

Let k → 0, we see that

µφ(t) = limk→0+

µφ(t, k)

=1

‖φ(·, t)‖2∫

Rd

[12|∇φ(x, t)|2 + Vd(x)φ2(x, t) + βdφ

4(x, t)]

dx.


This suggests us to consider the following continuous normalized gradient flow(CNGF):

φt =12∇2φ− Vd(x)φ− βd |φ|2φ + µφ(t)φ, x ∈ Rd, t ≥ 0, (104)

φ(x, 0) = φ0(x), x ∈ Rd with ‖φ0‖ = 1. (105)

In fact, the right hand side of (104) is the same as (92) if we view µφ(t) as aLagrange multiplier for the constraint (93). Furthermore for the above CNGF,as observed in [4], the solution of (104) also satisfies the following theoremwhich provides a mathematical justification for the algorithm (96)-(98):

Theorem 3. Suppose Vd(x) ≥ 0 for all x ∈ Rd and βd ≥ 0. Then the CNGF(104)-(105) is normalization conservation and energy diminishing, i.e.

‖φ(·, t)‖2 =∫

Rd

|φ(x, t)|2 dx = ‖φ0‖2 = 1, t ≥ 0, (106)

d

dtEβ(φ) = −2 ‖φt(·, t)‖2 ≤ 0 , t ≥ 0, (107)

which in turn implies

Eβ(φ(·, t1)) ≥ Eβ(φ(·, t2)), 0 ≤ t1 ≤ t2 < ∞.

In fact, it is easy to prove that eigenfunctions of the nonlinear eigenvalueproblem (92) under the constraint (93), critical points of the energy functionalEβ(φ) under the constraint (93) and steady state solutions of the CNGF (104)-(105) are equivalent.

Here we present a backward Euler finite difference (BEFD) scheme forfully discretizing the GFDN (96)-(98) to compute the ground state solutionsof BEC. For simplicity of notation we introduce the methods for the caseof one spatial dimension (d = 1). Generalizations to higher dimension arestraightforward for tensor product grids and the results remain valid withoutmodifications. In 1d, we choose an interval [a, b] with |a|, b sufficiently largeand spatial mesh size h = ∆x > 0 with h = (b−a)/M and M an even positiveinteger, and define grid points and time steps by

xj := a + j h, tn := n k, j = 0, 1, · · · ,M, n = 0, 1, 2, · · ·Let φn

j be the numerical approximation of φ(xj , tn). We use backward Eulerfor time discretization and second-order centered finite difference for spatialderivatives. The detail scheme is:

φ∗j − φnj

k=

φ∗j+1 − 2φ∗j + φ∗j−1

2h2− V1(xj)φ∗j − β1

(φn

j

)2φ∗j , 1 ≤ j ≤ M − 1,

φ∗0 = φ∗M = 0, φ0j = φ0(xj), j = 0, 1, · · · ,M,

φn+1j =

φ∗j‖φ∗‖ , j = 0, · · · ,M, n = 0, 1, · · · ; (108)


where the norm is defined as ‖φ∗‖2 = h∑M−1

j=1

(φ∗j

)2. When V1(x) ≥ 0, asobserved in [4], this scheme is monotone for β1 ≥ 0 and energy diminishingfor β1 = 0 for any time step k > 0. For non-rotating BEC with harmonicoscillator potential, the above scheme can be used to compute the ground(first excited) state provided that we chose the initial data φ0(x) as an evenpositive (odd) function, e.g. φ0(x) = 1

π1/4 e−x2/2 (φ0(x) = x√

2π1/4 e−x2/2). Figure

13 shows the ground state φg(x) and first excited state φ1(x) of BEC in 1dwith V1(x) = x2/2 in (86) for different β1. For more numerical results ofground states in 1d, 2d and 3d BEC, we refer [4, 9, 2].

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x

Φ g(x)

0 5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

φ 1(x)

Fig. 13. Stationary states of BEC in 1d for β1 =0, 3.1371, 12.5484, 31.371, 62.742, 156.855, 313.71, 627.42, 1254.8 (with decreas-ing peak). Left: ground state φg(x) (positive even function); Right: first excitedstate φ1(x) (odd function).

5.4 Time-splitting spectral method for GPE

To study the dynamics of BEC, one needs to solve the time-dependent GPE(86) numerically. Here we present the time-splitting spectral method intro-duced in [5, 3, 2] for the GPE (86). Similarly, we only introduce the methodin one space dimension (d = 1). Generalizations to d > 1 are straightforwardfor tensor product grids and the results remain valid without modifications.For d = 1, the equation (86) with homogeneous Dirichlet boundary conditionsbecomes

i∂ψ(x, t)

∂t= −1

2ψxx + V1(x)ψ + β1|ψ|2ψ, a < x < b, (109)

ψ(x, t = 0) = ψ0(x), a ≤ x ≤ b, ψ(a, t) = ψ(b, t) = 0, , t > 0.(110)


Let ψnj be the approximation of ψ(xj , tn).From time t = tn to t = tn+1,

the GPE (109) is solved in two splitting steps. One solves first

iψt = −12ψxx, (111)

for the time step of length k, followed by solving

i∂ψ(x, t)

∂t= V1(x)ψ(x, t) + β1|ψ(x, t)|2ψ(x, t), (112)

for the same time step. Equation (111) will be discretized in space by the sinespectral method and integrated in time exactly. For t ∈ [tn, tn+1], the ODE(112) leaves |ψ| invariant in t [7, 8] and therefore becomes

i∂ψ(x, t)

∂t= V1(x)ψ(x, t) + β1|ψ(x, tn)|2ψ(x, t) (113)

and thus can be integrated exactly. We combine the splitting steps via thefourth-order split-step method and obtain a fourth-order time-splitting sine-spectral method (TSSP4) for GPE (109). The detailed method is given by

ψ(1)j = e−i2w1k(V1(xj)+β1|ψn

j |2)) ψnj ,

ψ(2)j =

M−1∑

l=1

e−iw2kµ2l ψ

(1)l sin(µl(xj − a)),

ψ(3)j = e−i2w3k(V1(xj)+β1|ψ(2)

j |2)) ψ(2)j ,

ψ(4)j =

M−1∑

l=1

e−iw4kµ2l ψ

(3)l sin(µl(xj − a)), j = 1, 2, · · · ,M − 1,

ψ(5)j = e−i2w3k(V1(xj)+β1|ψ(4)

j |2)) ψ(4)j ,

ψ(6)j =

M−1∑

l=1

e−iw2kµ2l ψ

(5)l sin(µl(xj − a)),

ψn+1j = e−i2w1k(V1(xj)+β1|ψ(6)

j |2)) ψ(6)j ; (114)

where w1 = 0.33780 17979 89914 40851, w2 = 0.67560 35959 79828 81702,w3 = −0.08780 17979 89914 40851 and w4 = −0.85120 71979 59657 63405[50], and Ul, the sine-transform coefficients of a complex vector U = (U0, U1, · · · , UM )with U0 = UM = 0, are defined as

µl =πl

b− a, Ul =

2M

M−1∑

j=1

Uj sin(µl(xj − a)), l = 1, 2, · · · ,M − 1, (115)

with


0 2 4 6 8 100.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

σ x (or

|ψ(0

,t)|2 )

0

1

2

3

4

5

6 0

0.2

0.4

0.6

0.8

1

0

0.5

1

tx

|ψ|2

Fig. 14. Dynamics of 1d BEC. Left: Width of the condensate σx (‘—’) and centraldensity |ψ(0, t)|2 (‘- - -’); Right: Evolution of the density function |ψ|2.

0 5 10 15 200

2000

4000

6000

8000

10000

12000

14000

16000

18000

acollapse

= −6.7a0

−30a0

−250a0

Number of Atoms in Condensate

t [ms]

Fig. 15. Number of remaining atoms after collapsing a 85Rb condensate ofN0 = 16000 atoms. Collapse is achieved by ramping the scattering length lin-early from ainit = 7a0 (where a0 is the Bohr radius) to acollapse = −6.7a0,−30a0

and −250a0 in 0.1 [ms] as a function of time τevolve [ms] (labelled as t). The ‘*’and ‘o’ are taken from the experiment [18], the solid curves are our numerical so-lutions and the dashed curves are fitted to the experimental points: Ntotal(t) =Nremnant(τevolve) = N0

remnant + (N0 −N0remnant) · exp((tcollapse − τevolve)/tdecay) with

N0remnant = 7000, 5000, 1660; tcollapse = 8.6, 3.8, 1.1 [ms] and tdecay = 2.8, 2.8, 1.2

[ms] for acollapse = −6.7a0, −30a0, −250a0, respectively.


ψ0j = ψ(xj , 0) = ψ0(xj), j = 0, 1, 2, · · · , M. (116)

Note that the only time discretization error of TSSP4 is the splitting error,which is fourth order in k. As observed in [5, 3, 2], the scheme is explicit,unconditionally stable, of spectral order accuracy in space and fourth orderaccuracy in time, and conserves the position density. Furthermore, it is timereversible and time transverse invariant, just as they hold for the GPE itself.Figure 14 plots the condensate width σx(t) = ‖xψ‖, central density ψ(0, t)|2as functions of time and evolution of the density |ψ|2 in space-time for 1d GPE(86) with V1(x) = 22x2/2, β1 = 20 and ψ0(x) be the ground state solution of(86) with d = 1, V1(x) = x2/2 and β1 = 20. The above time-splitting spectralmethod can be easily extended to GPE with a quintic damping term for mod-eling a collapse and explosion BEC [3, 6] observed in experiment [18]. In theexperiment, the s-wave scattering length is changed by an external magneticfield near a Feshbach resonance of the 85Rb atoms, i.e. as = as(t). They startedfrom a stable condensate with a positive scattering length as = ainit > 0, thenchanged as from positive to negative as = acollapse < 0, and observed a seriescollapse and explosion. Figure 15 plots the number of remaining atoms aftercollapsing a 85Rb condensate of initially N0 = 16000 atoms with differentacollapse [6]. For more numerical results of dynamics in 1d, 2d and 3d BEC,we refer [2, 5, 6].

6 Coupling Quantum Boltzmann and Gross-Pitaevskiiequations

As already mentioned in the introduction the process of creating a Bose-Einstein condensate in a trap by means of evaporative cooling starts in aregime covered by the QBE and finishes in a regime where the Gross-Pitaevskii(GPE) equation is expected to be valid. The GPE is capable to describe themain properties of the condensate at very low temperatures, it treats the con-densate as a classical field and neglects quantum and thermal fluctuations. Asa consequence the theory breaks down at higher temperatures where the noncondensed fraction of the gas cloud is significant. An approach which allowsthe treatment of both condensate and noncondensate parts simultaneouslywas developed in [51].

The resulting equations of motion reduce to a generalized GPE for thecondensate wave function coupled with a semiclassical QBE for the thermalcloud:

i~∂

∂tψ(x, t) = − ~

2

2m∇2ψ(x, t) + V (x)ψ(x, t) (117)

+[U0(nc(x, t) + 2n(x, t))− iR(x, t)]ψ(x, t),∂F

∂t+

pm· ∇xF −∇xU · ∇pF = Q(F ) + Qc(F ). (118)


where nc(x, t) = |ψ(x, t)|2 is the condensate density, V (x) is the confining po-tential. The collision integral Q(F ) is the conventional QBE integral whereasQc(F ) describes collisions between condensate and non condensate particlesand is given by

σnc

m2π

∫

R9δ(mvc + p∗ − p′ − p′∗)δ(εc + ε∗ − ε′ − ε′∗)[δ(p− p∗)− δ(p− p′)

(119)−δ(p− p′∗)][F′F ′∗(1 + F∗)− F∗(1 + F ′)(1 + F ′∗)] dp∗dp

′dp′∗,

with σ = 8πa2s, ε = p2/2 + U(x, t) where U = V + 2U0(nc + n) is the mean

field potential. The presence of Qc(F ) leads to a change in the number ofcondensate particles. n(x, t) and R(x, t) are the non condensate density anda source term, respectively, which are defined as

n(x, t) =1

(2π~)3

∫F (x,p, t) dp, R(x, t) =

~2nc(2π~)3

∫Qc(F )dp. (120)

Note that for low temperatures T → 0 we have n,R → 0 and we recover theconventional GPE. The equations (117), (118) are normalized as Nc(0) = N0

c

and Nt(0) = N0t with

Nc(t) =∫

R3|ψ(x, t)|2 dx, Nt(t) =

∫

R3|n(x, t)|2 dx, t ≥ 0, (121)

where N0c and N0

t are the number of particles in the condensate and thermalcloud at time t = 0, respectively. It is easy to see from the equations (117),(118) that

dNt(t)dt

=1

(2π~)3

∫

R6Qc(F )dp dx = −dNc(t)

dt, t ≥ 0. (122)

As a consequence the total number of particles defined as Ntotal(t) = Nc(t) +Nt(t) ≡ N0

total = N0c + N0

t is obviously conserved. This set of equations hasbeen solved numerically in [32] by combining a Monte Carlo method for thekinetic equations with a Fourier spectral method for the GPE. We hope topresent results for such system based on coupling the methods derived in theprevious sections for the kinetic part and the GPE solver [5, 40].

Acknowledgement

The authors are grateful to Dieter Jaksch for stimulating discussions on thesubject of this work. This work was supported by the WITTGENSTEINAWARD 2000 of Peter Markowich, financed by the Austrian Research FundFWF and by the European network HYKE, funded by the EC as contractHPRN-CT-2002-00282.


References

1. A. Aftalion, and Q. Du, Vortices in a rotating Bose-Einstein condensate: Criticalangular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev.A, 64, 063603, (2001).

2. W. Bao, Ground states and dynamics of multi-component Bose-Einstein con-densates, arXiv: cond-mat/0305309.

3. W. Bao, D. Jaksch, An explicit unconditionally stable numerical methods forsolving damped nonlinear Schrodinger equations with a focusing nonlinearity,SIAM J. Numer. Anal., to appear (arXiv: math.NA/0303158).

4. W. Bao and Q. Du, Computing the ground state solution of Bose-Einsteincondensates by a normalized gradient flow, SIAM J. Sci. Comp., to appear(arXiv: cond-mat/0303241).

5. W.Bao, D.Jaksch, P.Markowich, Numerical solution of the Gross-PitaevskiiEquation for Bose-Einstein condensation, J. Comput. Phys., 187, 318 - 342,(2003).

6. W.Bao, D.Jaksch, P.Markowich, Three Dimensional Simulation of Jet Forma-tion in Collapsing Condensates, arXiv: cond-mat/0307344.

7. W. Bao, S. Jin and P.A. Markowich, On time-splitting spectral approximationsfor the Schrodinger equation in the semiclassical regime, J. Comput. Phys., 175,487-524, (2002).

8. W. Bao, S. Jin and P.A. Markowich, Numerical study of time-splitting spectraldiscretizations of nonlinear Schrodinger equations in the semi-clasical regimes,SIAM J. Sci. Comp., to appear.

9. W. Bao and W. Tang, Ground state solution of trapped interacting Bose-Einstein condensate by directly minimizing the energy functional, J. Comput.Phys., 187, 230-254, (2003).

10. D.Benedetto, F. Castella, R. Esposito, M. Pulvirenti, Some Considerations onthe derivation of the nonlinear Quantum Boltzmann Equation. MathematicalPhysics Archive, University of Texas, 03-19, (2003).

11. S.N. Bose, Plancks Gesetz and Lichtquantenhypothese, Z. Phys., 26, 178–181,(1924).

12. M.L. Chiofalo, S. Succi and M.P. Tosi, Ground state of trapped interactingBose-Einstein condensates by an explicit imaginary-time algorithm, Phys. Rev.E, 62, 7438-7444, (2000).

13. C. Buet, S. Cordier, Numerical method for the Compton scattering operator,Lecture Notes on the discretization of the Boltzmann equation, ed. N.Bellomo,World Scientific, (2002).

14. C. Buet, S. Cordier, P. Degond and M. Lemou, Fast algorithms for numerical,conservative, and entropy approximations of the Fokker-Planck equation, J.Comp. Phys., 133, 310-322, (1997).

15. S. Chapman and T. G. Cowling, The mathematical theory of non- uniformgases, Cambridge University Press, 1970. Third edition.

16. E. A. Cornell, J. R. Ensher and C. E. Wieman, Experiments in dilute atomicBose-Einstein condensation in Bose-Einstein Condensation in Atomic Gases,Proceedings of the International School of Physics Enrico Fermi Course CXL(M. Inguscio, S. Stringari and C. E. Wieman, Eds., Italian Physical Society,1999), pp. 15-66 (cond-mat/9903109).

17. P.J.Davis, P.Rabinowitz, Methods of numerical integration, Academic Press,(1975).


18. E.A. Donley, N.R. Claussen, S.L. Cornish, J.L. Roberts, E.A. Cornell and C.E.Wieman, Dynamics of collapsing and exploding Bose-Einstein condensates, Na-ture, 412, 295-299, 2001.

19. A.Einstein, Quantentheorie des einatomingen idealen gases, Stiz. PresussischeAkademie der Wissenshaften Phys-math. Klasse, Sitzungsberichte, 23, 1–14,(1925).

20. A.Einstein, Zur quantentheorie des idealen gases, Stiz. Presussische Akademieder Wissenshaften Phys-math. Klasse, Sitzungsberichte, 23, 18–25, (1925).

21. L. Erdos, M. Salmhofer, H.Yau, On the quantum Boltzmann equation, preprint2003

22. M. Escobedo, S. Mischler, Equation de Boltzmann quantique homogene: exis-tence et comportement asymptotique, C. R. Acad. Sci. Paris 329 Serie I, 593–598(1999)

23. M.Escobedo, S.Mischler, M.A.Valle, Homogeneous Boltzmann equation inquantum relativistic kinetic theory, Electronic Journal of DifferentialEquations, Monograph 04, 2003, 85 pages. (http://ejde.math.swt.edu orhttp://ejde.math.unt.edu).

24. M.Escobedo, S.Mischler, On a quantum Boltzmann equation for a gas of pho-tons, J. Math. Pures Appl., 9 80, 471–515, (2001).

25. C.W.Gardiner, D.Jaksch, P.Zoller, Quantum Kinetic Theory II: Simulation ofthe Quantum Boltzmann Master Equation, Phys. Rev. A, 56, 575, (1997)

26. C.W.Gardiner, P.Zoller, Quantum Kinetic Theory I: A Quantum Kinetic MasterEquation for Condensation of a weakly interacting Bose gas without a trappingpotential, Phys. Rev. A, 55, 2902, (1997),

27. C.W.Gardiner, P.Zoller, Quantum Kinetic Theory III: Quantum kinetic masterequation for strongly condensed trapped systems, Phys. Rev. A, 58, 536, (1998)

28. V.L. Ginzburg, L.P. Pitaevskii, Zh. Eksp. Teor.Fiz. 34, 1240 (1958) [Sov. Phys.JETP 7, 858 (1958)]

29. E.P. Gross, J. Math. Phys. 4, 195 (1963).30. E.P. Gross, Nuovo. Cimento., 20, 454, (1961).31. D.Jaksch, P.Markowich, L.Pareschi, M.Wenin, P.Zoller, Increasing phase-space

density by varying the trap potential, work in progress.32. B.Jackson, E.Zaremba, Dynamical simulations of trapped Bose gases at finite

temperatures, preprint 2002 (cond-mat/0106652).33. W. Ketterle, D.S. Durfee, D.M. Stamper-Kurn, Making, probing and under-

standing Bose-Einstein condensates, in Bose-Einstein condensation in atomicgases, Proceedings of the International School of Physics Enrico Fermi, CourseCXL, edited by M. Inguscio, S. Stringari, and C.E. Wieman (IOS Press, Ams-terdam, 1999), pp. 67176 (cond-mat/9904034).

34. L. Landau and E. Lifschitz, Quantum Mechanics: non-relativistic theory, Perg-amon Press, New York, (1977).

35. P. Leboeuf and N. Pavloff, Phys. Rev. A 64, 033602 (2001); V. Dunjko,V. Lorent, and M. Olshanii, Phys. Rev. Lett. 86, 5413 (2001).

36. M.Lemou, Multipole expansions for the Fokker-Planck-Landau operator, Nu-merische Mathematik, 78, 597–618, (1998).

37. E.H. Lieb, R. Seiringer and J. Yngvason, Bosons in a Trap: A Rigorous Deriva-tion of the Gross-Pitaevskii Energy Functional, Phys. Rev. A, 61, 3602, (2000).

38. X.Lu, On spatially homogeneous solutions of a modified Boltzmann equationfor Fermi-Dirac particles, J. Statist. Phys., 105, 353–388, (2001).


39. X.Lu, A modified Boltzmann equation for Bose-Einstein particles: isotropic so-lutions and long-time behavior, J. Statist. Phys., 98, 1335–1394, (2000).

40. P. Markowich, L.Pareschi, Fast, conservative and entropic numerical methodsfor the boson Boltzmann equation, preprint 2002.

41. L. Pareschi, Computational methods and fast algorithms for Boltzmannequations. Lecture Notes on the discretization of the Boltzmann equation, ed.N.Bellomo, World Scientific, (2002).

42. L. Pareschi, G. Russo and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comp. Phys, 165, 1–21, (2000).

43. L. Pareschi, G.Toscani and C. Villani, Spectral methods for the non cut-offBoltzmann equation and numerical grazing collision limit, Numerische Mathe-matik, (to appear).

44. L.P. Pitaevskii, Zh. Eksp. Teor. Fiz., 40, 646, (1961) (Sov. Phys. JETP, 13,451, (1961)).

45. D.V.Semikoz, I.I.Tkachev, Kinetics of Bose condensation, Physical Review Let-ters, 74, 3093–3097, (1995).

46. D.V.Semikoz, I.I.Tkachev, Condensation of bosons in the kinetic regime, Phys-ical Review D, 55, 489–502, (1997).

47. D.M.Stamper-Kurn, H.J.Miesner, A.P.Cdikkatur, S.Inouye, J.Stenger,W.Ketterle, Phys. Rev, Lett. 81, p.2194, (1988).

48. G. Strang, On the construction and comparison of difference schemes, SIAM J.Numer. Anal. 5, (1968) pp. 506.

49. J.T.M.Walraven, Quantum dyanmics of simple systems, edited by G.L.Oppo,S.L.Burnett, E.Riis and M.Wilkinson, Bristol 1996. (SUSSP Proceedings,vol.44).

50. H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett.A, 150, 262-268, (1990).

51. E.Zaremba, T.Nikuni, A.Griffin, J. Low Temp. Phys. 116, p. 277, (1999).

Documents

Quantum kinetic theory: modelling and numerics for Bose-Einstein