8/3/2019 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
8/3/2019 L. M. Morato- Dissipation and concentration of vorticity from Stochastic Quantization
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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).
Dissipation and concentration of vorticit – p. 2
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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.
Dissipation and concentration of vorticit – p. 3
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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 )
Dissipation and concentration of vorticit – p. 4
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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
Dissipation and concentration of vorticit – p. 5
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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)
Dissipation and concentration of vorticit – p. 6
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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.
Dissipation and concentration of vorticit – p. 7
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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
Dissipation and concentration of vorticit – p. 8
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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
Dissipation and concentration of vorticit – p. 9
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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]
Dissipation and concentration of vorticit – p. 10
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Numerical experiment (t=0)
ρ at time t = 0, E = 195 −∧ v = 2Ωo
Dissipation and concentration of vorticit – p. 11
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Numerical experiment (t=0.08)
ρ at time t = 0.08, E = 43 −∧ v
Dissipation and concentration of vorticit – p. 12
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Numerical experiment (t=0.14)
ρ at time t = 0.14, E = 17 −∧ v
Dissipation and concentration of vorticit – p. 13
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Numerical experiment (t=0.16)
ρ at time t = 0.16, E = 15 −∧ v
Dissipation and concentration of vorticit – p. 14
8/3/2019 L. M. Morato- Dissipation and concentration of vorticity from Stochastic Quantization
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Numerical experiment (t=0.19)
ρ at time t = 0.19, E = 11 −∧ v
Dissipation and concentration of vorticit – p. 15
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Numerical experiment (t=0.23)
ρ at time t = 0.23, E = 9 −∧ v
Dissipation and concentration of vorticit – p. 16
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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
Dissipation and concentration of vorticit – p. 17
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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
Dissipation and concentration of vorticit – p. 18
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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
Dissipation and concentration of vorticit – p. 19
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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
Dissipation and concentration of vorticit – p. 20
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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
Dissipation and concentration of vorticit – p. 21
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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))
Dissipation and concentration of vorticit – p. 22