Kinetic theory and

B-E condensationin weakly interacting gases

M. Escobedo

Universidad del Paıs Vasco

Bilbao.

Joint work with J. J. L. Velazquez.

1

PLAN

1.- Motivation: formation of BE condensate in a weakly interacting gas of bosons.

2.- Previous related results.

3.- Classical and weak solutions

4.- Main Theorem: blow up and delta formation in finite time.

5.- Final remark and questions.

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Formation of Bose Einstein condensate

Over time a comprehensive picture of the process of condensation:

3 distinct stages of non equilibrium dynamics

1.- A kinetic redistribution of particles towards lower energy modes in the noncondensed phase.

2.- Development of an instability that leads to the nucleation of the condensate.

3.- Build up of coherence and condensate growth.

Subject of some attention since mid 70’s up to now ...

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In stage 1: A description of a weakly interacting gas of bosons based on theBoltzmann equation is admitted to be adequate. For an homogeneous gas, thedistribution function of particles with momentum p at time t, n ≡ n(t, p).

∂n

∂t(t, p1) =

1

2

∫∫∫R9|M|2(2π)3δ(p1 + p2 − p3 − p4)×

×2πδ(ε1 + ε2 − ε3 − ε4)q(n)dp2 dp3 dp4

q (n) = n3n4(1 + n1)(1 + n2)− n1n2(1 + n3)(1 + n4)

εi = ε(pi) ≡|pi|2

2m

ni = n(t, ε(pi)), i = 1, 2, 3, 4.

|M|2 = (8πam−1)2

a is the scattering length of the Fermi pseudopotential V (r1−r2) = 4πam−1δ(r1−r2).

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• Equation first obtained by L. W. Nordheim in 1928. Derived since then byseveral authors with different methods and arguments, in presence or not ofcondensate:

A. I. Akhiezer, S. V. Peletminskij, Methods of Statistical Physics, Moscow, 1977

C. W. Gardiner & P. Zoller ’97, ’98, ’00

E. Zaremba, T. Nikuni & A. Griffin ’99 (ZNG)

M. Imamovic-Tomasovic & A. Griffin ’01

R. Baier, T. Stockkamp: ’05.

No rigorous deduction up to now.————//————

• Let us consider in all the remaining a simplified situation:

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Homogeneous isotropic gas

Homogeneous + isotropic gas:

n(t, ε(p)) ≡ f(t, ε)

∂f

∂t(t, ε1) =

∫∫D(ε1)

w (ε1, ε3, ε4) q(f)dε3dε4 ≡ Q(f)(t, ε1)

q (f) = f3f4(1 + f1)(1 + f2)− f1f2(1 + f3)(1 + f4)

fi = f(εi), i = 1, 2, 3, 4,

ε2 = ε3 + ε4 − ε1

D (ε1) = (ε3, ε4) : ε3 > 0, ε4 > 0, ε3 + ε4 ≥ ε1 > 0

w (ε1, ε3, ε4) =min

(√ε1,√ε2,√ε3,√ε4

)√ε1

.

All the contants have been absorbed in time.

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EquilibriaGiven M > 0 and E > 0 there exists a unique equilibrium F such that

M(F ) :=

∫ ∞0

√εF (ε)dε = M, and E(F ) :=

∫ ∞0

ε3/2F (ε)dε = E

Moreover, for some constant C0 > 0 (explicitly known):

If M ≤ C0E3/5: there is no condensate in the equilibrium:

F (ε) =(eβ(ε−µ) − 1

)−1

, β > 0, µ ≤ 0.

If M > C0E3/5: there is a condensate in the equilibrium. F is such that

√εF (ε) =

√ε(eβε − 1

)−1+ n δ(ε), β > 0, n > 0.

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Previous results

1.- E. Levich & V. Yakhot Study a Boltzmann type equation:

∂f

∂t(t, ε) = Q(f)(t, ε) + Q(f)(t, ε)

Q(f) is our collision integral

Q(f) describes collisions of bosons with a heat bath of fermions.

1.-A (Phys. Rev. B.’77). Q(f) + Q(f) drive f close to an equilibrium of Q(f).

Then Q(f) is dominant: Self similar Delta formation in infinite time is deduced.

(In a homogeneous system condensation is signified by the presence of a Diracdelta in the momentum distribution n(t, p) at zero momentum.)

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1.-B In a second paper (J. Low Temp.’77). When f >> 1: neglect Q (quadraticin f) in front of Q (cubic in f).

- Obtain an approximated equation that is explicitely solvable.- Prove self similar Delta formation in finite time:

√εf(t, ε) Cδ(ε), t→ t∗.

for some constant C > 0 and finite time t∗ > 0. But they admit that very strongapproximations’ were donne...

2.-D. W. Snoke & J. P. Wolfe (Phys. Rev B ’89) Perform numerical calculationsof the equation. “The calculations successfully model the evolution of a Bose gasvery far from equilibrium up to the moment just before condensation”.

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3.-B. V. Svistunov, (J. Mosc. Phys. Soc. ’91) Discusses condensate formationin weakly interacting dilute Bose gas using ideas form weak wave turbulence.Based on formal scaling arguments, he proposed that the distribution function ofparticles f(t, ε) follows the self similar form:

f(t, ε) ∼ (t∗ − t)−72 h

(ε

(t∗ − t)3

)where h(x) ∼ x−7/6 as x→∞.

At ε = 0 : limt→t∗

f(t, 0) =∞

This was continued in works by Y. Kagan, B. V. Svistunov and G. V. Shlyapnikovin JETP’92, Y. Kagan, B. V. Svistunov JETP ’94.

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4.-D. V. Semikoz and I. I. Tkachev (PRL ’95, PRD ’97) Investigated Svistunov’sscenario performing numerical calculations.They obtained slightly different exponents.

5.-H. Spohn (Physica D ’10) Describes the mechanism of how the condensateis generated and annihilated on the level of the Boltzmann-Nordheim kineticequation. The formation of a condensate through a finite time self similar blowup is considered in a global scenario running from the pre condensation to thepost condensation stage.

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Remarks

1.- Finite time blow up of solutions+delta formation: interesting conjecture basedon formal asymptotics.

2. “Supported” by numerical calculations. But these never too conclusive forblowing up solutions.

3. Question: what is the meaning of a delta formation? Is it the manifestation ofthe BE condensation? Or just a dynamical process in the equation without anydirect link with the BEC?

Our goal: To prove one of the points of the scenario suggested by Svistunov andadopted by others: the finite time blow up and Dirac mass formation for someslutions.

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What kind of solutions?1.- Classical-solutions. These are functions with some properties of integrabilityand boundedness with respect to t > 0 and ε > 0 of the form:

supt∈(T1,T2), ε>0

|h(t, ε)|(1 + ε)γ <∞, for some γ > 0

and that satisfy the equation for almost every value of t > 0 and ε > 0.We denote:

||h(t)||L∞(R+;(1+ε)γ) = supε>0|h(t, ε)|(1 + ε)γ

2.- Measure-solutions. The Nordheim equation may be solved in the space ofmeasures. Actually, some of its stationary solutions are measures:

√εF (ε) =

√ε(eβ(ε) − 1

)−1

+ n δ(ε)

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More generally, using the symmetries of the collision integral Q(f) with respectto the variables εi, we notice that if

g (t, ε) =√εf (t, ε)

then, for every ϕ ∈ C20 ([0,∞)):∫ ∞

0

Q(f)(t, ε)ϕ(ε)√εdε =

1

2

∫∫∫(R+)

3

w(ε1, ε2, ε3)Qϕ√ε1ε2ε3

g1 g2 g3 dε1dε2dε3dt+

+1

2

∫ T

0

∫∫∫(R+)3

w(ε1, ε2, ε3)Qϕ√ε1ε2

g1 g2 dε1dε2dε3

where Qϕ = ϕ (ε3) + ϕ (ε1 + ε2 − ε3)− 2ϕ (ε1) .

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So, f is a weak solution if

g (t, ε) =√εf (t, ε) ∈ C([0, T ) ;M+(R+)︸ ︷︷ ︸

positive measures

)

and satisfies:

−∫R+g0 (ε)ϕ (0, ε) dε =

∫ T

0

∫R+g∂tϕdεdt+

+1

2

∫ T

0

∫(R+)

3

g1g2g3w√ε1ε2ε3

Qϕdε1dε2dε3dt+1

2

∫ T

0

∫(R+)3

g1g2w√ε1ε2

Qϕdε1dε2dε3dt

for every ϕ ∈ C20 ([0, T ) ; [0,∞)).

The existence of global weak solutions was proved by X. Lu in J. Stat. Phys ’05.

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Existence of Classical Solutions

Theorem. Suppose that f0 ∈ L∞ (R+; (1 + ε)γ) with γ > 3. There exists

T > 0, depending only on ‖f0 (·)‖L∞(R+;(1+ε)γ), and there exists a classical

solution, f ∈ L∞ ([0, T ) ;L∞ (R+; (1 + ε)γ)).

The solution f satisfies (mass & energy conservation):∫ ∞0

f (t, ε) ε12dε =

∫ ∞0

f0 (ε) ε12dε,

∫ ∞0

f (t, ε) ε32dε =

∫ ∞0

f0 (ε) ε32dε

This solution f can be extended to a maximal time interval (0, Tmax) withTmax ≤ ∞. If Tmax <∞ we have:

limt→T−max

‖f (t, ·)‖L∞(R+) =∞.

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The proof uses a fixed point argument. Similar arguments to those in T.Carleman’s ’57 paper for classical Boltzmann equation. Based on writting theequation as:

∂tf1 + a (t) f1 =

∫ ∞0

∫ ∞0

f3f4 (1 + f1 + f2)w dε3dε4

a (t, ε1) =

∫ ∞0

∫ ∞0

f2 (1 + f3 + f4)wdε3dε4

• THEN: If the initial data f0 is a bounded function decaying at infinity like(1 + ε)−γ for some γ > 3 THEN there is a solution f(t) that remains a boundedfunction of ε with the same decay at infinity.

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Main Theorem. There exist θ∗ > 0 such that, for all M > 0, E > 0,ν > 0, γ > 3, there exists ρ0 = ρ0 (M,E, ν) > 0, K∗ = K∗ (M,E, ν) > 0satisfying the following property. For any f0 ∈ L∞ (R+; (1 + ε)

γ) such that

M(f0) = M , E(f0) = E and

•• sup0≤ρ≤ρ0

[min

inf

0≤R≤ρ

1

νR3/2

∫ R

0

f0(ε)√εdε,

1

K∗ρθ∗

∫ ρ

0

f0(ε)√εdε

]≥ 1,

the unique classical solution f ∈ L∞loc ([0, Tmax) ;L∞ (R+; (1 + ε)γ)), with initial

data f0, is defined on a maximal existence time Tmax <∞ and satisfies:

lim supt→T−max

‖f (t)‖L∞(R+) =∞.

Moreover, there exists T ∗ ≥ Tmax such that:

supTmax≤t≤T

∫0

√εf(t, ε)dε > 0 ∀T > T∗.

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Remark The above Theorem means that initial data f0 ∈ L∞ (R+; (1 + ε)γ),

with a sufficiently large density around ε = 0, blow up in finite time and form aDirac mass at the origin.The condition (••) means that there exists ρ ∈ (0, ρ0) satisfying:

∫ R

0

f0 (ε)√εdε ≥ νR3

2 for 0 < R ≤ ρ ,

∫ ρ

0

f0 (ε)√εdε ≥ K∗ (ρ)

θ∗ , (1)

The first condition is satisfied if for example f0(ε) ≥ 3ν2 when 0 ≤ ε < ρ for some

ρ > 0 small.

The second condition in (1) holds if the distribution f0 has a mass sufficientlylarge in a ball with radius ρ for some ρ sufficiently small.

Since θ∗ might be small, the first condition in (1) does not necessarily implies thesecond.

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Corollary. Suppose that f0 ∈ L∞ (R+; (1 + ε)γ) with γ > 3.

Let f ∈ L∞loc ([0, Tmax) ;L∞ (R+; (1 + ε)γ)) be the classical solution of the

Nordheim equation with initial data f0, where Tmax is the maximal existencetime. Suppose that:

M(f0) > C0E35(f0).

Then:Tmax <∞,

and

limt→T−max

||f(t)||L∞ = +∞.

Morever, there exists T∗ ≥ Tmax such that:

supTmax≤t≤T

∫0

√εf(t, ε)dε > 0, ∀T > T∗.

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This seem to relate the finite time Dirac mass formation in the equation with theB-E condensation:

All the solutions with initial data f0 whose mass M(f0) and energy E(f0) aresuch that the corresponding equilibrium contains a condensate, blow up and forma Dirac mass in finite time.

BUT:

The condition for the finite time blow up/condensation result is a local condition:

•• sup0≤ρ≤ρ0

[min

inf

0≤R≤ρ

1

νR3/2

∫ R

0

f0(ε)√εdε,

1

K∗ρθ∗

∫ ρ

0

f0(ε)√εdε

]≥ 1,

This is possible even if M(f0) < C0E(f0)3/5 for which no BE condensate isexpected.

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Some arguments in the proof

• Simplest example: f ′(t) = fp(t), t > 0, f(0) = a > 0, p > 1.

Blow up follows from explicit expression: f(t) =(

a1−a(p−1)t

)− 1p−1

.

• The second simpler blow up is for the heat equation:

∂u

∂t−∆u = up(t), t > 0, x ∈ Ω

u(t, x) = 0, x ∈ ∂Ω; u(0, x) ≥ 0, p > 1.

If the solution was global, multiply by ϕ first eigenvalue of −∆ on Ω and definef(t) =

∫Ωu(t, x)ϕ(x)dx satisfies:

f ′(t) ≥ Cfp(t)− λ1f(t), t > 0, f(0) = a > 0, p > 1

for some constant C = C(Ω) > 0. Impossible if f is global...

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We first try to apply similar argument to the Nordheim equation. Consider thecubic (main) part of the collision integral. For all suitable f :∫∫∫

R+q3(f)(ε1)w(ε1, ε2, ε3)ϕ(ε1)ε1dε1dε3dε4 =

=

∫∫∫(R+)3

f1f2f3 Gϕ(ε1, ε2, ε3)dε1dε2dε3

for some function Gϕ and where q3(f) = f3f4(f1 + f2)− f1f2(f3 + f4). Then,

d

dt

∫ ∞0

f(t, ε1)ϕ(ε1)ε1dε1 =

∫∫∫(R+)3

f1f2f3 Gϕ(ε1, ε2, ε3)dε1dε2dε3 +

+1

2

∫∫∫(R+)3

f1f2Qϕ(ε1, ε2, ε3)w(ε1, ε2, ε3)√ε1dε1dε2dε3

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1© Monotonicity of the cubic part of the collision integral.

• If ϕ convex then Gϕ (ε1, ε2, ε3) ≥ 0

Then the cubic term is nonnegative. If we had:∫∫∫(R+)3

f1f2f3 Gϕ(ε1, ε2, ε3)dε1dε2dε3 ≥ C(∫

R+f(t, ε)εϕ(ε)dε

)3

that could lead to something like:

f ′ ≥ C1f3 − C2f

2, for some C1, C2 positive constants

and would lead to the blow up for large initial data.

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Difficulty• Gϕ (ε1, ε2, ε3) vanishes along the diagonal

(ε1, ε2, ε3) ∈ (R+)

3; ε1 = ε2 = ε3

.

• The distributions f such that√εf(ε) = δ(ε) solve the equation.

Then: if f(t, ε) concentrates around one single point then f1f2f3 Gϕ is very smalland we can not have:∫∫∫

(R+)3f1f2f3 Gϕ(ε1, ε2, ε3)dε1dε2dε3 ≥ C

(∫R+f(t, ε)εϕ(ε)dε

)3

The blow up happens because:

− If√εf(t, ε) is “far from a Dirac measure”, then (ε1, ε2, ε3) is far from the

diagonal and Gϕ(ε1, ε2, ε3) does not vanish.

− The quadratic terms prevent√εf(t, ε) to get close to a Dirac measure.

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We must change the strategy of the proof. The same proof works to show Diracmass formation in finite time.

We argue by contradiction: suppose that we have a solution f on time interval[0, T ). Obtain a contradiction if T is “too large”

The details use measure theory and “classical” arguments in regularity theory forpartial differential equations.

We present one of the typical Lemmas.

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We divide [0, T ) in two subsets as follows.

For any real number b > 1 define, for all k = 1, 2, . . . :

Ik (b) = b−k(b−1, 1

], I(E)

k = Ik−1 (b) ∪ Ik (b) ∪ Ik+1 (b)

|0

|1

|1b

|1b2

|1b3

|1b4

|1b5

|1b6

|1b7

I(E)1 (b) in red; I(E)

5 (b) in blue

Pb=

A ⊂ [0, 1] : A =⋃j

Ikj (b) for some set of indexes kj ⊂ 1, 2, ...

.

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If A =

∞⋃j=1

Ikj (b) define: A(E) =

∞⋃j=1

I(E)kj

(b) .

Given 0 < δ < 23, define η = min

(13 −

δ2

), δ6

.

For any function h ∈ L∞(0, 1), h ≥ 0 at least one of the following is satisfied:

(i) There exists an interval Ik (b) such that:

∫I(E)k

(b)

h(ε)dε ≥ (1− δ)∫ 1

0

h(ε)dε

(ii) There exists two sets U1,U2 ∈ Pb such that U2 and U (E)1 are disjoint and:

min

∫U1

h(ε)dε,

∫U2

h(ε)dε

≥ η

∫ 1

0

h(ε)dε.

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Conclusion1.- Finite time blow up and Dirac mass formation have been proved to take placefor some solutions of the Nordheim equation.

2.- This happens for all initial data whose mass and energy correspond to anequilibrium with condensate.

3. We do not know if the blow up and Dirac mass formation are self similar likein Svistunov’s scenario.

4. Finite time blow up and Dirac formation may also happen for “sub critical”initial data.

5. What is the precise relation of this finite time blow up with the B-Econdensation?

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