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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

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


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


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


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


Waterbag

p

��

��

∆ θ

∆p

θ

M0 =sin∆θ

∆θ

U = limN→∞

H

N=

(∆p)2

6+

1 − (M0)2

2


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


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.


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


Separation of time scales

Initial Condition

Vlasov’s Equilibrium

Boltzmann’s Equilibrium

τv = O(1)

τc = N δ

Violentrelaxation

Collisionalrelaxation

?

?


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)


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〉


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.


Vlasov fluid-I


Vlasov fluid-II


Stirring


Allowed moves

YESNO !!


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


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


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)


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).


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....


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


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


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’


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


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)


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


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

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.


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)


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).


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


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.


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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