Adventures of a Long-Range Walker Thierry DAUXOIS CNRS & ENS Lyon 1
Stefano Ruffo Adventures of a long-range walker, born 13th May 1954 60th birthday 2
Studying Links between Statistical Mechanics and Nonlinear Dynamics Fermi-Pasta-Ulam -Tsingou Problem N p 2 2 + 1 2( x i − x i +1 ) 2 + β � i 12( x i − x i +1 ) 4 H = i =1 Classical simplification of 1D Heat conduction Questions: • Existence of thermodynamic limit for statistical properties of a dynamical system? • Equipartition threshold in nonlinear Hamiltonian systems • Role of localized excitations in these systems 3
Article 1 4
Distribution of characteristic Lyapunov exponents in the thermodynamic limit 5
Article 2 6
Model: Hamiltonian Mean-Field (HMF) N N p 2 2 − 1 � � i H = cos( θ i − θ j ) 2 N i =1 i,j =1 Simplification of: • 1D charged sheets model, 1D gravitation • Hamiltonian for plasma-wave, or Free Electron Laser Simple model, Mean Field, Introducing m = 1 � e iθ n , one obtains H = K − N 2 | m | 2 N n 7
Caloric curve 0.6 Equilibrium N=500 QSS T 0.5 0.4 0.4 0.5 0.6 0.7 0.8 0.9 U Solid line: Canonical results at equilibrium Circles: Microcanonical numerical simulations Ensemble Inequivalence ? 8
At Equilibrium 9
Article 3 10
Ensemble inequivalence: BEG model � N � 2 N � � i − J S 2 H = ∆ S i with S i = ± 1 , 0 2 N i =1 i =1 simple model, mean-field, with phase transition, on a lattice. Ferromagnetic states: S i = 1 , ∀ i , or S i = − 1 , ∀ i ⇒ E F = (∆ − J/ 2) N Paramagnetic states: S i = 0 , ∀ i ⇒ E P = 0 ∆ defines the energy difference between ferro. and para states. Canonical ensemble: minimization F = E − TS at T = 0 → minimization of E Paramagnetic state is the most favorable if E F > E P ⇒ ∆ > J/ 2, Phase transition (PT) at ∆ = J/ 2, which is first order since there is a sudden jump of magnetization from ferro. to para. state. 11
Elementary features of the phase diagram � N � 2 N � i − J � S 2 H = ∆ S i 2 N i =1 i =1 T ✻ 1 st order PT ∆ ✲ 0 ✻ J Ferromagnetic state Paramagnetic state 2 12
Elementary features of the phase diagram � N � 2 N � i − J � S 2 H = ∆ S i 2 N i =1 i =1 T ✻ 1 st order PT ∆ ✲ 0 ✻ J Ferromagnetic state Paramagnetic state 2 For vanishingly small ∆, one recovers the Curie-Weiss Hamiltonian. 13
Curie-Weiss Hamiltonian � N � 2 H = − J � S i 2 N i =1 Extensivity : For a given intensive magnetization m = � i S i /N , if one doubles the number of spins the energy doubles. Additivity : � 2 = − JN � J + N E + = − 2( N/ 2) 2 4 � 2 = − JN � J − N E − = − 2( N/ 2) 2 4 and ⇒ E + + E − � = E � N � 2 = 0 E = − J 2 − N 2 N 2 This model is extensive but non additive . 14
Curie-Weiss Hamiltonian Such a system has a second order phase transition when T c = 2 J/ 3 T ✻ ✲ 2 nd order PT 2 J/ 3 1 st order PT ∆ ✲ 0 ✻ J Ferromagnetic state Paramagnetic state 2 PT of different orders on the T and ∆ axis, one expects a transition line separating the low T ferro phase from the high T para phase. 15
Ensemble inequivalence: BEG model � N � 2 N � i − J � S 2 H = ∆ S i with S i = ± 1 , 0 2 N i =1 i =1 simple model, mean-field, with phase transition, on a lattice. • Microcanonical: N + + N − + N 0 = N N ! Ω ( N + , N − , N 0 ) = ⇒ S = k B ln Ω N + ! N − ! N 0 ! � 2 m 2 � m = N + − N − q = N + + N − ∆ q − J ⇒ E = N and N N Equilibrium state: maximization of S ( E, m ) with respect to m . � Ω( q, m ) e − βE ( q,m ) • Canonical: Z ( β, m ) = q Equilibrium state: minimization of F ( β, m ) with respect to m . Barr´ e, Mukamel, Ruffo, Phys. Rev. Lett. 87 , 030601 (2001). 16
Caloric Curve Branches with negative specific heat correspond to local maxima of F ( β, m ), that the constraint of constant energy stabilize in the microcanonical ensemble. 17
Inequivalence of ensemble Landau Theory of Phase Transition Microcanonical ensemble (Power serie expansion of S ) Tricritical point is ∆ c = 0 . 4624 ... and β c = 3 . 0272. Canonical ensemble (Power serie expansion of F ) Tricritical point is ∆ c = ln 4 / 3=0.4621... and β c = 3. • Both points although very close do not coincide. The microcanonical critical line extends beyond the canonical one. • This feature which is a clear indication of ensemble inequivalence was first found in the BEG model (Barr´ e, Mukamel, Ruffo 2001) and later confirmed for gravitational models (Chavanis 2002) • The non coincidence of microcanonical and canonical tricritical points is a generic feature as proven by Bouchet and Barr´ e (2005) 18
A wide range of models Model Variable Ensemble Negative Ergodicity Comput. Inequivalence cv Breaking Entropy BEG Discrete Y Y Y Y 3 states Potts Discrete Y Y N Y Ising L+S Discrete Y Y Y Y N ∗ α -Ising Discrete Y N Y HMF Continuous N N N Y XY L+S Continuous Y Y Y Y N ∗ α -HMF Continuous N N N Generalized XY Continuous Y Y Y Y Mean-Field φ 4 N ∗ Continuous Y N Y Colson-Bonifacio Continuous N N N Y Point vortex Continuous Y Y Y Y Quasi-geostrophic Continuous Y Y Y Y SGR Continuous Y Y Y Y Stefano was involved in all related studies of these models. 19
A wide range of models Model Variable Ensemble Negative Ergodicity Comput. Inequivalence cv Breaking Entropy BEG Discrete Y Y Y Y 3 states Potts Discrete Y Y N Y Ising L+S Discrete Y Y Y Y N ∗ α -Ising Discrete Y N Y HMF Continuous N N N Y XY L+S Continuous Y Y Y Y N ∗ α -HMF Continuous N N N Generalized XY Continuous Y Y Y Y Mean-Field φ 4 N ∗ Continuous Y N Y Colson-Bonifacio Continuous N N N Y Point vortex Continuous Y Y Y Y Quasi-geostrophic Continuous Y Y Y Y SGR Continuous Y Y Y Y Stefano: chairman of the HMF club. 20
A wide range of models Model Variable Ensemble Negative Ergodicity Comput. Inequivalence cv Breaking Entropy BEG Discrete Y Y Y Y 3 states Potts Discrete Y Y N Y Ising L+S Discrete Y Y Y Y N ∗ α -Ising Discrete Y N Y HMF Continuous No N N Y XY L+S Continuous Y Y Y Y N ∗ α -HMF Continuous N N N Generalized XY Continuous Y Y Y Y Mean-Field φ 4 N ∗ Continuous Y N Y Colson-Bonifacio Continuous N N N Y Point vortex Continuous Y Y Y Y Quasi-geostrophic Continuous Y Y Y Y SGR Continuous Y Y Y Y No ensemble inequivalence for the HMF model? 21
Caloric curve 0.6 Equilibrium N=500 QSS T Antoni,Ruffo 0.5 Phys Rev. E 1995 0.4 0.4 0.5 0.6 0.7 0.8 0.9 U Solid line: Microcanonical and Canonical results at equilibrium Circles: Microcanonical numerical simulations Origin of the paradox ? 22
Dynamics matters 23
Numerical Simulations Evolution of the order parameter for different N -values 0 . 35 ← Boltzmann Equilibrium 0 . 3 0 . 25 0 . 2 M ( t ) Non trivial scaling law 0 . 15 t QSS ∼ N 1 . 7 0 . 1 0 . 05 ← Quasi-stationary state 0 −1 0 1 2 3 4 5 6 7 8 log 10 t t →∞ before lim lim N →∞ � = lim N →∞ before lim t →∞ 24
Questions to be addressed • Can we explain theoretically these numerical facts ? – Dynamical ensemble inequivalence – Order of limits – Algebraic Relaxation • Is usual statistical mechanics sufficient ? 25
Kinetic Theory For LRI, the single particle time-dependent density function: N f d ( θ, p, t ) = 1 � δ ( θ − Θ j ( t )) δ ( p − P j ( t )) , N j =1 θ, p : Eulerian coordinates of the phase space and Θ j , P j : Lagrangian coordinates of the N -particles ∂f d ∂t + p∂f d ∂θ − ∂v ∂f d ∂p = 0 . Klimontovich Eq. ∂θ � d θ ′ d p ′ V ( θ − θ ′ ) f d ( θ ′ , p ′ , t ) , where v ( θ, t ) = N • Derivation is exact , even for a finite number of particles N . • This equation contains the information about the orbit of every single particle which is far more than necessary but is a useful starting point for approximations. 26
Vlasov equation Consider a large number of initial conditions, close to the same macroscopic state. + 1 √ f d ( θ, p, t ) = � f d ( θ, p, t ) � δf ( θ, p, t ) . N � �� � f 0 ( θ,p,t ) � ∂δv � ∂f 0 ∂t + p∂f 0 ∂θ − ∂ � v � ∂f 0 1 ∂δf = . ∂θ ∂p N ∂θ ∂p • For short-range interactions, the r.h.s. leads to the collision term of the Boltzmann equation, while the third term is negligible. • For long-range interactions, the r.h.s is of order 1 /N (finite N effects), while the third term is the leading term ( collective effects). 2 N ODE are thus replaced by only 1 PDE. 27
Next Order: Lenard-Balescu Equation Restricting to homogeneous f 0 , a stable stationary solution of the Vlasov equation, we get � ∂δv � ∂f 0 ∂t = 1 ∂δf N ∂θ ∂p At the level 1 /N , the r.h.s can be determined using solutions for δv and δf of the collisionless dynamics, i.e. linearized Vlasov equation . • For any 1D LRI, Vlasov stable homogeneous distribution functions do not evolve on timescales of order smaller or equal to N • Explanation of the dynamical ensemble inequivalence 28
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