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Quasi-stationary-states inHamiltonian mean-field dynamics
STEFANO RUFFO
Dipartimento di Energetica “S. Stecco”, Universita di Firenze, and INFN,
Italy
Statistical Mechanics Day III, The Weizmann
Institute of Science, June 16, 2010
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.1/35
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Plan
Hamiltonian Mean Field (HMF) model
Quasi-stationary-states (QSS)
Klimontovich and Vlasov equations
Lynden-Bell theory
Non equilibrium phase transition
Mean-field equilibria
Application to the free electron laser
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HMF model
H =N
∑
i=1
p2
i
2+
1
2N
N∑
i,j=1
(1 − cos(θi − θj))
Magnetization M = limN→∞
„
PNi=1
cos θi
N,
PNi=1
sin θi
N
«
= (Mx, My)
Energy U = limN→∞
HN
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Equations of motion
θi =∂H
∂pi= pi
pi = −∂H
∂θi= − 1
N
N∑
j=1
sin(θi − θj)
θi = pi
pi = −M sin(θi − φ)
tan(φ) = My
Mx
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Equilibrium phase transition
0.0 0.4 0.8 1.2U
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9U
0.0
0.1
0.2
0.3
0.4
0.5
0.6
T
Theory (c.e.) N=100N=1000N=5000N=20000 N=20000 n.e.
Latora, Rapisarda & Ruffo − Lyapunov instability and finite size...
Fig. 1 revised
Uc = 3/4 = 0.75, T = (∂S/∂U)−1 = limN→∞
∑
i p2i /N
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Waterbag
p
������������������������������������������������
������������������������������������������������
∆ θ
∆p
θ
M0 =sin∆θ
∆θ
U = limN→∞
H
N=
(∆p)2
6+
1 − (M0)2
2
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.6/35
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Equilibrium
-3.14 3.14theta-1.2
1.2
p
N=10000, U=0.63, a=1.0, t=100, sample=1000
0 0.12
PDF
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Quasi-stationary states
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
−1 0 1 2 3 4 5 6 7 8
M(t)
log10t
(a)
U = 0.69, from left to rightN = 102, 103, 2 × 103, 5 × 103, 104, 2 × 104.Initially ∆θ = π, hence M0 = 0.
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.8/35
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Power law
2
3
4
5
6
7
1 2 3 4 5
b(N
)
logN
(b)
Power law increase of the lifetime, exponent 1.7
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Separation of time scales
Initial Condition
Vlasov’s Equilibrium
Boltzmann’s Equilibrium
τv = O(1)
τc = N δ
Violentrelaxation
Collisionalrelaxation
?
?
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Klimontovich equation
H =
NX
i=1
p2i
2+ U(θi)
U(θ1, .., θN ) =1
N
NX
i<j
V (θi − θj)
Discrete, one-particle, time dependent density function
fd(θ, p, t) =1
N
NX
i=1
δ(θ − θi(t))δ(p − pi(t))
Klimontovich equation∂fd
∂t+ p
∂fd
∂θ− ∂v
∂θ
∂fd
∂p= 0
where
v(θ, t) =
Z
dθ′dp′V (θ − θ′)fd(θ′, p′, t)
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From Klimontovich to Vlasov
fd = 〈fd(θ, p, t)〉 +1√N
δf(θ, p, t) = f(θ, p, t) +1√N
δf(θ, p, t)
where the average 〈·〉 is taken over initial conditions.
v(θ, t) = 〈v〉(θ, t) +1√N
δv(θ, t)
where
〈v〉(θ, t) =
Z
dθ′dp′V (θ − θ′)f(θ′, p′, t)
Using Klimontovic equation, one gets
∂f
∂t+ p
∂f
∂θ− ∂〈v〉
∂θ
∂f
∂p=
1
N〈∂δv
∂θ
∂δf
∂p〉
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HMF Vlasov equation
∂f
∂t+ p
∂f
∂θ− dV
dθ
∂f
∂p= 0 ,
V (θ)[f ] = 1 − Mx[f ] cos(θ) − My[f ] sin(θ) ,
Mx[f ] =
∫
f(θ, p, t) cos θdθdp ,
My[f ] =
∫
f(θ, p, t) sin θdθdp .
Specific energye[f ] =
∫
(p2/2)f(θ, p, t)dθdp + 1/2 − (M2x + M2
y )/2 andmomentum P [f ] =
∫
pf(θ, p, t)dθdp are conserved.
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Vlasov fluid-I
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.14/35
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Vlasov fluid-II
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Stirring
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.16/35
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Allowed moves
YESNO !!
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.17/35
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Coarse graining for two-level distr.
MicrocellMacrocell
ν microcells of volume ω in a macrocell. A macroscopic configuration has ni microcells
occupied with level f0 in the i-th macrocell (the remaining are occupied with level 0). The
distribution is coarse-grained on a macrocell, f(θ, p). The total number of occupied
microcells is N , such that the conserved total mass is mass =R
f(θ, p)dθdp = Nωf0
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.18/35
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Lynden-Bell entropy
SLB(f) = − 1
ω
∫
dpdθ
[
f
f0
lnf
f0
+
(
1 − f
f0
)
ln
(
1 − f
f0
)]
.
f(θ, p) =2
∑
i=1
ρ(θ, p, ηi) ηi ,
η1 = 0, η2 = f0
W ({ni}) =N !
∏
i ni!×
∏
i
ν!
(ν − ni)!.
ρi(f0) = ni/ν = fi/f0
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Maximal Lynden-Bell entropy states
f(θ, p) =f0
eβ(p2/2−My [f ] sin θ−Mx[f ] cos θ)+λp+µ + 1.
f0x√β
Z
dθeβM·mF0
“
xeβM·m
”
= 1
f0x
2β3/2
Z
dθeβM·mF2
“
xeβM·m
”
= U +M2 − 1
2
f0x√β
Z
dθ cos θeβM·mF0
“
xeβM·m
”
= Mx
f0x√β
Z
dθ sin θeβM·mF0
“
xeβM·m
”
= My
M = (Mx, My), m = (cos θ, sin θ).F0(y) =
R
exp(−v2/2)/(1 + y exp(−v2/2))dv,F2(y) =
R
v2 exp(−v2/2)/(1 + y exp(−v2/2))dv.f0 = 1/(4∆θ0∆p0)
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Velocity PDF
-1,5 -1 -0,5 0 0,5 1 1,5
f QS
S(p
)0,01
0,1
a)
-1,5 -1 -0,5 0 0,5 1 1,50,01
0,1
b)
-1,5 -1 -0,5 0 0,5 1 1,50,01
0,1
c)
-1,5 -1 -0,5 0 0,5 1 1,50
0,1
0,2
0,3
0,4
0,5 d)
p
|M0| = sin(∆θ0)/∆θ0 = 0.3, 0.5, 0.7. U = 0.69. All QSS arehomogeneous (M = 0).
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.21/35
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Non equilibrium phase transition
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 5 10 15 20 25 30 35 40 45 50 55
M(t
)
t
U=0.50
U=0.54
U=0.58
U=0.62
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.5 0.55 0.6 0.65 0.7 0.75 0.8
h fp
U
(a)
LEFT: N = 103
RIGHT: First peak height as a function of U for increasingvalues of N (102, 103, 104, 105). The transition line is atU = 7/12 = 0.583....
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.22/35
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Tricritical point
0 0.2 0.4 0.6 0.8 1M
0
0.6
0.65
0.7
0.75
U
1ST
order2
ND order
Tricritical point
0.04 0.08 0.12 0.16
0.595
0.6
0.605
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.23/35
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Magnetization plot
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
0.55
0.6
0.65
0.7
0.75
M0
U
0.1
0.2
0.3
0.4
0.5
0.6
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.24/35
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Negative heat capacity
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63
’temp_1000000_0.05’’temp_10000_0.05’
’temp_1000_0.05’
0.24
0.26
0.28
0.3
0.32
0.34
0.36
0.38
0.4
0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63
’temp_1000000_0.3’’temp_10000_0.3’
’temp_10000.3’
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.25/35
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Analytical results
0 0.2 0.4 0.6 0.8 1U
0
0.2
0.4
0.6
0.8
1
T
Homogeneous branch
Uc=3/4
Uc=5/8
Polytropicn
c=1
Isothermaln=∞q=1
qc=3
F. Staniscia et al. arXiv:1003.0631, P. H. Chavanis and A. Campa arXiv:1001.2109v1
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.26/35
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Mean-field equilibria
ǫ =p2
2− H cos θ
H constant.
P (ǫ; ∆θ, ∆p) =1
4∆θ∆p
Z +∆θ
−∆θdθ
Z +∆p
−∆pdp δ(
p2
2− H cos θ − ǫ)
Then
PH(θ, p;∆θ, ∆p) =1
4∆θ∆p
P (ǫ, ∆p)
Q(ǫ, ∆p)
with Q(ǫ, ∆p) a normalization.Self consistency
m(∆θ, ∆p) =
Z +π
−πdθdp cos θPm(θ, p;∆θ, ∆p)
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.27/35
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Numerical verification
10-1 100 101�
p
10-3
10-2
10-1
100m
� �
=1
Rotators th.Rotators sim.HMF sim.
10-1 100 101�
p
10-3
10-2
10-1
100
m
� �=2
Rotators th.Rotators sim.HMF sim.
P. de Buyl, D. Mukamel and SR, in progress
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.28/35
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Free Electron Laser
y
x
z
Colson-Bonifacio model
dθj
dz= pj
dpj
dz= −Aeiθj − A
∗e−iθj
dA
dz= iδA +
1
N
X
j
e−iθj
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.29/35
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Quasi-stationary states
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
z
0 500 1000 1500 20000.4
0.5
0.6
-
Lase
r int
ensi
ty (a
rb. u
nits
)
-zLa
ser i
nten
sity
1
23
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 Hamiltonian mean-field dynamics – p.30/35
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Vlasov equation
In the N → ∞ limit, the single particle distribution functionf(θ, p, t) obeys a Vlasov equation.
∂f
∂z= −p
∂f
∂θ+ 2(Ax cos θ − Ay sin θ)
∂f
∂p,
∂Ax
∂z= −δAy +
1
2π
∫
f cos θ dθdp ,
∂Ay
∂z= δAx − 1
2π
∫
f sin θ dθdp .
with A = Ax + iAy =√
I exp(−iϕ)
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.31/35
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Vlasov equilibria
Lynden-Bell entropy maximization
SLB(f) = −Z
dpdθ
„
f
f0ln
f
f0+
„
1 − f
f0
«
ln
„
1 − f
f0
««
.
SLB(ε, σ) = maxf ,Ax,Ay
[SLB(f)|H(f , Ax, Ay) = Nε;
Z
dθdpf = 1; P (f , Ax, Ay) = σ].
f = f0e−β(p2/2+2A sin θ)−λp−µ
1 + e−β(p2/2+2A sin θ)−λp−µ.
Non-equilibrium field amplitude
A =q
A2x + A2
y =β
βδ − λ
Z
dpdθ sin θf(θ, p).
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.32/35
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Results
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50
0.5
1
1.5
2
δ
La
ser
Inte
nsi
ty,
Bu
nch
ing
Intensity
Bunching Threshold
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bunching
Intensity
La
ser
Inte
nsi
ty,
Bu
nch
ing
ε
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.33/35
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Conclusions
Mean-field Hamiltonian systems with many degrees of freedom display interestingstatistical and dynamical properties.
Non equilibrium quasi-stationary states arise “naturally" from water-bag initialconditions. Their life-time increases with a power of system size.
Vlasov (non collisional) equation correctly describes the “initial" dynamics.
Lynden-Bell maximum entropy principle provides a theoretical approach toquasi-stationary states.
The results are derived for mean-field systems, but some of them are expected to bevalid also for decaying interactions.
Quasi-stationary-states in Hamiltonian mean-field dynamics – p.34/35
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References
M. Antoni and S. Ruffo, Clustering and relaxation in long range Hamiltonian dynamics, Phys. Rev.E, 52, 2361 (1995).
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).
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).
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. Campa, T. Dauxois and S. Ruffo : Statistical mechanics and dynamics of solvable models with
long-range interactions, Physics Reports, 480, 57 (2009).
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