Dynamics and Thermodynamics of Systems with Long Range Interactions Thierry DAUXOIS ´ Ecole Normale Sup´erieure de Lyon (France)

In collaboration with

J. Barr´e, F. Bouchet, D. Mukamel and S. Ruffo. also A. Antoniazi, P. Chavanis, D. Fanelli, G. De Ninno, T. Tatekawa, Yamaguchi. 1

PLAN I. Introduction II. Microcanonical Statistical Mechanics via Large deviations methods III. Ergodicity breaking IV. Conclusion and Perspectives

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Why long-range forces are important ? • Methods to describe N particles interacting via a gravitational potential in 1/r strongly depends on N . – N = 2 (Kepler) ⇒ Exact solution – N = 3 − 103

⇒ Numerical Solution

– N = 104 − 1011 ⇒ Statistical Solution • Statistical behaviors are totally different from usual systems (neutral gaz, atomic lattices, ...) because of the long-range interaction: ensembles inequivalence, CV < 0, . . . ⇒ It is necessary to come back to the basis. • Interesting dynamical properties ⇒ stability or metastability ? • Broad spectrum of applications 3

Extensivity 6= Additivity Curie-Weiss Hamiltonian

!2 ÃN 1 X H=− Si N i=1

Extensivity: one variable is extensive if proportional to the number of elements N whereas intensive variables are constant.

Additivity: E+ = E− =

− N1 − N1

and E=

− N1

¡

+ N2 − N2

¡

¡N 2

−

¢2 ¢2 N 2

= − N4 = − N4

¢2

⇒ E+ + E− 6= E =0

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Definition of a LONG-RANGE system Potential energy of one single particle located in the center of an homogeneous sphere interacting with V (r) ∼ 1/rα

Z

R

1 4πr dr ρ α = 4πρ r

Z

R

2

U= ε

r ε

2−α

dr ∝

£

¤

R r3−α ε

∼ R3−α

• α > 3: Contribution of the surface negligible • α ≤ 3: Divergence of the integral ⇒ Important surface effects Conclusion: In dimension d, total energy will be additive when potential energy will behave as 1 α V (r) ∼ α with >1 r d

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Applications: V (r) ∼

1 rα

with α < d

Large systems: α

α/d

Gravity

1

1/3

Long-range

Coulomb

1

1/3

Long-range with Debye-screening

Dipole

3

1

Limiting case

2D Flows

0

0

Logarithmic Interactions

Fracture

2

1

Stress Field around the tip

Small systems:

range ∼ size

Clusters Nuclear Physics BE Condensation Coupling Wave-Matter Dauxois, Ruffo, Arimondo, Wilkens, Lecture Notes in Physics 602 (2002).

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Construction of canonical ensemble Probability for system 1 to get an energy E1 is ∝

Ω1 (E1 ) Ω2 (E2 ) dE1

=

Ω1 (E1 ) Ω2 (E − E1 ) dE1 Ω (E ) eS2 (E − E1 ) dE

=

1

1

(S2 (E) − E1

=

Ω1 (E1 ) e

∝

Ω1 (E1 ) e−βE1 dE1

|

{z

Ω(E) number of microstates ←Additivity 1

∂S2 + ...) ∂E dE1

if

E1 ¿ E

}

canonical distribution

⇒ non-additive systems will present a strange behavior in contact with a thermal bath.

⇒ The microcanonical ensemble is well defined but difficult to use whereas canonical ensemble is not defined ! 7

Ensemble Inequivalence Entropy always increasing but, possibly, with a convex intruder. s

ε2

ε1

ε

• Coexistence region: E0 = xE1 + (1 − x)E2 ⇒ S0 = xS1 + (1 − x)S2 • When the system additive (short-range interactions), phase separation leads to a state with a larger entropy. ⇒ S concave and the microcanonical specific heat non-negative. 8

Ensemble Inequivalence s

ε2

ε1

ε

• When the system is non-additive (Long-range interactions), the entropy follow the homogeneous system curve. • Possibility of non concave entropy, negative specific heat, negative susceptibility, temperature jumps • while in the canonical ensemble, T 2 Cv = hE 2 i − hEi2 ≥ 0 9

Importance of the being able to compute directly the Entropy

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PLAN I. Introduction II. Microcanonical Statistical Mechanics via Large deviations methods III. Ergodicity breaking IV. Conclusion and Perspectives

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Microcanonical Statistical Mechanics: Large Deviations What is the PDF of the average MN = variables Xi ?

1 N

P i

Xi of the random

Cram´er’s theorem ensures that P (MN = x) ∼ exp (−N I(x)) where I(x) = sup (λ · x − ln Ψ(λ)) λ∈Rd

and Ψ(λ) = heλ·X i is the generating function R Hint to proof in d = 1: P (MN = x) =

dµ(X1 )...dµ(XN ) δ(MN − x)

Laplace representation of δ + saddle point integration.

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J. Stat. Phys. (2005).

Entropy and Free energy Step 1: Express the Hamiltonian in terms of global variables γ e N (γ(ωN )) + RN (ωN ) HN (ωN ) = H ωN a phase space configuration Step 2: Compute the entropy functional in terms of the global variables using Cram`er’s theorem 1 ln ΩN (γ) N →∞ N

s(γ) = lim

with ΩN (γ) the number of microscopic configurations with γ fixed. Step 3: Obtain ³ the microcanonical´ and canonical variational pbs e N (γ)/N p = E where p is chosen to make E intensive S(E) = sup s(γ) | H γ

F (β) = inf (β0 h(γ) − s(γ)) γ

where β0 = βN

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p−1

and h(γ) =

e

p lim HN (γ)/N . N →∞

Infinite range 3 states Potts model HN

N J X =− δSi ,Sj 2 i,j=1

with Si = a, b, c.

Step 1: Local random variables: Xk = (δSk ,a , δSk ,b , δSk ,c ) Appropriate global variable γ = (na , nb , nc ) where ni = is the fraction of spins in state i. One gets HN

JN 2 2 eN (na + n2b + n2c ) = H =− 2

which leads to p = 2 and RN (ωN ) = 0. 14

1 N

P

k δSk ,i

Step 2: One therefore gets the generating function 1 X ¡ λa δS,a +λb δS,b +λc δS,c ¢ e Ψ(λa , λb , λc ) = 3 S=a,b,c

¢ 1 ¡ λa λb λc = e +e +e 3 and the large deviation functional I(γ) =

sup (λa na + λb nb + λc nc − ln Ψ(λa , λb , λc ))

λa ,λb ,λc

= na ln na + nb ln nb + (1 − na − nb ) ln(1 − na − nb ) + ln 3 The thermodynamic entropy density is finally given by s(γ) = −I(γ) + ln N . One thus recovers the result of the combinatorial approach. 15

Step 3: Microcanonical entropy ´ ³ e N (γ)/N p = E S(E) = sup s(γ) | H γ

¯ J¡ 2 ¢ 2 2 ¯ = sup (−na ln na −nb ln nb −nc ln nc − na + nb + nc = E) 2 na ,nb

Canonical free energy F (β) = inf (β0 h(γ) − s(γ)) γ

=

inf

na ,nb ,nc

¡

¢ βJ ¡ 2 2 2 na + nb + nc na ln na +nb ln nb +nc ln nc − 2 ¯ ¢ ¯n a + n b + n c = 1 16

Caloric Curve

• Negative specific heat region while first order Phase Transition at β t . • Maxwell’s construction • Illustration that S(E) is not always the Legendre transform of F (β) 17

Inf/Sup vs Sup/Inf

(Leyvraz & Ruffo 2002, Touchette 2007)

Ω(E, m) being the number of microstates corresponding to a macrostate characterized by E and m, one defines s(E, m) = kB ln Ω(E, m)

and ϕ(β, m) = inf [βE − s(E, m)] E

Microcanonical stat. mech.: S(E) = sup [s(E, m)] m

Canonical stat. mech.: ϕ(β) = inf [βE − S(E)] E

(Legendre Transform)

Let us consider the Legendre-Fenchel transform of ϕ(β) h i S ∗∗ (E) = inf [βE − ϕ(β)] = inf βE − inf ϕ(β, m) = inf sup [βE − ϕ(β, m)] β

while

m

β

β

m

S(E) = sup [S(E, m)] = supinf [βE − ϕ(β, m)] m

m

β

One recover the well known result: S ∗∗ corresponds to the concave envelope of the original function S, ⇒ both functions coincide only if the entropy S is concave! 18

Large deviation method applied to

Barr´ e et al, J. Stat. Phys. 119, 677 (2005).

• discrete variables and infinite range N X J – Ising model: HN = N (1 − Si Sj ) i,j=1 J – 3 states Potts model: HN = − 2N

N X

δSi ,Sj

i,j=1

• continuous variables and infinite range N X C X p2i + cos(θi − θj ) – Hamiltonian Mean-Field HN = 2 2N i,j i=1 – Colson-Bonifacio’s model for Free Electron Laser • slowly decreasing interactions. – α-Ising model: HN =

J

N 1−α

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N X 1 − Si Sj α |i − j| i,j=1

PLAN I. Introduction II. Microcanonical Statistical Mechanics via Large deviations methods III. Ergodicity breaking IV. Conclusion and Perspectives

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Connexity of the attainable parameter space Consider two magnetic subsystems with the same potential energy but with two different magnetization values m1 and m2 .

It is possible to get any magnetization m by just combining the two subsystems, such that m = λm1 + (1 − λ)m2 , while the potential energy is kept constant. Systems with short-range interactions are defined on a convex region of their extensive parameter space ⇒ Additivity required: What about long-range interactions syst. ? 21

Model: Hamiltonian Mean-Field (HMF) H=

N X p2 i=1

N X 1 i ± cos(θi − θj ) 2 2N i,j=1

Simplification of: • 1D charged sheets model, 1D gravitation • Hamiltonian for plasma-wave or Free Electron Laser

Simple model, Mean Field, with appropriate characteristics

“+ sign ” antiferromagnetic while “– sign” ferromagnetic 22

See Stefano Ruffo’s talk this afternoon for a careful presentation of its • dynamical properties • applications

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Generalized version

1 X iθn e = m eiφ , let’s consider Introducing M = N n · 2 ¸ N 2 4 X pn m m H= +N −K . 2 2 4 n=1 • K = 0: Antiferromagnetic HMF • Low K: the antiferromagnetic coupling is dominant which favors the non-magnetized state. • Large K: the Ferromagnetic coupling is dominant which favors the magnetized state. It exists a range of the parameter K for which the model exhibits a first order phase transition between a paramagnetic phase at high energies and a ferromagnetic phase at low energies. In both phases, the accessible magnetization interval exhibits gaps. 24

MicrocanonicalStatistical Mechanics

X iθ 1 − → n  M = e   N   n   1 X 2 u= pn Step 1: global variables are N  n  X  1   pn v =   N

magnetization temperature total momentum

n

eN γ = (u, v, Mx , My ) ⇒ H

m2 m4 =u+ −K 2 4

Step 2: Generating function, Large deviation functional and finally the entropy

where sK (E, m) = 1 ln 2

s (E, m) = sK (E, m) + sconf (m) , ¡ ¢ ¡ I ¢−1 m2 m4 E−

2

+K

4

and sconf (m) = −

1 I0

(m) + ln (I0 (φ(m)))

Step 3: Microcanonical entropy S(E) = supm [s(E, m)] . 25

Evolution of the entropy when the energy E decreases from top to bottom.

The picture demonstrates that gaps in the accessible states develop as the energy is lowered. 26

Origin of the gaps m2 m4 non-negative kinetic energy ⇒ E ≥ V (m) = −K . 2 4

Local dynamics cannot make the system cross from segment to another ⇒ Ergodicity is thus broken even for a finite system Borgonovi et al (2004); Mukamel et al (2005).

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Dynamical consequences

⇒ the system can exhibit a stable ferromagnetic phase within the paramagnetic region, and conversely a disordered phase within the magnetically ordered region. 28

Conclusion • Equilibrium Stat. Mech. Wealth of unexpected results: – Ensembles inequivalence – Negative specific heat – Ergodicity breaking Large deviation techniques applied to systems with LR int. Barr´e, Bouchet, TD, Ruffo, J. Stat. Physics, 119, 677 (2005). ⇒

⇒ Phase space gaps and ergodicity breaking for LR int. syst. Bouchet, TD, Mukamel, Ruffo, arXiv:0711.0268 (2007).

• Out-of-Equilibrium Stat. Mech. – Quasi-stationary states – Algebraic relaxations – Non-gaussian distributions

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(see Stefano Ruffo’s talk)

The Dream Team

Julien ´ BARRE

Freddy BOUCHET

(JAD, Nice)

(INLN, Nice)

David

Stefano

MUKAMEL

RUFFO

(Weizmann)

(Firenze)

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Les Houches Summer School, August 2008 • Statistical physics – Equilibrium: D. Mukamel (Israel) – Non-Equilibrium: J. Kurchan (France) – Transport: D. Dubin (USA) • Mathematical aspects – Equilibrium tools (Large deviations): R. S. Ellis (USA) – Kinetic theory: F. Castella (France) • Applications – Self-gravitating systems: T. Padmanabhan (India) – Fluid mechanics (2D, stratified or rotating) B. Turkington (USA) – Wave-particles interactions: D. Escande (France) – Dipolar effects in condensed matter: S. Bramwell (UK) – Synchronization: A. Pikovsky (Germany) Web site: http://perso.ens-lyon.fr/thierry.dauxois/LesHouchesSummerSchool2008.htm

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