Quasi-stationary states in systems with long-range interactions - Infn

Quasi-stationary states in systemswith long-range interactionsSTEFANO RUFFODipartimento di Energetica “S. Stecco” and INFNCenter for the Study of Complex Dynamics (CSDC), FirenzeStatistical Mechanics and Field Theory, Bari20-22 September 2006Quasi-stationary states in systems with long-range interactions – p.1/22

PlanHamiltonian Mean Field (HMF) modelQuasi-stationary statesNon equilibrium phase transitionLynden-Bell entropyMaximum entropy principleApplication to the free electron laserQuasi-stationary states in systems with long-range interactions – p.2/22

HMF modelH XY =N∑i=1p 2 i2 + 12NN∑i,j=1(1 − cos(θ i − θ j ))Inspired byGravitational and charged sheet modelsWave-particle interactionsQuasi-stationary states in systems with long-range interactions – p.3/22

Quasi-stationary statesInitial water-bag of (semi) width ∆θ 0 = π and height ∆p 0(which determines the energy U = H/N)Magnetization M = ( ∑ Ni=1 cos θ i/N, ∑ Ni=1 sin θ i/N)0.3570.30.25(a)6(b)M(t)0.20.15b(N)540.10.0530−1 0 1 2 3 4 5 6 7 8log 10tLEFT: U = 0.69, from left to rightN = 10 2 , 10 3 , 2 × 10 3 , 5 × 10 3 , 10 4 , 2 × 10 421 2 3 4 5logNRIGHT: Power law increase of the lifetime, exponent 1.7Quasi-stationary states in systems with long-range interactions – p.4/22

Phase transition0.70.60.5U=0.50U=0.540.7M(t)0.40.30.2U=0.620.100 5 10 15 20 25 30 35 40 45 50 55LEFT: N = 10 3tU=0.58hfp0.60.50.40.30.20.1(a)00.5 0.55 0.6 0.65 0.7 0.75 0.8URIGHT: First peak height as a function of U = H/N forincreasing values of N (10 2 , 10 3 , 10 4 , 10 5 )Quasi-stationary states in systems with long-range interactions – p.5/22

HMF Vlasov equation∂f∂t + p∂f ∂θ − dVdθ∂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 energye[f] = ∫ (p 2 /2)f(θ,p,t)dθdp + 1/2 − (M 2 x + M 2 y)/2 andmomentum P[f] = ∫ pf(θ,p,t)dθdp are conserved.Quasi-stationary states in systems with long-range interactions – p.6/22

Vlasov simulationsQuasi-stationary states in systems with long-range interactions – p.7/22

Lynden-Bell entropyAssume that the initial distribution f(θ,p, 0) takes only twovalues (0,f 0 ) (“water bag"). Time evolution can only modifythe shape of the boundary of the “water-bag", conservingthe area inside it. Hence, the distribution remains two-levelas time evolves. Coarse-graining amounts to perform alocal average of f inside a given cell, which gives ¯f. The“mixing" entropy per particle associated with ¯f is(Lynden-Bell, 1967)∫s( ¯f) = −dpdθ[ ¯ff 0ln ¯ff 0+(1 − ¯f ) (ln 1 − ¯f )].f 0 f 0Lynden-Bell guesses that the initial evolution (“violentrelaxation") is characterized by a maximization of this“fermionic" entropy with given constraints (e.g. energy,momentum, ...).Quasi-stationary states in systems with long-range interactions – p.8/22

Derivation of Lynden-Bell entropyFermionic principle: If the initial single-particle distribution is two-level (0, f 0 ), it remains twolevel during time evolution (Liouville theorem).Other invariants (Casimirs) of the Vlasov equation exist.Divide the bounded phase space into a finite number of macrocells. Each macrocell containsν microcells of size h. Let n i be the number of microcells occupied by the level f 0 in the i thmacrocell and N the total number of occupied microcells.Then the number of microstates compatible with the macrostate n i isW({ ¯f}) = W({n i }) = QN!i n i! × Y iν!(ν − n i )! ,with ¯f the coarse-grained distribution. The mixing (Lynden-Bell) entropy isS LB = ln[W({ ¯f})].Quasi-stationary states in systems with long-range interactions – p.9/22

Maximal Lynden-Bell entropy states¯f(θ, p) = f 0e −β(p2 /2−M y [ ¯f] sin θ−M x [ ¯f] cos θ)−λp−µ1 + e −β(p2 /2−M y [ ¯f] sin θ−M x [ ¯f] cos θ)−λp−µ .f 0f 0f 0f 0Zx √βZx2β 3/2 Zx √βZx √βdθe βM·m F 0“xe βM·m” = 1dθe βM·m F 2“xe βM·m” = e + M2 − 12dθ cos θe βM·m F 0“xe βM·m” = M xdθ sin θe βM·m F 0“xe βM·m” = M yM = (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 systems with long-range interactions – p.10/22

Velocity PDFsingle particle distribution|M 0|=0.3|M 0|=0.50.10.1f(p)0.01-1.5 -1 -0.5 0 0.5 1 1.50.01-1.5 -1 -0.5 0 0.5 1 1.50.10.1|M 0|=0.70.01-1.5 -1 -0.5 0 0.5 1 1.50.01-1.5 -1 -0.5 0 0.5 1 1.5p|M 0|=0.9|M 0 | = sin(∆θ 0 )/∆θ 0Non Gaussian velocity distributionsQuasi-stationary states in systems with long-range interactions – p.11/22

Tricritical pointU0,75 0,750,7 0,70 0,05 0,1 0,150,5850,580,65 0,650,610,6050,60,5950,590,6 0,61 ST order lineU 2 ND order linec=7/12tricritical point0,55 0,550 0,2 0,4 0,6 0,8 1|M| 0Quasi-stationary states in systems with long-range interactions – p.12/22

Features of the phase transition0,60,5a)|M 0 |=0.8|M 0 |=0.55b)|M 0 |=0.1|M 0 |=0.010,30,25f QSS(p)0,40,30,2|M| QSS0,20,150,10,10,050-1,5 -1 -0,5 0 0,5 1 1,5p00,6 0,7 0,8 0,9|M| 0Quasi-stationary states in systems with long-range interactions – p.13/22

Addition of a short range contributionH =N∑i=1p 2 i2 + 12NN∑i,j=1[1 − cos (θ i − θ j )]−KN∑i=1cos (θ i+1 − θ i ) .LEFT: M(t) vs. t; K = 0.05, U = 0.71, N = 1024,...,32768RIGHT: increase of the lifetime; K = 0, 0.0025, 0.05, 0.1Quasi-stationary states in systems with long-range interactions – p.14/22

HMF core-halo structureRefinements of maximum entropy methods should take intoaccount the "true" dynamics.Quasi-stationary states in systems with long-range interactions – p.15/22

Free Electron LaseryColson-Bonifacio modelxzdθ jdzdp jdzdAdz= p j= −Ae iθ j− A ∗ e −iθ j= iδA + 1 NXje −iθ jQuasi-stationary states in systems with long-range interactions – p.16/22

Quasi-stationary states0.8Laser intensity (arb. units)0.60.40.2Laser intensity0.60.51230.40 500 1000 1500 2000- z00 20 40 60 80 100 120 140 160 180 200N = 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 toBoltzmann-Gibbs equilibrium on a time O(N). The quasi-stationary state is a Vlasovequilibrium, sufficiently well described by Lynden-Bell’s Fermi-like distributions.-zQuasi-stationary states in systems with long-range interactions – p.17/22

Vlasov equationIn the N → ∞ limit, the single particle distribution functionf(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 systems with long-range interactions – p.18/22

Vlasov equilibriaLynden-Bell entropy maximizationZS LB ( ¯f) = −dpdθ„ ¯fln ¯f „+ 1 − ¯f « „ln 1 − ¯f ««.f 0 f 0 f 0 f 0S LB (ε, σ) =Zmax [S LB ( ¯f)|H( ¯f, A x , A y ) = Nε;¯f,A x ,A ydθdp ¯f = 1; P( ¯f, A x , A y ) = σ].¯f = f 0Non-equilibrium field amplitudee −β(p2 /2+2A sin θ)−λp−µ1 + e −β(p2 /2+2A sin θ)−λp−µ .A =qA 2 x + A 2 y =ββδ − λZdpdθ sin θ ¯f(θ, p).Quasi-stationary states in systems with long-range interactions – p.19/22

Results20.7Intensity0.6Laser Intensity, Bunching1.510.5BunchingIntensityThresholdLaser Intensity, Bunching0.50.40.30.2Bunching0.10-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5δ00 0.05 0.1 0.15 0.2 0.25εQuasi-stationary states in systems with long-range interactions – p.20/22

ConclusionsMean-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 system size.Vlasov (non collisional) equation correctly describes the “initial" dynamics ofmean-field Hamiltonians.Lynden-Bell maximum entropy principle provides a theoretical approach toquasi-stationary states.Collective phenomena of wave-particle interactions (free electron laser) are the resultof a Lynden-Bell maximum entropy principle.Quasi-stationary states in systems with long-range interactions – p.21/22

ReferencesT. Dauxois, S. Ruffo, E. Arimondo and M. Wilkens (Eds.), Dynamics and statistics of systemswith long-range interactions, Springer Lecture Notes in Physics 602 (2002).Y.Y. Yamaguchi, J. Barré, F. Bouchet, T. Dauxois, S. Ruffo, Stability criteria of the Vlasovequation and quasi-stationary states of the HMF model, Physica A, 337, 36 (2004).J. Barré, T. Dauxois, G. De Ninno, D. Fanelli, S. Ruffo:Statistical theory of high-gainfree-electron laser saturation, Phys. Rev. E, Rapid Comm., 69, 045501 (R) (2004).M. Antoni, A. Torcini and S. Ruffo First-order microcanonical transitions in finite mean-fieldmodels, Europhysics Letters, 66, 645 (2004).D. Mukamel, S. Ruffo and N. Schreiber, Breaking of ergodicity and long relaxation times insystems with long-range interactions, Phys. Rev. Lett., 95, 240604 (2005).A. Campa, A. Giansanti, D. Mukamel and S. Ruffo, Dynamics and thermodynamics of rotatorsinteracting with both long and short range couplings , Physica A, 365, 177 (2006)A. Antoniazzi, D. Fanelli, J. Barré, P.-H. Chavanis, T. Dauxois and S. Ruffo, A maximumentropy principle explains quasi-stationary states in systems with long range interactions: the exemple ofthe HMF model, preprint (2006)Quasi-stationary states in systems with long-range interactions – p.22/22