Exponential convergence to the equilibrium for the stochastic Gross-Pitaevskii equation€¦ ·...

Exponential convergence to the equilibrium for thestochastic Gross-Pitaevskii equation

R. Fukuizumi 1

A. de Bouard 2 and A. Debussche 3

1RCPAM, GSIS, Tohoku University, Japan2CMAP, Ecole Polytechnique, France

3IRMAR, ENS Cachan Bretagne, France

Mathematical and Physical models of Nonlinear Optics @ IMA

R. Fukuizumi (Tohoku Univ.) October 31, 2016 1 / 29

Outline

1 IntroductionPhysical Motivation

2 Gibbs equilibriumGibbs measureMain resultsGibbs measure and the stationary solutionInvariance of the Gibbs measure

3 Local existence

4 Global existence for ρ a.e. initial data

5 Convergence to the Gibbs equilibriumPoincaré inequality

6 Final remarks


Introduction Physical Motivation

Outline



3 Local existence



6 Final remarks



Bose-Einstein Condensation (BEC)

A state of matter of a dilute gas of bosons cooled to temperatures veryclose to absolute zero; a large amount of bosons occupy the lowestquantum state, where macroscopic quantum phenomena can be observed.Predicted in 1924-25, Realization in 1995.

87Rb, 23Na, 7Li, 1H, 85Rb, 41K, 4He, 133Cs, 174Yb,

52Cr, 40Ca, 84Sr : Example of the dilute gases of bosons



Dynamics of BEC at nite temperature

At the zero temperature T = 0, all the atoms are well-presented by asingle condensate function and the coherent evolution of the wavefunction is described by the standard GP equation

At higher temperature, all spontaneous and incoherent process (for ex.interaction with thermal cloud) may not be neglected

The eect of such incoherent elements are implemented by adding adisspation and a noise to the GP equation, called 'Stochastic GPequation,' which is well suited for vortex formulation in BEC (Weileret al. Nature (2008))

dψ = P− i

~LGPψdt +

G (x)

kBT(µ− LGP)ψdt + dWG (x , t)

where

LGP = − ~2

2m∇2 + V (x) + g |ψ|2, 〈dW ∗G (s, y), dWG (t, x)〉 = 2G (x)δt−sδx−y



P.B. Blakie et al., Advanced in Physics (2008); introduce moredetailed energy cut-o

The equation evolves the system to the grand-canonical equilibriumdistribution.

M.Kobayashi-L.Cugliandolo (arXiv:1606.03262); interested in a quenchdynamics in BEC

Phase transition leads to the formation of topological defects (i.e.vortices)Investigation of the thermal equilibrium (Gibbs equilibrium) gives theclassication (Universality class) of the type of phase transitionsStudy of the equilibrium properties around the phase transition pointsusing, as a model, Stochastic GP equation with V (x) = 0, G (x) = 1under a periodic boundary condition

Remark that essentially, nite dimension models are analyzed in Physics.Our aim: Study these models from a mathematical point of view, (i.e., ininnite dimension). In particular, interested in the Gibbs equilibrium (thecase of a space-time white noise).



We consider mathematically (for the moment in 1d):

(Ω,F ,P) : propability space endowed with ltration (Ft)t≥0The equation :

dX = (i + γ)(∂2xX − V (x)X − λ|X |2X )dt +√γdW ,

X (0) = X0, t > 0, x ∈ R

where γ > 0. Assume V (x) = x2 and λ = 1 (defocusing).

(∂2x − x2)hk = −λ2khk with λk =√2k + 1, k ∈ N. The eigenfunctions

hk(x) are known as the Hermite functions.

W (t) : cylindrical Wiener pocess on L2(R,C), i.e.

W (t, x) =∑k∈N

βk(t)hk(x), t > 0, x ∈ R

where βk(t)k∈N : a sequence of C-valued independent Brownianmotions.


Gibbs equilibrium Gibbs measure

Outline



3 Local existence



6 Final remarks



Hamiltonian for the case of γ = 0:

H(u) =1

2

∫R|(−∂2x + x2)1/2u|2dx +

1

4

∫R|u|4dx

The Gibbs measure:

ρ(du) = Γe−H(u)du

= Γe−14

∫R |u|

4dxe−12((−∂2x+x2)u,u)L2du = Γe−

14

∫R |u|

4dxµ(du)

where Γ is the normalizing constant, may be justied in Lp(R), p > 2(see later) taking the limit N →∞ of the Gaussian measure onR2(N+1) dened by

dµN :=N∏k=0

λ2k2π

e−λ2k2(a2

k+b2

k)dakdbk

if we write u =∑

k(ak + ibk)hk with (ak , bk) ∈ R2.

µ := limN→∞ µN can be interpreted as the law of random variable∑k∈N

√2

λkgk(ω)hk(x) with L(gk) = NC(0, 1)



Known results

Burq, Tzvetkov and Thomann, (γ = 0, λ = ±1, V (x) = x2, 1d)Ann. Inst. Fourier (Grenoble) (2013)

Construction of Gibbs measure, Global Cauchy theory in the negativeSobolev space.

Barton-Smith, (γ 6= 0, λ = ±1, V (x) = 0, bounded domain D, any d)Nonlinear dier. equ. appl.(2004)

Existence and Uniqueness of invariant measure on Lp(D) (p ∈ [2,∞))for a not too small γ 6= 0

E.A. Carlen, J. Fröhlich and J. Lebowitz, (regular noise, λ = −1,V (x) = 0, periodic boundary condi., 1d) (arXiv:1409.2327)

Construction of the grand-canonical Gibbs measure (i.e. with amodied Hamiltonian) and exponential convergence by Dirichlet formapproach under some assumptions on the noise

Our result is new on the points/no restriction on γ, the exp-convergencewith a space-time white noise


Gibbs equilibrium Main results

Outline



3 Local existence



6 Final remarks


Gibbs equilibrium Main results

Recall that our equation isdX = (i + γ)(∂2xX − x2X − |X |2X )dt +

√γdW ,

X (0) = X0, t > 0, x ∈ R.

Let p ≥ 3, X0 ∈ Lp(R), and γ > 0.

Theorem

There exists a set O ⊂ Lp(R) such that ρ(O) = 1, and such that for

X0 ∈ O there exists a unique solution X (·) ∈ C ([0,∞), Lp(R)) a.s.

Let Pt be the transition semigroup for the equation, i.e.Ptφ(y) = E(φ(X (t, y))), y ∈ O, t ≥ 0.

Theorem

Let φ ∈ L2((Lp, dρ),R), and φ =∫Lpφ(y)dρ(y). Then Ptφ(·) converges

exponentially to φ in L2((Lp, dρ),R), as t →∞ ; more precisely,∫Lp|Ptφ(y)− φ|2dρ(y) ≤ e−γt

∫Lp|φ(y)− φ|2dρ(y).


Gibbs equilibrium Gibbs measure and the stationary solution

Outline



3 Local existence



6 Final remarks



Gibbs measure and the stationary solution

Let us denote by

Z∞(t) =√γ

∫ t

−∞e−(t−s)(i+γ)(−∂

2x+x

2)dW (s),

the solution of

dZ = (i + γ)(∂2x − x2)Zdt +√γdW

which is stationary, i.e. such that L(Z∞(t)) = L(Z∞(0)).

Write Z∞(t) using the basis hkk ,

Z∞(t) =√γ∑k∈N

(∫ t

−∞e−(t−s)(i+γ)λ

2kdβk(s)

)hk

The law of Z∞(t) equals to the Gaussian measure µ, since

L(√

γ

∫ 0

−∞es(i+γ)λ

2kdβk(s)

)= NC

(0,

2

λ2k

)R. Fukuizumi (Tohoku Univ.) October 31, 2016 14 / 29


Support of the measure

Let p > 2. Note that by Minkowski inequality, for q ≥ p, one has

|Z∞(t)|Lqω(Lpx ) ≤ |Z∞(t)|Lpx (Lqω).

Since Z∞ is Gaussian, to compute |Z∞|Lqω when q = 2m, it suces tocompute the L2ω norm:

E(|Z∞(t, x)|2) ≤ γ∑k∈N

E∣∣∣ ∫ t

−∞e−λ

2k(t−s)(i+γ)dβk(s)

∣∣∣2|hk(x)|2

≤ 2γ∑k∈N

∫ t

−∞e−2λ

2kγ(t−s)ds |hk(x)|2 ≤

∑k∈N

|hk(x)|2

λ2k.



For q = 2m with 2m ≥ p, we have

E(|Z∞(t, x)|2m) ≤ CmE(|Z∞(t, x)|2)m ≤ Cm

(∑k

|hk(x)|2

λ2k

)mand

|Z∞(t)|L2mω (Lpx )≤ Cm

∣∣∣∑k∈N

|hk(x)|2

λ2k

∣∣∣1/2Lp/2x

≤ Cm

(∑k

|hk(x)|2Lpx

λ2k

)1/2It is known the decay of hk in Lp (Yajima-Zhang, CMP 2001): for

p ≥ 4, |hk |Lp(R) ≤ Cpλ−1/6k , and by interpolation,

if 2 ≤ p ≤ 4, |hk |Lp(R) ≤ Cpλ− 1

3(1− 2

p)

k .

Remind that λ2k = 2k + 1 and the series converges for p > 2, i.e.,Z∞ ∈ L2m(Ω; Lp) for any m ≥ p/2 > 1;i.e. Z∞ ∈ Lp a.s. i.e. ρ(Lp) = 1 for p > 2.


Gibbs equilibrium Invariance of the Gibbs measure

Outline



3 Local existence



6 Final remarks



Formal proof for the invariance of the Gibbs measure

We write the equation under the form

dX = −J∇xH(X )dt − γ∇xH(X )dt +√γdW , X (0) = y ∈ Lpx

where

J =

(0 −11 0

): R2 → R2.

Let Pt be the transition semigroup, i.e,Ptφ(y) = E(φ(X (t, y))) = 〈φ, µt,x〉 for φ ∈ Cb(Lp) withµt,x = L(X (t, y)).

〈Ptφ, ρ〉 = Ptφ(y) provided L(y) = ρ. Thus P∗t ρ = µt,x .i.e. ρ is invariant i P∗t ρ = ρ.



Let L be the generater for the semigroup Pt :

Lφ(y) = limt→0

Ptφ(y)− φ(y)

t.

For the invariance of ρ it is enough to show∫Lp

(Lφ)(y)ρ(dy) = 0

On the other hand, the generator L is given by (cf. Kolmogorovequation)

(Lφ)(y) = γ∆yφ(y)−γ(∇yφ(y),∇yH(y))L2x − (∇yφ(y), J∇yH(y))L2x



∫Lp

(Lφ)(y)ρ(dy) =

∫Lp

[γ∆yφ(y)− γ(∇yφ(y),∇yH(y))L2

−(∇yφ(y), J∇yH(y))L2]e−H(y)dy

= γ

∫Lp

∆yφ(y)e−H(y)dy + γ

∫Lp

(∇yφ(y),∇y (e−H(y))

)L2dy

−∫Lp

(∇yφ(y), J∇y (e−H(y))

)L2dy = 0.

The last term is zero, and the second term is the opposite sign of therst term by I/P.

This calculation can be justied using a nite dimensionalapproximation.

This invariance holds for both purely dissipative case and mixed case.


Local existence

Local existence

Write X = v + Z∞ with v satisfying the deterministic PDE:

∂tv = (i + γ)(∂2xv − x2v − |v + Z∞|2(v + Z∞)), t > 0, x ∈ R

Solve locally, then obtain a sol. X in C ([0,T ∗), Lp), p ≥ 3(Ginibre-Velo, CMP. 1997) using the estimates on the linear semigroupobtained by Mehler's formula:

|et(i+γ2)(∂2x−x2)f |Lr (R) ≤ Ct−12l |f |Ls , t > 0 small,

0 ≤ 1

r=

1

s− 1

l, 1 ≤ s, l ≤ ∞.

Energy methods give a global bound in Lp, but it requires somerestriction on the parameter γ. (not too small γ 6= 0)


Global existence for ρ a.e. initial data

ρ-a.e. Global existence (formal)

Let T ≥ T ∗, and p ≥ 3. Wish to show that there exists CT such that∫Lp

E(

supt∈[0,T∗)

|X (t,X0)|Lp)ρ(dX0) ≤ CT .

Mild form for X :

X (t) = et(i+γ)(∂2x−x2)X0 − et(i+γ)(∂

2x−x2)Z∞(0)

+

∫ t

0

e(t−s)(i+γ)(∂2x−x2)(|X |2X )(s)ds + Z∞(t).

By the regularity of Z∞: supt |Z∞(t)|Lp ≤ MT , we have∫Lp

E(

supt∈[0,T∗)

|X (t,X0)|Lp)dρ(X0)

≤ C

∫Lp

E|X0|Lpdρ(X0) + CE∫Lp

∫ T

0

|X (s)|3L3pdsdρ(X0) + MT .


Global existence for ρ a.e. initial data

Here the invariance of ρ implies∫LpPtφ(X0)dρ =

∫Lpφ(X0)dρ

with φ(X0) = |X0|3L3p . Therefore∫ T

0

∫Lp

E|X (s,X0)|3L3pdρ(X0)ds =

∫ T

0

∫LP|X0|3L3pdρ(X0)ds

≤∫ T

0

∫LP|X0|3L3pdµ(X0)ds

which is equivalent to∫ T0

E(|Z∞(s)|3L3p

)ds since L(Z∞(s)) = µ, andis bounded by CT .

Thusρ(X0 ∈ Lp; sup

t∈[0,T∗)|X (t,X0)|Lp < +∞) = 1.

This formal argument can be justied by an nite dimensionalapproximation.


Convergence to the Gibbs equilibrium

Convergence to the equilibrium (formal)

The proof of the convergence is based on a Poincaré inequality: Let γ > 0.For any φ ∈ C 1

b (Lp), the following inequality is satised.∫Lp|∇yφ(y)|2L2ydρ(y) ≥ √γ

∫Lp|φ(y)− φ|2dρ(y), (1)

where φ =∫Lpφ(y)dρ(y).

Assume (1). Let u(t, y) = Ptφ(y) and φ = 0. Then, if L is thegenerator of Pt , then

dudt

= Lu.

L satises (again by the Kolmogorov equation)

Lu2(y) = 2√γ|∇yu|2L2 + 2uLu(y).


Convergence to the Gibbs equilibrium

Invariance of the measure ρ implies

0 =

∫LpLu2(y)dρ(y)

= 2√γ

∫Lp|∇yu|2L2dρ(y) + 2

∫Lpu(Lu)(y)dρ(y)

= 2√γ

∫Lp|∇yu|2L2dρ(y) +

d

dt

∫Lp|u(t)|2dρ(y).

hence by (1)d

dt|u(t)|2L2(Lp ,dρ) ≤ −2γ|u|

2L2(Lp ,dρ),

from which it follows

|u(t)|L2(Lp ,dρ) ≤ e−γt |φ|L2(Lp ,dρ).


Convergence to the Gibbs equilibrium Poincaré inequality

Outline



3 Local existence



6 Final remarks



Proof of Poincaré inequality (formal)

This inequality is a property of the measure only, so that we can proveit by the purely dissipative case, for which ρ is also an invariantmeasure.

dY = γ(∂2xY − x2Y − |Y |2Y )dt +√γdW

This is again a formal argument, we need a nite dimensionalapproximation

Denote ηh := DyY (t, y).h; then ηh satisesdη = γ(∂2xη − x2η − γ|Y |2Y − 2γ(Re(Y η)Y )dt,η(0) = h ∈ L2(R).

Thanks to dissipative property, ηh satises

|ηh(t)|L2 ≤ e−γt |h|L2 . (2)



let Rt be the transition semigroup, and let g(t, y) = Rtφ(y) forφ ∈ C 1

b (Lp) and y ∈ Lp. The above exp.decay of ηh and theinvariance of the measure ρ implies∫

Lp|∇yg(y)|2L2dρ(y) ≤ e−2γt

∫Lp

∣∣∣Rt |∇yφ|L2(y)∣∣∣2dρ(y)

≤ e−2γt∫LpRt(|∇yφ|2L2)(y)dρ(y)

= e−2γt∫Lp|∇yφ|2L2(y)dρ(y).

since (∇yg(t, y), h)L2 = E((∇yφ(Y (t, y)), ηh)L2)On the other hand, by the Kolmogorov equation and the invariance ofthe measure (as above),

d

dt|g(t, y)|2L2(Lp ,dρ) = −2√γ|∇yg(y)|2L2(Lp ,dρ)

Note that g(0, y) = φ(y) and that, thanks to dissipative property,|g(t)|L2(Lp ,dρ) → 0 as t →∞. Then integrating in time and tendingt →∞ concludes the Poincaré inequality.


Final remarks

Final remarks

Our results may be extended to

Stochastic complex Ginzburg-Landau eq. without the harmonicpotential (but with a periodic boundary condi., or on a boundeddomain with Dirichlet condi.)

with the chemical potential

global existence for all initial data

2d or 3d case, but for the renormalized Hamiltonian

with a rotating term; Lz := −i(x∂y − y∂x),

L(u) =1

2

∫R3

|(−H)1/2u|2 + Re

∫R3

(uΩLzu) +1

4

∫R3

|u|4

The focusing case (λ = −1)


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Exponential convergence to the equilibrium for the stochastic Gross-Pitaevskii equation€¦ ·...

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