Quasi-stationary states in mean-field dynamics STEFANO RUFFO Dipartimento di Energetica “S. Stecco” Universita` di Firenze and INFN
Entropy production, transport, chaos and turbulence, IHP, Paris, Nov. 5-9 (2007) Center for the Study of Complex Dynamics University of Florence
Quasi-stationary states in mean-field dynamics – p.1/35
Plan Hamiltonian Mean Field (HMF) model Quasi-stationary states Non equilibrium phase transition Lynden-Bell entropy Maximum entropy principle Application to the free electron laser
Quasi-stationary states in mean-field dynamics – p.2/35
HMF model H=
N X p2
N X 1 i (1 − cos(θi − θj )) + 2 2N
i=1
Magnetization M = limN →∞ Energy U = limN →∞
H N
Temperature T = limN →∞
„P
i,j=1
N i=1
cos θi , N
PN
sin θi N
i=1
«
p2 i N
P
i
Quasi-stationary states in mean-field dynamics – p.3/35
Equilibrium phase transition Fig. 1
revised
Latora, Rapisarda & Ruffo − Lyapunov instability and finite size...
1.0
0.6 0.5
T
0.4 0.3 0.2 0.1
0.8 0.6 0.4 0.2 0.0 0.0 0.4 0.8 1.2
U Theory (c.e.) N=100 N=1000 N=5000 N=20000 N=20000 n.e.
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
U
Uc = 3/4 = 0.75
Quasi-stationary states in mean-field dynamics – p.4/35
Waterbag p ∆θ
111111 000000 000000 111111 ∆p 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
θ
sin ∆θ ∆θ (∆p)2 1 − (M0 )2 H = + U = lim N →∞ N 6 2 M0 =
Quasi-stationary states in mean-field dynamics – p.5/35
Equilibrium N=10000, U=0.63, a=1.0, t=100, sample=1000
p
1.2
-1.2 -3.14
theta
3.14
PDF 0
0.12
Quasi-stationary states in mean-field dynamics – p.6/35
Quasi-stationary states-I 0.35 0.3
(a)
M(t)
0.25 0.2 0.15 0.1 0.05 0
−1
0
1
2
3
4
5
6
7
8
log10t
U = 0.69, from left to right N = 102 , 103 , 2 × 103 , 5 × 103 , 104 , 2 × 104 . Initially ∆θ = π , hence M0 = 0.
Quasi-stationary states in mean-field dynamics – p.7/35
Quasi-stationary states-II 7
b(N)
6
(b)
5 4 3 2
1
2
3
4
5
logN
Power law increase of the lifetime, exponent 1.7
Quasi-stationary states in mean-field dynamics – p.8/35
HMF Vlasov equation ∂f ∂f dV ∂f +p − =0 ∂t ∂θ dθ ∂p
,
V (θ)[f ] = 1 − Mx [f ] cos(θ) − My [f ] sin(θ) , Z Mx [f ] = f (θ, p, t) cos θdθdp , Z My [f ] = f (θ, p, t) sin θdθdp .
Specific R energy e[f ] = (p2 /2)f (θ, p, t)dθdp + 1/2 − (Mx2 + My2 )/2 and R momentum P [f ] = pf (θ, p, t)dθdp are conserved.
Quasi-stationary states in mean-field dynamics – p.9/35
Vlasov fluid-I
Quasi-stationary states in mean-field dynamics – p.10/35
Vlasov fluid-II
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Comparison of Vlasov with N -body
Quasi-stationary states in mean-field dynamics – p.12/35
Stirring
Quasi-stationary states in mean-field dynamics – p.13/35
Allowed moves YES
NO !!
Quasi-stationary states in mean-field dynamics – p.14/35
Lynden-Bell entropy Lynden-Bell guesses that the initial evolution (“violent relaxation") is characterized by a maximization of a “fermionic" entropy with given constraints (e.g. energy, momentum, ...). ¯ ¯ ¯ ¯ f f f f ¯ s(f ) = − dpdθ ln 1 − . ln + 1 − f0 f0 f0 f0 Z
Quasi-stationary states in mean-field dynamics – p.15/35
Maximal Lynden-Bell entropy states f¯(θ, p) =
f0 f0 f0 f0
x √ β
f0 2 ¯ ¯ eβ(p /2−My [f ] sin θ−Mx [f ] cos θ)+λp+µ
Z
dθe
βM·m
+1
.
” “ βM·m =1 F0 xe
Z ” “ M2 − 1 x βM·m βM·m =e+ dθe F2 xe 3/2 2 2β Z “ ” x βM·m βM·m √ dθ cos θe F0 xe = Mx β Z ” “ x βM·m βM·m = My dθ sin θe F0 xe √ β
M = (Mx , My ), m = (cos θ, sin θ). R F0 (y) = exp(−v 2 /2)/(1 + y exp(−v 2 /2))dv, R 2 F2 (y) = v exp(−v 2 /2)/(1 + y exp(−v 2 /2))dv. f0 = 1/(4∆θ0 ∆p0 )
Quasi-stationary states in mean-field dynamics – p.16/35
Velocity PDF a)
fQSS(p)
0,1
0,01
b) 0,1
-1,5
-1
-0,5
0
0,5
1
1,5
c)
0,01
-1,5
-1
-0,5
0
0,5
1
1,5
d)
0,5 0,4 0,3
0,1
0,2 0,1 0,01
0
-1,5
-1
-0,5
0
0,5
1
1,5
-1,5
-1
-0,5
0
0,5
1
1,5
p |M0 | = sin(∆θ0 )/∆θ0 = 0.3, 0.5, 0.7. U = 0.69. All QSS are homogeneous (M = 0). Non Gaussian velocity distributions
Quasi-stationary states in mean-field dynamics – p.17/35
Non equilibrium phase transition 0.7
U=0.50 0.5
U=0.54
0.4
U=0.58
0.7 0.6 0.5
0.3
0.4
hfp
M(t)
0.6
0.2
U=0.62
0.3 0.2
0.1
0.1
0 0
5
10 15 20 25 30 35 40 45 50 55
t
0 0.5
(a) 0.55
0.6
0.65
0.7
0.75
0.8
U
LEFT: N = 103 RIGHT: First peak height as a function of U for increasing values of N (102 , 103 , 104 , 105 ). The transition line is at U = 7/12 = 0.583....
Quasi-stationary states in mean-field dynamics – p.18/35
Tricritical point-I 0.75 0.605
0.6
0.7
U
0.595
0.65
0.04
0.08
0.12
0.16
ST
1 order ND 2 order Tricritical point
0.6 0
0.2
0.4
M0
0.6
0.8
1
Quasi-stationary states in mean-field dynamics – p.19/35
Tricritical point-II N=100000
U
0.75
0.5 0
M0
1
M_QSS 0
1
Quasi-stationary states in mean-field dynamics – p.20/35
First and second order
0.5
0.5 b)
3
MQSS
0.4 0.3 0.2
a)
0.3 3
0.2 0.1
0.1 0 0.57
0.4
MQSS
N=10 4 N=10 6 N=10 Theory
0.58
0.59
0.6
U
0.61
0.62
0 0.56
N=10 4 N=10 6 N=10 Theory
0.57 0.58 0.59
0.6
0.61 0.62
U
Quasi-stationary states in mean-field dynamics – p.21/35
Metastability
20 15
P( M )
20
(a)
(b)
15
M0=0.08
10
10
5
5
0
0
0.1
0.2
0.3
20 15
0.4
20
5
10
0
0.1
0.2
0.3
0.1
0.2
0.3
0.4
0
0
0.1
0.4 (d)
M0=0.1
30
10
0
0
40
(c)
M0=0.09
0
M0=0.0848
0.2
0.3
0.4
M
U = 0.6 Dashed (solid) lines stand for N = 105 (N = 106 )
Quasi-stationary states in mean-field dynamics – p.22/35
Negative heat capacity 0.38
0.4 ’temp_1000000_0.05’ ’temp_10000_0.05’ ’temp_1000_0.05’
0.36
’temp_1000000_0.3’ ’temp_10000_0.3’ ’temp_10000.3’ 0.38
0.34 0.36 0.32 0.34
0.3 0.28
0.32
0.26
0.3
0.24 0.28 0.22 0.26
0.2 0.18 0.56
0.57
0.58
0.59
0.6
0.61
0.62
0.63
0.24 0.56
0.57
0.58
0.59
0.6
0.61
0.62
0.63
Quasi-stationary states in mean-field dynamics – p.23/35
Free Electron Laser y
z x
Colson-Bonifacio model dθj dz dpj dz dA dz
=
pj
=
−Aeiθj − A∗ e−iθj
=
1 X −iθj iδA + e N j
Quasi-stationary states in mean-field dynamics – p.24/35
Quasi-stationary states
0.6
0.4
0.6
Laser intensity
Laser intensity (arb. units)
0.8
0.2
3 2
0.4
0
0
20
40
1
0.5
60
0
500
80
100
-z
1000
120
1500
140
160
-z
2000
180
200
N = 5000 (curve 1), N = 400 (curve 2), N = 100 (curve 3) On a first stage the system converges to a quasi-stationary state. Later it relaxes to Boltzmann-Gibbs equilibrium on a time O(N ). The quasi-stationary state is a Vlasov equilibrium, sufficiently well described by Lynden-Bell’s Fermi-like distributions.
Quasi-stationary states in mean-field dynamics – p.25/35
Vlasov equation In the N → ∞ limit, the single particle distribution function f (θ, p, t) obeys a Vlasov equation. ∂f ∂f ∂f = −p + 2(Ax cos θ − Ay sin θ) ∂z ∂θ ∂p Z 1 ∂Ax f cos θ dθdp , = −δAy + ∂z 2π Z ∂Ay 1 f sin θ dθdp . = δAx − ∂ z¯ 2π √ with A = Ax + iAy = I exp(−iϕ)
,
Quasi-stationary states in mean-field dynamics – p.26/35
Vlasov equilibria Lynden-Bell entropy maximization SLB (f¯) = −
SLB (ε, σ) =
Z
„ ¯ „ « „ «« f f¯ f¯ f¯ dpdθ ln + 1− ln 1 − . f0 f0 f0 f0
max [SLB (f¯)|H(f¯, Ax , Ay ) = N ε;
f¯,Ax ,Ay
f¯ = f0
Z
dθdpf¯ = 1; P (f¯, Ax , Ay ) = σ].
2 /2+2A sin θ)−λp−µ
e−β(p 1+
2 e−β(p /2+2A sin θ)−λp−µ
.
Non-equilibrium field amplitude A=
q
β A2x + A2y = βδ − λ
Z
dpdθ sin θf¯(θ, p).
Quasi-stationary states in mean-field dynamics – p.27/35
Results 0.7
Intensity
2
Laser Intensity, Bunching
Laser Intensity, Bunching
0.6
Intensity 1.5
1 Threshold
Bunching
0.5
0.5
0.4
Bunching
0.3
0.2
0.1
0
-2
-1.5
-1
-0.5
0
δ
0.5
1
1.5
2
2.5
0
0
0.05
0.1
ε
0.15
0.2
0.25
Quasi-stationary states in mean-field dynamics – p.28/35
More waves dθj dz dpj dz
= pj = −
X h
Fh (Ah eihθj − A∗h e−ihθj )
dAh Fh X −ihθj e = Fh bh = dz N
,
j
Quasi-stationary states in mean-field dynamics – p.29/35
Two waves-I 3
2,5
Z3
2
F3 1,5 1
Z1
0,5
0
0
0,2
0,4
0,6
0,8
1
F1
Quasi-stationary states in mean-field dynamics – p.30/35
Two waves-II 2
|A1|
0,5
2
|A3|
0,4 0,3 0,2 0,1 0
0
100
200
300
400
500
0
100
200
300
400
500
3 2,5
SN
2 1,5 1 0,5
z
Quasi-stationary states in mean-field dynamics – p.31/35
Collective optical field enhancement 106 Rb atoms at T = 1µK can be modelled by the Colson-Bonifacio equations (Zimmermann, Tubingen).
High finesse optical cavity 11111111 00000000 00000000 11111111 00000000 11111111 00000000Rb atoms 11111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 0000000 1111111 00000000 11111111 00000000 11111111 00000000 11111111
PUMP
Backscattering
11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
11111111 00000000 At higher tempetures and adding "molassas" viscosity becomes effective (S. Slama et al, 2007).
Quasi-stationary states in mean-field dynamics – p.32/35
Conclusions Mean-field Hamiltonian systems with many degrees of freedom display interesting statistical and dynamical properties. Non equilibrium quasi-stationary states arise “naturally" from water-bag initial conditions. Their life-time increases with system size. Vlasov (non collisional) equation correctly describes the “initial" dynamics of mean-field Hamiltonians. Lynden-Bell maximum entropy principle provides a theoretical approach to quasi-stationary states. Collective phenomena of wave-particle interactions (free electron laser) are the result of a Lynden-Bell maximum entropy principle.
Quasi-stationary states in mean-field dynamics – p.33/35
References-I Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model, Physica A, 337, 36 (2004). J. Barré, T. Dauxois, G. De Ninno, D. Fanelli, S. Ruffo:Statistical theory of high-gain free-electron laser saturation, Phys. Rev. E, Rapid Comm., 69, 045501 (R) (2004). A. Antoniazzi, G. De Ninno, A. Guarino, D. Fanelli and S. Ruffo, Wave-particle interaction: from plasma physics to the free electron laser, J. Phys. CS, 7, 143 (2005). A. Antoniazzi, Y. Elskens, D. Fanelli, S. Ruffo, Statistical mechanics and Vlasov equation allows for a simplified Hamiltonian description of single-pass free electron laser saturated dynamics, European Physical Journal B, 50, 603 (2006). A. Campa, A. Giansanti, D. Mukamel and S. Ruffo, Dynamics and thermodynamics of rotators interacting with both long and short range couplings , Physica A, 365, 177 (2006) A. Antoniazzi, D. Fanelli, J. Barré, P.-H. Chavanis, T. Dauxois and S. Ruffo, A maximum entropy principle explains quasi-stationary states in systems with long range interactions: the exemple of the HMF model,
Phys. Rev. E, 75, 011112 (2007).
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References-II A. Antoniazzi, F. Califano, D. Fanelli, S. Ruffo, Exploring the thermodynamic limit of Hamiltonian models, convergence to the Vlasov equation, Phys. Rev. Lett, 98 150602 (2007). A. Antoniazzi, D. Fanelli, S. Ruffo and Y.Y. Yamaguchi, Non equilibrium tricritical point in a system with long-range interactions, Phys. Rev. Lett., 99, 040601 (2007). A. Antoniazzi, R. S. Johal, D. Fanelli and S. Ruffo, On the origin of quasi-stationary states in models of wave-particle interaction, Comm. Nonlin. Sci. Num. Sim., 13, 2 (2008).
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