L. M. Morato- Dissipation and concentration of vorticity from Stochastic Quantization

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Dissipation and concentration of vorticity from StochasticQuantization

L. M. Morato

Universita di Verona

Verona,16 09 2009

Dissipation and concentration of vorticit – p. 1



CANONICAL QUANTIZATION

Quantum Hamiltonian (α := coupling parameter)

H =

N i=1

−

2

2m2

i + Φ(ri)

+ Φint(r1, .....,rN , α)

( H is bounded from below)

i ∂ tΨ =−

2

2m2 + Φα,N

tot

Ψ

where := (1, . . . ,N ) and Φα,N tot :=

N

i=1Φ(ri) + Φint(r1, . . . , rN , α).




STOCHASTIC QUANTIZATION BY LAGRANGIAN VARIATIONAL PRINCIPLE

The basic object is the classical lagrangian

L[qcl] =

N i=1

12

m(qcli )2(t) − Φ(qcli (t)) − Φint(qcl1 (t), ...,qclN (t), α)

qcl := classical N - body configuration.

Quantization comes from requiring that the configuration of the system evolves as a Markov

diffusion (with diffusion coefficient equal to

m) and that it makes stationary the mean classical

action

- Eulerian ( or " stochastic control" ) approach, with pre-regularization of the action : Guerra andM., Phys. Rev. D ’83.

- Lagrangian ( or "path-wise") approach, with self-regularization of the action : M. , Phys. Rev. D

’85.




diffusion

i) q is a solution of the stochastic differential equation

dq(t) = b(q(t), t)dt +m

1

2

dW (t)

ii) The drift b, is smooth

The Brownian Motion W models "quantum fluctuations"

ρ := time dependent probability density of the configuration

V := current velocity field

b = V +

2m ln ρ

and the continuity equation holds

∂ρ∂t

= − · (ρ V )




Eq. of motion in the Lagrangian approach

d = 3 ) ( ρ, V )

∂ tρ = − · (ρ V )

∂ tV + (V · ) V − 2

2m22√ρ√

ρ

+

+

2m( ln ρ + ) ∧ (∧ V ) = − 1

mΦ

d = 3N (ρ, V )

∂ tρ = − · ρ V ∂ tV +

V ·

V −

2

2m22

√ρ√

ρ

k

+

+

2m

3N p=1

(∂ p ln ρ + ∂ p)

∂ kV p − ∂ pV k

= −1

m ∂ kΦα,N

tot k = 1, . . . , 3N




Equations of motion in Schrödinger form

d = 3 ) (for simplicity of notations)

There exists S s.t. Ψ := ρ1

2 eı

S , A := mV −S

ı∂ tΨ = 1

2m

(ı

+A

)2Ψ + ΦΨ

∂ tA = b∗ ∧ (∧A) −

2m∧ (∧A)

b∗ :=1

m[

S −A−

2ln|Ψ|2] (Loffredo, M., JMP ’89)




Relaxation to dynamical equilibrium

Energy Theorem (Loffredo and M., JMP ’89; extended in 2007)( d = 3 for simplicity of notations)

E [ρ, V ] := R3N

1

2mV

2 +1

2mU

2 + Φ ρ dr

with U :=

2m ln ρ (osmotic velocity).

d

dtE [ρ, V ] =

−

2 R3N

(∧

V )2ρdr

NOTE: Irrotational solutions conserve the energy and

E =

<Ψ

,HΨ

>

For generic initial data Schrödinger solutions act as an attracting set, which corresponds to

DYNAMICAL EQUILIBRIUM.




SCHRÖDINGER VERSUS ROTATIONAL SOLUTIONS

Schrodinger solutions

Rotational solutions

t

Σ

(ρt, vt)t>0

Note

• Schrödinger solutions :singular velocity field in cor-

respondence of nodes of the

wave function.

• Rotational solutions : smooth




Non trivial behavior of vorticity 1

1) For ρ > 0 : the vorticity ∧A does not go to zero monotonically(firstly conjectured by Guerra in 1992)

“gaussian solutions to the bidimensional harmonic oscillator”

(M. and Ugolini, AHP ’94) ( global existence and center manifold )

Φ := Φ(r) =1

2k2

r2

ρ(r, t) =A

π

exp(

−Ar

2), V (r, t) = arr

−αrθ

Ω

12

10

8

64

2

0

a

64

20

-2-4

-6-8A

76543210

t = 0

trajectory

t

E

20151050

250

200

150

100

50

0

energy vs. time

t

2Ω

20151050

24

20

16

12

8

4

0

vorticity vs. time




Non trivial behavior of vorticity 2

2) The vorticity can concentrate in the zeroes of the density

Non gaussian solutions to the bidimensional harmonic oscillator, numerical results:

[Caliari, Inverso and M. 2004, New J. Phys., Vol 6, no. 69,http://www.iop.org/EJ/journal/NJP]


http://www.iop.org/EJ/journal/NJP

http://www.iop.org/EJ/journal/NJP



Numerical experiment (t=0)

ρ at time t = 0, E = 195 −∧ v = 2Ωo




Numerical experiment (t=0.08)

ρ at time t = 0.08, E = 43 −∧ v





ρ at time t = 0.14, E = 17 −∧ v





ρ at time t = 0.16, E = 15 −∧ v





ρ at time t = 0.19, E = 11 −∧ v





ρ at time t = 0.23, E = 9 −∧ v




Cubic Schrödinger equation with vorticity

(Caliari,Loffredo , M. and Zuccher : New.J. Phys. (10) 2008) ( for the connection with the N -body

problem see some results in Loffredo and M. Journal of Physics A: Mathematical and Theoretical 40 2007)

Ψ := ρ1

2 eı

S ,

A:= mV

−S

ı ∂ tΨ = 1

2m(ı + A)2Ψ + (g|Ψ|2 + Φ)Ψ

∂ tA = b∗ ∧ (∧A) −

2m∧ (∧A)

b∗ :=1

m[S −A−

2 ln |Ψ|2]

Extension of the energy theorem ( dissipation independent ofg ! )

E A[ρ,S,t] :=

(S −A)2

2+

2

2m( log ρ)2 + Φ +

g

2ρ

ρ

d

dt E A[ρ,S,t]dr =

−

2 (

∧A)2ρdr




exp

c = 100c = 10

c = 0

x

ρ ( x ,

0 ,

0 )

86420-2-4-6-8

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Profiles of the two-dimensional ground

state densities ( t=0) for different values of

the coupling constant

c = 100c = 10

c = 0

t

E ( t )

0.30.250.20.150.10.050

400

350

300

250

200

150

100

50

0

Energy as a function of time




exp1

−× v

t = 0, ρ

x

−

×

v ( x ,

0 ,

t )

ρ

( x ,

0 ,

t )

20.20

20.15

20.10

20.05

20.00

19.95

19.90

19.85

19.8086420-2-4-6-8

0.160

0.140

0.120

0.100

0.080

0.060

0.040

0.020

0.000

Density versus vorticity (c = 0)

−× v

t = 0 .237, ρ

x

−

×

v ( x , 0

, t )

ρ ( x ,

0 ,

t )

5.00

4.00

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00

-4.0086420-2-4-6-8

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

−× v

t = 0 .248, ρ

x

−

×

v ( x , 0

, t )

ρ ( x ,

0 ,

t )

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00

-4.0086420-2-4-6-8

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

−× v

t = 0 .268, ρ

x

−

×

v ( x , 0

, ¯ t )

ρ ( x ,

0 ,

¯ t )

300.0

250.0

200.0

150.0

100.0

50.0

0.0

-50.0

-100.0

-150.086420-2-4-6-8

0.025

0.020

0.015

0.010

0.005

0.000




exp2

−× v

t = 0, ρ

x

−

×

v ( x ,

0 ,

t )

ρ

( x ,

0 ,

t )

20.20

20.15

20.10

20.05

20.00

19.95

19.90

19.85

19.8086420-2-4-6-8

0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Density versus vorticity (c = 100)

−× v

t = 0 .193, ρ

x

−

×

v ( x , 0

, t )

ρ ( x ,

0 , t )

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00

-4.00

-5.0086420-2-4-6-8

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

−× v

t = 0 .202, ρ

x

−

×

v ( x , 0

, t )

ρ ( x ,

0 , t )

3.00

2.00

1.00

0.00

-1.00

-2.00

-3.00

-4.0086420-2-4-6-8

0.030

0.025

0.020

0.015

0.010

0.005

0.000

−× v

t = 0 .212, ρ

x

−

×

v ( x , 0

, ¯ t )

ρ ( x ,

0 ,

¯ t )

800.0

600.0

400.0

200.0

0.0

-200.0

-400.0

-600.086420-2-4-6-8

0.018

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

-0.002




exp3

c = 100c = 10

c = 0

t

ρ ( 0 ,

0 ,

t )

0.30.250.20.150.10.050

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Density at the origin as a function of time

c = 100c = 10

c = 0

t

−

×

v

0.30.250.20.150.10.050

20

15

10

5

0

Vorticity (− ∧ v) at the origin as a func-

tion of time




comment

Application to conceptually simple experiments where the condensation occurs after the stirringof a normal cloud ?

( P.C. Haljan, I. Coddington, P. Engels and E.A. Cornell*, Phys. Rev. Lett. 87, 210403 (2001))


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L. M. Morato- Dissipation and concentration of vorticity from Stochastic Quantization