Quasi-stationary-states in Hamiltonian mean-field dynamics STEFANO RUFFO Dipartimento di Energetica S. Stecco, Università 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 Quasi-stationary-states in Hamiltonian mean-field dynamics p.2/35
HMF model H = N i=1 p 2 i 2 + 1 2N N i,j=1 (1 cos(θ i θ j )) Magnetization M = lim N PNi=1 cos θ i N, Energy U = lim N H N P Ni=1 sin θ i N «= (M x, M y ) Quasi-stationary-states in Hamiltonian mean-field dynamics p.3/35
Equations of motion θ i = H p i = p i ṗ i = H θ i = 1 N N j=1 sin(θ i θ j ) θ i = p i ṗ i = M sin(θ i φ) tan(φ) = M y M x Quasi-stationary-states in Hamiltonian mean-field dynamics p.4/35
Equilibrium phase transition Fig. 1 revised Latora, Rapisarda & Ruffo Lyapunov instability and finite size... T 0.6 0.5 0.4 0.3 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.4 0.8 1.2 U 0.2 0.1 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 U c = 3/4 = 0.75, T = ( S/ U) 1 = lim N i p2 i /N Quasi-stationary-states in Hamiltonian mean-field dynamics p.5/35
Waterbag p θ 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 p θ M 0 = sin θ θ U = lim N H N = ( p)2 6 + 1 (M 0) 2 2 Quasi-stationary-states in Hamiltonian mean-field dynamics p.6/35
Equilibrium 1.2 N=10000, U=0.63, a=1.0, t=100, sample=1000 p -1.2-3.14 theta 3.14 PDF 0 0.12 Quasi-stationary-states in Hamiltonian mean-field dynamics p.7/35
Quasi-stationary states 0.35 0.3 (a) 0.25 M(t) 0.2 0.15 0.1 0.05 0 1 0 1 2 3 4 5 6 7 8 log 10 t U = 0.69, from left to right N = 10 2, 10 3, 2 10 3, 5 10 3, 10 4, 2 10 4. Initially θ = π, hence M 0 = 0. Quasi-stationary-states in Hamiltonian mean-field dynamics p.8/35
Power law 7 6 (b) b(n) 5 4 3 2 1 2 3 4 5 logn Power law increase of the lifetime, exponent 1.7 Quasi-stationary-states in Hamiltonian mean-field dynamics p.9/35
Separation of time scales Initial Condition Violent relaxation τ v = O(1) Vlasov s Equilibrium Collisional relaxation τ c = N δ Boltzmann s Equilibrium Quasi-stationary-states in Hamiltonian mean-field dynamics p.10/35
Klimontovich equation H = NX i=1 p 2 i 2 + U(θ i) U(θ 1,.., θ N ) = 1 N NX V (θ i θ j ) i<j Discrete, one-particle, time dependent density function f d (θ, p, t) = 1 N NX δ(θ θ i (t))δ(p p i (t)) i=1 Klimontovich equation where v(θ, t) = f d t + p f d θ v θ Z f d p = 0 dθ dp V (θ θ )f d (θ, p, t) Quasi-stationary-states in Hamiltonian mean-field dynamics p.11/35
From Klimontovich to Vlasov f d = f d (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 Quasi-stationary-states in Hamiltonian mean-field dynamics p.12/35
HMF Vlasov equation f t + p f θ dv dθ f p = 0, V (θ)[f] = 1 M x [f] cos(θ) M y [f] sin(θ), M x [f] = f(θ,p,t) cos θdθdp, M y [f] = f(θ,p,t) sin θdθdp. Specific energy e[f] = (p 2 /2)f(θ,p,t)dθdp + 1/2 (M 2 x + M 2 y)/2 and momentum P[f] = pf(θ,p,t)dθdp are conserved. Quasi-stationary-states in Hamiltonian mean-field dynamics p.13/35
Vlasov fluid-i Quasi-stationary-states in Hamiltonian mean-field dynamics p.14/35
Vlasov fluid-ii Quasi-stationary-states in Hamiltonian mean-field dynamics p.15/35
Stirring Quasi-stationary-states in Hamiltonian mean-field dynamics p.16/35
Allowed moves YES NO!! Quasi-stationary-states in Hamiltonian mean-field dynamics p.17/35
Coarse graining for two-level distr. Macrocell Microcell ν microcells of volume ω in a macrocell. A macroscopic configuration has n i microcells occupied with level f 0 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 = Nωf 0 Quasi-stationary-states in Hamiltonian mean-field dynamics p.18/35
Lynden-Bell entropy S LB ( f) = 1 ω dpdθ [ f f 0 ln f f 0 + ( 1 f ) ( ln 1 f )]. f 0 f 0 f(θ,p) = 2 i=1 ρ(θ,p,η i )η i, η 1 = 0, η 2 = f 0 W({n i }) = N! i n i! i ν! (ν n i )!. ρ i (f 0 ) = n i /ν = f i /f 0 Quasi-stationary-states in Hamiltonian mean-field dynamics p.19/35
Maximal Lynden-Bell entropy states f(θ, p) = f 0 e β(p2 /2 M y [ f] sin θ M x [ f] cos θ)+λp+µ + 1. f 0 f 0 f 0 f 0 Z x β x 2β 3/2 Z x β Z x β dθe βm m F 0 xe βm m = 1 Z dθe βm m F 2 xe βm m = U + M2 1 2 dθ cos θe βm m F 0 xe βm m = M x dθ sin θe βm m F 0 xe βm m = M y M = (M x, M y ), m = (cos θ, sin θ). F 0 (y) = R exp( v 2 /2)/(1 + y exp( v 2 /2))dv, F 2 (y) = R v 2 exp( v 2 /2)/(1 + y exp( v 2 /2))dv. f 0 = 1/(4 θ 0 p 0 ) Quasi-stationary-states in Hamiltonian mean-field dynamics p.20/35
Velocity PDF a) b) 0,1 0,1 f QSS (p) 0,01 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 0,5 d) 0,4 0,3 0,2 0,1 0,01-1,5-1 -0,5 0 0,5 1 1,5 p 0-1,5-1 -0,5 0 0,5 1 1,5 M 0 = sin( θ 0 )/ θ 0 = 0.3, 0.5, 0.7. U = 0.69. All QSS are homogeneous (M = 0). Quasi-stationary-states in Hamiltonian mean-field dynamics p.21/35
Non equilibrium phase transition 0.7 0.6 0.5 U=0.50 U=0.54 0.7 M(t) 0.4 0.3 0.2 U=0.62 0.1 0 0 5 10 15 20 25 30 35 40 45 50 55 LEFT: N = 10 3 t U=0.58 hfp 0.6 0.5 0.4 0.3 0.2 0.1 (a) 0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 U RIGHT: First peak height as a function of U for increasing values of N (10 2, 10 3, 10 4, 10 5 ). The transition line is at U = 7/12 = 0.583... Quasi-stationary-states in Hamiltonian mean-field dynamics p.22/35
Tricritical point 0.75 0.605 0.7 0.6 0.595 U 0.65 0.04 0.08 0.12 0.16 0.6 1 ST order 2 ND order Tricritical point 0 0.2 0.4 0.6 0.8 1 M 0 Quasi-stationary-states in Hamiltonian mean-field dynamics p.23/35
Magnetization plot 0.6 0.75 0.5 0.7 U 0.4 0.65 0.3 0.6 0.2 0.55 0.1 0.5 0 0.1 0.2 0.3 0.4 0.5 M 0.6 0.7 0.8 0.9 1 0 Quasi-stationary-states in Hamiltonian mean-field dynamics p.24/35
Negative heat capacity 0.38 0.36 temp_1000000_0.05 temp_10000_0.05 temp_1000_0.05 0.4 0.38 temp_1000000_0.3 temp_10000_0.3 temp_10000.3 0.34 0.36 0.32 0.3 0.34 0.28 0.32 0.26 0.3 0.24 0.28 0.22 0.2 0.26 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 Hamiltonian mean-field dynamics p.25/35
Analytical results 1 0.8 0.6 U c =3/4 T 0.4 0.2 Isothermal n= q=1 Polytropic n c =1 q c =3 U c =5/8 Homogeneous branch 0 0 0.2 0.4 0.6 0.8 1 U 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
Mean-field equilibria ǫ = p2 2 H cos θ H constant. P(ǫ; θ, p) = 1 4 θ p Z + θ θ dθ Z + p p dp δ( p2 2 H cos θ ǫ) Then with Q(ǫ, p) a normalization. Self consistency P H (θ, p; θ, p) = 1 4 θ p P(ǫ, p) Q(ǫ, p) m( θ, p) = Z +π π dθdp cos θp m (θ, p; θ, p) Quasi-stationary-states in Hamiltonian mean-field dynamics p.27/35
m m Numerical verification 10 0 =1 10 0 =2 10-1 10-1 10-2 10-3 Rotators th. Rotators sim. HMF sim. 10-1 10 0 10 1 p 10-2 Rotators th. Rotators sim. HMF sim. 10-3 10-1 10 0 10 1 p P. de Buyl, D. Mukamel and SR, in progress Quasi-stationary-states in Hamiltonian mean-field dynamics p.28/35
Free Electron Laser y Colson-Bonifacio model x z dθ j dz dp j dz da dz = p j = Ae iθ j A e iθ j = iδa + 1 N X j e iθ j Quasi-stationary-states in Hamiltonian mean-field dynamics p.29/35
Quasi-stationary states 0.8 Laser intensity (arb. units) 0.6 0.4 0.2 Laser intensity 0.6 0.5 1 2 3 0.4 0 500 1000 1500 2000 - z 0 0 20 40 60 80 100 120 140 160 180 200 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. - z Quasi-stationary-states in Hamiltonian mean-field dynamics p.30/35
Vlasov equation In the N limit, the single particle distribution function f(θ,p,t) obeys a Vlasov equation. f z A x z A y z = p f θ + 2(A x cos θ A y sin θ) f p = δa y + 1 f cosθ dθdp, 2π = δa x 1 f sin θ dθdp. 2π, with A = A x + ia y = I exp( iϕ) Quasi-stationary-states in Hamiltonian mean-field dynamics p.31/35
Vlasov equilibria Lynden-Bell entropy maximization Z S LB ( f) = dpdθ f ln f + 1 f «ln 1 f ««. f 0 f 0 f 0 f 0 S LB (ε, σ) = Z max [S LB ( f) H( f, A x, A y ) = Nε; f,a x,a y dθdp f = 1; P( f, A x, A y ) = σ]. f = f 0 Non-equilibrium field amplitude e β(p2 /2+2A sin θ) λp µ 1 + e β(p2 /2+2A sin θ) λp µ. A = q A 2 x + A 2 y = β βδ λ Z dpdθ sin θ f(θ, p). Quasi-stationary-states in Hamiltonian mean-field dynamics p.32/35
Results 2 0.7 Intensity 0.6 Laser Intensity, Bunching 1.5 1 0.5 Bunching Intensity Threshold Laser Intensity, Bunching 0.5 0.4 0.3 0.2 Bunching 0.1 0-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2.5 δ 0 0 0.05 0.1 0.15 0.2 0.25 ε Quasi-stationary-states in Hamiltonian mean-field dynamics p.33/35
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 a power of system size. Vlasov (non collisional) equation correctly describes the initial" dynamics. Lynden-Bell maximum entropy principle provides a theoretical approach to quasi-stationary states. The results are derived for mean-field systems, but some of them are expected to be valid also for decaying interactions. Quasi-stationary-states in Hamiltonian mean-field dynamics p.34/35
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). Quasi-stationary-states in Hamiltonian mean-field dynamics p.35/35