Dynamics and Statistical Mechanics of (Hamiltonian) many body systems with long-range interactions. A. Rapisarda

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1 SMR SCHOOL and CONFERENCE on COMPLEX SYSTEMS and NONEXTENSIVE STATISTICAL MECHANICS 31 July - 8 August 2006 Dynamics and Statistical Mechanics of (Hamiltonian) many body systems with long-range interactions A. Rapisarda Dipartimento di Fisica e Astronomia and INFN sezione di Catania, Università di Catania Italy

2 School and Conference on Complex Systems and Nonextensive Statistical Mechanics July 30 - August , ICTP Trieste Dynamics and Statistical Mechanics of (Hamiltonian) many body systems with long-range interactions A. Rapisarda Dipartimento di Fisica e Astronomia and INFN sezione di Catania, Università di Catania Italy

3 School and Conference on Complex Systems and Nonextensive Statistical Mechanics July 30 - August , ICTP Trieste main collaborators Alessandro Pluchino Vito Latora Filippo Caruso Salvo Rizzo Dipartimento di Fisica e Astronomia and INFN sezione di Catania, Università di Catania Italy Dipartimento di Fisica e Astronomia and INFN sezione di Catania, Università di Catania Italy Scuola Superiore di Catania and Scuola Normale di Pisa, Italy ENAV Firenze and Dipartimento di Fisica e Astronomia Università di Catania, Italy Group web page:

4 3 Plan of the lectures 1 st lecture Part 1 Dynamics and Thermodynamics of the Hamiltonian Mean Field (HMF) model: a simple model of fully-coupled particles on the unitary ring or classical spins, whose behavior seems to be paradigmatic for a large class of nonextensive systems: Equilibrium case Part 2 Dynamical anomalies in the out-equilibrium regime and the role of initial conditions: Quasi-Stationary States & Negative specific heat Chaos suppression & slow dynamics Anomalous diffusion & Lèvy walks 2 nd lecture Non Gaussian velocity PDF, power law relaxation, aging Generalized HMF model: the α-xy model Links to Tsallis generalized statistics Importance of initial conditions Interpretation of the QSS regime in terms of a glassy phase Part 3 Non Hamiltonian systems: a) The Kuramoto model b) Coupled map lattices with noise c) Soc models for earthquakes dynamics Conclusions

5 4 Motivation Why one should study long-range interacting systems... Long-range interactions are important for phase transitions in finite size systems, for example: fragmenting nuclei and atomic clusters For understanding how one can treat statistically selfgravitating objects and plasmas. Long-range correlations are also frequently observed in out-of-equilibrium and complex systems In general long-range interactions pose fundamental problems to standard statistical mechanics, so one needs new statistical tools to treat them properly

6 5 The HMF model The Hamiltonian Mean Field (HMF) model H 2 p 1 i 2 2N N N = + i i= 1 i, j= 1 [1 co s( ϑ ϑ )] j Antoni and Ruffo PRE 52 (1995) 2361 The system has an infinite range force It is a useful paradigmatic model to study Hamiltonian long-range interacting (nonextensive) systems as for example astrophysical systems, but also fragmenting nuclei and atomic clusters

7 6 Phase transition at equilibrium The The model model can can be be seen seen as as N classical classical interacting interacting spins spins or or particles particles moving moving on on the the unit unit circle. circle. One One can can define define the the total total magnetization magnetization M as an order parameter M 1 N mi N i = 1 = where where the the single single spin spin is is M as an order parameter m = (cos ϑ,sin ϑ ) i i i The The model model shows shows a second-order second-order phase phase transition, transition, passing passingfrom froma clustered clusteredphase phasetoto a homogeneous homogeneousone one as as a function functionof of energy energy M=1 M=1 M=0 M=0 clustered phase for U<U clustered phase for U<U c c homogeneous phase for U>U homogeneous phase for U>U c c

8 7 Equilibrium solution By using the saddle point method, one gets for the free energy F 2 1 2π β y βf = log + max + log 0 ) 2 β 2 y 2β ( 2πI ( y ) where β = and k k B 1T B =1 is the Boltzmann constant y I1 Then one gets the consistency equation 0 (1), β I = 0 I1 where M = and I is the modified Bessel function of ith order. I i Solving eq.(1) one gets the exact canonical equilibrium expression U = 0 E N ( βf ) = β 1 = + 2β ( 1 M ) Caloric curve

9 8 Critical behavior of the model The model has a second order phase transition. The critical point is at 4 M β U c = Close to the critical point β Hence M vanishes with the classical critical mean field exponent 1/2 T c Close to the critical point one gets for U On the other hand, the specific heat C V ( T c ) = 5 2 and C V = 1 2 and for = 1 2 β ( β 2) β β C V ( T) C V T c + β c + ( U ) = T T > T c α 1 2 is with α = 0

10 9 Dynamics vs Thermodynamics The model can be studied also dynamically by means of microcanonical numerical simulations. One can compare dynamical aspects with thermodynamical ones. One can study finite size effects. One can also study how the system relax to equilibrium

11 10 Dynamics The equations of motion are: θ i t pi t = p i = M sinθ + M cosθ x i y i The potential is connected to the magnetization M as N N V = 1 M M 1 M 2 + = ( ) x y ( ) The equations are solved numerically by using a fourth order simplectic algorithm (Yoshida, Physica A 150 (1990) 262). E Energy is conserved with an error smaller than =10 5 for a number of time steps 6 E 10

12 11 Critical behavior of the model for finite sizes Microcanonical simulations follow the canonical prediction, even for N=100 1/2 ε

13 12 Equilibrium Good agreement between exact canonical solution and numerical microcanonical simulations at equilibrium for various sizes N of the system Latora, Rapisarda and Ruffo PRL 80 (1998) 692, Physica D 131 (1999) 38

14 13 Dynamics at Equilibrium One finds a maximum of the largest Lyapunov exponent (LLE) in connection to the critical point, where both the fluctuations in kinetic energy and temperature and the specific heat present a peak Latora, Rapisarda and Ruffo Physica D 131 (1999) 38

15 14 Dynamics at Equilibrium: scaling of the LLE λ N for U > U c λ1 U 1 2 for U << U c

16 15 Lyapunov spectra at Equilibrium In Hamiltonian systems with N degrees of freedom λ i λ = 2 N i+1 At low energy only a few degrees of freedom are active. (positive part only)

17 16 Lyapunov spectra at Equilibrium No significant change in the shape of the spectra across the critical point is observed

18 17 Kolmogorov Sinai entropy S KS = N i= 1 λ i with λ i > 0 A peak close to the critical point is found also for S KS

19 18 Scaling laws fot the Kolmogorov Sinai entropy 3 4 U <<U S KS U c S KS N 1 5 U >Uc

20 19 Antiferromagnetic behavior of HMF The HMF model can have also an antiferromagnetic behavior if one considers H = K The general canonical solution for V ± V is 2 ( 1 ) U 1 ε = + 2β 2 M with ε = ±1

21 20 Dynamics at Equilibrium One has a different behavior of the Largest Lyapunov exponent and the KS entropy in the ferromagnetic and antiferromagnetic case

22 21 LLE in the thermodynamical limit In the thermodynamic limit, the LLE goes to zero for the whole energy range in the antiferromagnetic case, while it remains finite, for energies smaller than the critical one (Uc=0.75), in the ferromagnetic one. In the latter case it goes to zero for overcritical energies as 1/ 3 λ N 1

23 22 LLE in the antiferromagnetic case Both the LLE and the KS entropy go to zero as in the antiferromagnetic case λ1 U with U * * = U + 1 2

24 23 Equilibrium PDFs for the HMF model In the continuum limit, considering the one-body distribution function F, the evolution of the HMF model is described by the Vlasov equation F t + F V F p ϑ ϑ p = 0 Supposing a factorization of the distribution function F = f ( p) g( ϑ, t) One gets the stationary equilibrium solution 2 1 p 2 T f = f 0 e and 2π T 1 ( M / T ) where g0 = 2πI, 0 In the overcritical region I 0 g 1 M = 0 g = 2π = g 0 e M cos( ϑ φ ) T z σ 0( z) g e 2 ϑ 2 In the low energy region 2 2 I φ is the Bessel function e 2πz is the phase of M 1 8z 1 2πσ A. Latora, RAPISARDA Rapisarda ICTP 2006 and Ruffo Physica D 131 (1999) 38

25 24 Comparison with numerical pdfs At low energy

26 25 Comparison with numerical pdfs at equilibrium At the critical point

27 26 Anomalous dynamics HMF model in the out-of-equilibrium case

28 27 OUT-OF-EQUILIBRIUM CASE When the system is started with initial conditions very far from equilibrium.. we observe many dynamical anomalies, in particular in an energy range below the critical point.

29 28 Negative specific heat C V 1 T = < 0 U V In a region before the critical point the specific heat becomes negative: the temperature decreases, by increasing the energy density. This phenomenon has been observed in multifragmentation nuclear reactions and atomic clusters, but also in selfgravitating stellar objects, i.e. for nonextensive systems. See for example: Thirring, Zeit. Physik 235 (1970) 339 Lynden-Bell, Physica A 263 (1999) 293 D.H.E.Gross, Microcanonical Thermodynamics: Phase transitions in Small systems, World Scientific (2001). M. D Agostino et al, Phys. Lett. B 473 (2000) 279 Schmidt et al, Phys. Rev. Lett. 86 (2001) 1191

30 29 Quasi Stationary States U=0.69 Equilibrium The system lives for a very long time in a metastable quasi stationary state (QSS), whose temperature defined as 2< K> T = N is smaller than the equilibrium one. QSS regime The larger N, the longer the QSS lifetime. The temperature T tends to 0.38 for N

31 30 QSS: initial evolution QSS QSS regime is isreached almost immediately

32 31 QSS and numerical accuracy QSS QSS do do not not depend on on the the accuracy of of the the integration

33 32 QSS lifetime and Temperature The QSS lifetime diverges with the size N The QSS temperature goes to T inf =0.38 (for U=0.69) as N Notice that being M 2 = T + 1 2U = T one has M 2 1/3 N 0.38 Latora, Latora, Rapisarda Rapisarda and and Tsallis Tsallis PRE PRE (2001) (2001)

34 15 The energy density U=0.69 The majority of of the the results refer to tothe the case U=0.69, where anomalies are are most evident

35 34 Force in the QSS regime In the QSS regime the force F i on the ith spin goes to zero with the size N being F i = M x sin θ + M i y cosθ i Thus the QSS dynamics is slower and slower by increasing N. For N the system remains frozen in the QSS regime

36 35 Vanishing Lyapunov exponents λ QSS N 1/9 In the QSS regime the largest Lyapunov exponent tends to zero as the size of the system tends to infinity This scaling can be obtained considering that λ ( ) 1/3 M N = N 2/3 1/3 1/9 Latora, Rapisarda, Tsallis Physica A 305 (2002) 129

37 36 Importance of the order in the limits Simulations show that, going towards the thermodynamic limit, it is very crucial the time order of the size limit and time limit to infinity In general, the two limits do not commute: N t t N

38 37 Anomalous diffusion in the QSS regime In general one has for the mean square displacement σ 2 () t t α α = 1 α 1 Normal diffusion Anomalous diffusion Latora, Latora, Rapisarda Rapisarda and and Ruffo, Ruffo, PRL PRL (1999) (1999) In our case we get superdiffusion with an exponent α=1.38 in correspondence of the QSS regime.

39 38 Anomalous diffusion: typical dynamics Phase space dynamics of of a typical particle (spin) Lèvy Lèvy walks walks QSS Equilibrium Normal Normal random random walks walks

40 39 Anomalous diffusion The The cross-over times, times, from fromanomalous to tonormal normaldiffusion, coincide with withthe the relaxation times times QSS Equilibrium

41 40 Lévy walks: walking and trapping time PDFs For a one dimensional system which shows sticking and flying particles with constant velocity, one finds: P walk ( t) P trap ( t) µ t ν t In our case we find with α = 2+ ν µ when 2 < µ < 3 ν < 2 See Klafter and Zumofen PRE 49 (1994) 4873

42 41 Time evolution of velocity PDFs QSS QSS regime regime Equilibrium Equilibrium regime regime

43 42 Non-Gaussian velocity PDF Velocity PDFs in the QSS regime for different sizes N of the system

44 43 Role of initial conditions for QSS We have recently studied the nature of the anomalous QSS regime by starting from different initial magnetizations with 0 M 1 (considering in all cases a uniform momenta: water bag). distribution in Dynamical anomalies depend on the initial conditions. The most interesting are those observed for initial magnetization M 1

45 44 QSS for different initial conditions: M1 vs. M0

46 Dependence of QSS on the initial magnetization Pluchino, Latora, Rapisarda Physica A 338 (2004) 60

47 46 Correlations in phase space for different IC M1 IC M0 IC Structures Structures in in phasespace phasespace appear, appear, in in the the QSS QSS regime regime for for M1IC, M1IC, but but not not for for M0IC M0IC

48 47 Dynamical evolution for different initial conditions M1 IC M0 IC

49 Dynamical evolution for different initial conditions M1 IC M0 IC

50 Dynamical evolution for different initial conditions M1 IC M0 IC

51 Dynamical evolution for different initial conditions M1 IC M0 IC

52 Dynamical evolution for different initial conditions M1 IC M0 IC

53 Dynamical evolution for different initial conditions M1 IC M0 IC

54 Dynamical evolution for different initial conditions M1 IC M0 IC

55 Dynamical evolution for different initial conditions M1 IC M0 IC

56 Dynamical evolution for different initial conditions M1 IC M0 IC

57 Dynamical evolution for different initial conditions M1 IC M0 IC

58 Dynamical evolution for different initial conditions M1 IC M0 IC

59 Dynamical evolution for different initial conditions M1 IC M0 IC

60 Dynamical evolution for different initial conditions M1 IC M0 IC

61 Dynamical evolution for different initial conditions M1 IC M0 IC

62 Dynamical evolution for different initial conditions M1 IC M0 IC

63 Dynamical evolution for different initial conditions M1 IC M0 IC

64 Dynamical evolution for different initial conditions M1 IC M0 IC

65 Dynamical evolution for different initial conditions M1 IC M0 IC

66 Dependence of µ-space structures on the i.c.

67 66 Entropy and free energy for QSS Entropy Free energy Also the entropy and the free energy show a different dynamical evolution for the two initial conditions M1IC and M0IC

68 67 Relaxation to equilibrium Exponential decay Power-law decay One can study the process of relaxation to equilibrium by means of velocity correlation functions, < () t (0) > < () t >< (0) > Ct (,0) = P P P P σ() t σ(0) where P( ) (,,... ) t = p1 p2 p N is the velocity vector, the brackets indicate an average over the events, and σ () t is the standard deviation at time t. Correlations show a power law decay for M1IC and an exponential decay for M0IC

69 68 Aging and strong memory effects For M1 IC, the system, in going towards equilibrium, shows strong memory effects and aging, i.e. the correlation functions depend on t and on the waiting time t w < P( t+ tw) P( tw) > < P( t+ tw) >< P( tw) > Ct ( + tw, tw) = σ( t+ t ) σ( t ) w w Montemurro, Montemurro, Tamarit Tamarit and and Anteneodo Anteneodo PRE PRE (2003) (2003) Pluchino, Pluchino, Latora Latora and and Rapisarda, Rapisarda, Physica Physica D D (2004) (2004)

70 69 Dynamical frustration M1IC M0IC M=1 M=1 initial initial conditions: conditions: - - Competition Competition between between clusters clusters of of rotating rotating particles particles in in the the QSS QSS regime regime - - Each Each cluster cluster tries tries to to capture capture the the maximum maximum number number of of particles particles in in order order to to reach reach the the final final equilibrium equilibrium configuration: configuration: This competition leads to a DYNAMICAL FRUSTRATION in the QSS regime! Equilibrium M=0 M=0 initial initial conditions: conditions: No No competition competition and and no no frustration frustration are are present present At At equilibrium equilibrium only only one one big big rotating rotating cluster cluster is is present present for for both both the the IC IC

71 Critical cluster size distribution µ-space PDF t=0 t=50 t=500 Cumulative Number of Clusters The cumulative cluster size distribution is a power law for the QSS regime, which is robust for the entire plateau

72 Cluster size distribution vs initial conditions x -1.6 x -1.6

73 Cluster formation vs initial conditions t=200 M1 ic ic M0 ic ic

74 Cluster formation vs initial conditions M095 ic M08 ic

75

76 75 The generalized HMF model α-xy XY model The HMF model has been also generalized to study the dynamic and thermodynamic behavior as a function of the range of the interaction H = N i= 1 p 2 i N [ ] N 1 cos( θ i θ j) α i j r ij Anteneodo and Tsallis, PRL 80 (1998) 5313 Campa, Giansanti and Moroni, PRE 62 (2000) 303 Tamarit and Anteneodo, PRL 84 (2000) 208 Campa, Giansanti and Moroni, J. Phys. A 36 (2003) For For α d α this generalized model reduces to HMF. one has interaction only among nearest neighbour spins.

77 α-xy model and nonextensive effects Anomalies depend in a crucial way on the range of the interaction The lifetime τ of the QSS does not diverge for all α see A. Campa et al. Physica A 305 (2002) 137 Decreasing the range of the interaction, i.e. diminishing nonextensivity this anomalous behaviour disappears: α Relaxation is very fast ( τ e ) No negative specific heat is observed ( α > 0)

78 Long-range Lennard-Jones potential Another example which support the long-range nature of the interaction as the origin of negative specific heat is the 2-d Lennard-Jones gas with attractive potential V α r Borges and Tsallis Physica A 305 (2002)148 Again decreasing the range of the interaction, i.e. diminishing non extensivity this anomalous specific heat disappears. In correspondence they also find non-boltzmann velocity distributions Further similar examples have been found in self-gravitating systems, see Sota et al PRE 64 (2001)

79 < = N j i ij r ij r C V α α σ σ 12 More precisely the potential energy is ( ) α α ε α α α = 12 / 12 1 N b C σ is the diameter of the particles, b is a constant and ε is the energy scale. For α=6 one gets the usual Lennard-Jones

80 79 Interpretration of QSS regime The anomalous QSS regime is the effect of non extensivity or, in other words, of the long-range character of the interaction. These anomalies seems be connected thermostatistics to Tsallis Very recently a link with a glassy phase has also been found

81 80 Tsallis generalized thermostatistics In the last decade a lot of effort has been devoted to understand if thermodynamics can be generalized to nonequilibrium complex systems In particular one of these attempts is that one started by Constantino Tsallis with his seminal paper J. Stat. Phys. 52 (1988) 479 For recent reviews see for example: C. Tsallis, Nonextensive Statistical Mechanics and Thermodynamics, Lecture Notes in Physics, eds. S. Abe and Y. Okamoto, Springer, Berlin, (2001); Proceedings of NEXT2001, special issue of Physica A 305 (2002) eds. G. Kaniadakis, M. Lissia and A Rapisarda; C. Tsallis, A. Rapisarda, V. Latora and F. Baldovin in "Dynamics and Thermodynamics of Systems with Long-Range Interactions", T. Dauxois, S. Ruffo,E. Arimondo, M. Wilkens eds., Lecture Notes in Physics Vol. 602, Springer (2002) 140; ``Nonextensive Entropy - Interdisciplinary Applications'', C. Tsallis and M. Gell-Mann eds., Oxford University Press (2003). A. Cho, Science 297 (2002) 1268; S. Abe and A.K. Rajagopal, Science 300, (2003)249; A. Plastino, Science 300 (2003) 250; V. Latora, A. Rapisarda and A. Robledo, Science 300 (2003) 250. For a regularly updated bibliografy:

82 Tsallis conjecture for for nonextensive systems C.Tsallis, Braz. Jour.. of Phys.. 29 (1999) 1

83 82 Generalized velocity pdfs q p 1 (1 q) 2T

84 83 q-exponential decay of C(t,0) The The decay decay of of the the velocity velocity correlation correlation function function can can be be reproduced reproduced very very well well by by means means of of the the generalized generalized q-exponential q-exponential 1 1 Ae ( x) = A[1 + (1 q) x] q q In In our our case case x=-t/τ x=-t/τ.. Within Within a a generalized generalized Fokker-Plank Fokker-Plank equation equation which which generates generates Tsallis Tsallis q-exponential q-exponential pdfs pdfs [1], [1], one one can can extract extract the the following following relation relation between between the the exponent γ exponent of of the the anomalous anomalous diffusion diffusion and and q q 2 γ = 3 q [1] [1] Tsallis Tsallis and and Buckman Buckman PRE PRE (1996) (1996) R2197 R2197 In In our our case γ case = , = , thus thus we we expect expect q= , q= , which which is is confirmed confirmed by by the the fit fit in in the the figure figure for for M1IC. M1IC. On On the the other other hand, hand, for for M0IC M0IC the the decay decay is is almost almost exponential. exponential.

85 62 q-exponential decay also for different i.c.

86 85 q-exponential decay for aging Also for the aging behavior, the power law decay of the correlation functions, after a proper rescaling, can be reproduced with a q- exponential function. In this case we get q=1.65.

87 64 q vs initial Magnetization

88 γ 63 Anomalous diffusion for different i.c. γ

89 65 Correlation decay vs range α

90 66 Summary of most recent results regarding HMF model and q-statistics Anomalous diffusion vs q-exponential decay Rapisarda Rapisarda and and Pluchino, Pluchino, Europhysics Europhysics News News (2005) (2005)

91 90 Most recent results Some unpublished results on recent criticism

92 Recent criticism A few comments on recent criticism regarding anomalous diffusion in the HMF model and q-statistics

93 It seems that velocity pdfs can be described using Lynden- Bell entropy within a Vlasov approach.. There are however several discrepancies..due to dynamical effects and initial conditions Antoniazzi et al cond-mat/

94 It seems that velocity pdfs can be described using Lynden- Bell entropy within a Vlasov approach.. There are however several discrepancies..due to dynamical effects and initial conditions Antoniazzi et al cond-mat/

95 Latora, Rapisarda Tsallis PRE 2001 (1 q) P( p) 1 2T p q

96 Anomalous diffusion is a finite size effect? Antoniazzi et al cond-mat/

97 Anomalous diffusion is a finite size effect? Moyano and Anteneodo cond-mat/

98 Moyano and Anteneodo cond-mat/

99 Anomalous diffusion is a finite size effect? Moyano and Anteneodo cond-mat/

100 It is not completely clear if anomalous diffusion disappears at large N at the moment. However initial conditions are extremely important even for large N Tsallis, Rapisarda, Pluchino in preparation

101 Single realizations can have a very different dynamics. So a good statistics is needed even for large N. Tsallis, Rapisarda, Pluchino in preparation

102 Tsallis, Rapisarda, Pluchino in preparation

103 A few notes on recent criticisms regarding anomalous diffusion in the HMF model and q-statistics Recently a dynamical transition as a function of the initial magnetization was claimed by Chavanis

104 A few notes on recent criticisms regarding anomalous diffusion in the HMF model and q-statistics Recently a dynamical transition as a function of the initial magnetization was claimed by Chavanis See Chavanis cond-mat/ For an initial magnetization M>0.897 (if U=0.69) The homogeneous Linden- Bell distribution becomes unstable and the system can be trapped into an incomplete mixed state, where Tsallis statistics is a possible explanation

105 A few notes on recent criticisms regarding anomalous diffusion in the HMF model and q-statistics Recently a dynamical transition as a function of the initial magnetization was claimed by Chavanis See Chavanis cond-mat/ unstable For an initial magnetization M>0.897 (if U=0.69) The homogeneous Linden-Bell distribution becomes unstable and the system can be trapped into an incomplete mixed state, where Tsallis statistics is a possible explanation stable Antoniazzi et al cond-mat/

106 A few notes on recent criticisms regarding anomalous diffusion in the HMF model and q-statistics Several numerical simulations confirm this transition, but further investigation is needed U=0.69 N=100000

107 Also Morita and Kaneko recently found anomalous collective dynamics which cannot be explained with Vlasov states

108

109 Microscopic dynamics can be very different according to the initial conditions (as for other fully-coupled systems, for example the Kuramoto model) and different theoretical approaches can be applied Most probably the Vlasov equation cannot explain all the anomalies found, which on the other hand, are found also in other models with long-range interactions Anomalies have a clear dynamical origin. q-statistics provides a coherent interpretations scenario and at the same time does not excludes other

110 109 Glassy phase in HMF Ferromagnetic Phase: Paramagnetic Phase: Glassy Phase: FRUSTRATION & QUENCHED DISORDER M 0 M 0 M 0 M N 1 = s i N i = 1

111 110 The polarization p One can introduce the elementary polarization : τ 1 < s i >= si () t dt τ i.e. the temporal average, over a time interval t, of the successive positions of each elementary spin vector. The modulus of the elementary polarization has to be furtherly averaged over the quenched disorder of the N spin configurations, to finally obtain the polarization p: p 1 = < > 1 N si N i = 1

112 92 Polarization & Glassy phase Ferromagnetic Phase: Paramagnetic Phase: Glassy Phase: One can consider the polarization p as a new order parameter in order to measure the freezing of the spins M 0 p 0 M 0 p 0 M 0 p 0

113 112 Polarization and QSS In the HMF model we can use the polarization p in order to characterize the QSS regime as a Glassy Phase, related with the dynamical frustration between clusters: PHASE p M Ferromagnetic 0 0 Paramagnetic 0 0 Glassy phase 0 0 Pluchino, Latora, Rapisarda, Phys. Rev. E 69 (2004)

114 113 p and M in the QSS regime vs N In the QSS regime the magnetization goes to zero with the system size as M N 6 while the polarization remains constant p

115 114 Dependence of p on the integration time The The polarization polarizationp does doesnot notdepend dependon on the the integration integrationtime time interval intervalinside inside the the QSS QSS regime regime

116 115 p and M at equilibrium N=10000 The The polarization p and and the the magnetization M coincide at at equilibrium

117 102 Polarization vs initial conditions Polarization is very robust as a function of N and time (within the QSS regime).

118 103 Polarization vs initial conditions Polarization is very sensitive to the initial conditions: it goes rapidly to zero when the initial magnetization is less than 1.

119 Minimization of the Temperature 68 Monte Carlo simulations A. Pluchino,G.Andronico, A. Rapisarda, Physica A 349 (2005) 143 Standard Metropolis

120 105 Glassy thermodynamics for the QSS regime A. Pluchino PhD thesis (2005) and cond-mat/ Physica A (2006) in press Let us start from the following effective spin-glass Hamiltonian N 1 H = J ijs i s j (1) i (cos i,sin i) 2 s with = ϑ ϑ i, j= 1 Using the following distribution probability for the interaction J ij : with average and variance ( Jij J0 ) 2 1 2σ J pj ( ij ) = ( 2 πσ J ) e ij ij ij σ J J = J0 = 1/ N, J J = = 1/ N This distribution implies a random coupling among spins with a probability on average J 0 =1/N which simulates a glassy behavior similar to what we observe in the QSS regime. In the thermodynamic limit we have no interaction and the system remains frozen for ever. In the thermodynamic limit the Hamiltonian (1) reduces to the potential part of the HMF Hamiltonian. One can treat the kinetic part considering a heat bath with T=2K/N General HMF Hamiltonian : 2 N N p 1 i H = K+ V = + Jijsi s 2 2N 2 i= 1 i, j= 1 j

121 75 Comparison with numerical data By applying the replica method, after some standard calculations one can extract from the Hamiltonian (1), the following selfconsistent equation for the spinglass order parameter r I1 βr p/2 psg = 1 β r drexp p 2 0 I 0 βr p/2 Which can be compared with numerical molecular dynamics calculation of the polarization in the thermodynamic limit of the QSS regime considering T = T(N ) = T QSS The comparison is good and gives further support to the connection between glassy systems and the QSS regime of the HMF model

122 Schematic summary

123 Summary for the HMF model Summarizing, the Hmf model represents a paradigmatic model for nonextensive (long range interacting or finite ) systems, as for example self-gravitating objects, nuclear and atomic systems. Several dynamical metastable anomalies are present: negative specific heat, very slow dynamics, anomalous diffusion, power-law relaxation, fractal-like structures, vanishing of lyapunov exponents, ergodicity breaking and aging This metastable anomalous behavior can become stable if the infinite size limit is performed before the infinite time limit: the two limits do not commute. There are links to Tsallis generalized statistics. It is possible to treat the QSS regime in terms of a glassy phase. But one can find similarities with other non Hamiltonian systems.

124 Some references on the HMF model Antoni and Ruffo, Phys. Rev. E 52 (1995) 2361 Latora, Rapisarda, Ruffo, Phys. Rev. Lett. 80 (1998) 698. Physica D 131 (1999) 38 Phys. Rev. Lett. 83 (1999) 2104 Latora, Rapisarda, Tsallis, Phys. Rev. E 64 (2001) Dauxois, Latora, Rapisarda, Ruffo,Torcini, Lectures Notes in Phys., Springer 602 (2002). Tsallis, Rapisarda, Latora, Baldovin, Lectures Notes in Phys., Springer 602 (2002). Montemurro, Tamarit, Anteneodo Phys. Rev E (2003) Yamaguchi, Yamaguchi, Barrè, Barrè, Bouchet, Bouchet, Dauxois, Dauxois, Ruffo, Ruffo, Physica Physica A (2004) (2004) Pluchino, Pluchino, Latora, Latora, Rapisarda, Rapisarda, Physica Physica D (2004) (2004) 315; 315; Phys. Phys. Rev. Rev. E E (2004) (2004) , , Physica Physica A (2004) (2004) 60, 60, Cont. Cont. Mech. Mech. and and Therm. Therm (2004) (2004) 245, 245, cond-mat/ cond-mat/ and and cond-mat/ , cond-mat/ , Europhysics Europhysics News News (2005) (2005) Physica Physica A (2006) (2006) Most recent results Tamarit, Tamarit, Maglione, Maglione, Anteneodo, Anteneodo, Stariolo, Stariolo, Phys. Phys. Rev. Rev. E (2005) (2005) For the generalized version of the HMF model see: Anteneodo and Tsallis, Phys. Rev. Lett. 80 (1998) 5313; Tamarit and Anteneodo, Phys. Rev. Lett. 84 (20 Campa, Giansanti and Moroni, J. Phys. A 36 (2003) 6897.

125 124 Non Hamiltonian systems The Kuramoto Model Coupled Logistic maps with noise Soc models for earthquakes dynamics

126 125 Non Hamiltonian systems The Kuramoto model

127 The Kuramoto Model (Kuramoto 1975) Eqs. for the N coupled oscillators Order parameter mean field equation Phase transition Global synchronization Incoherent phase

128 112 Metastability in the Kuramoto Model A. Pluchino, A. Rapisarda, Physica A 365 (2006) 184

129 113 Metastability in the Kuramoto Model Also for the Kuramoto model metastability seems to diverge with the system size

130 Coupled maps on a lattice

131 102 Coupled Maps on a lattice We consider a model of Coupled Logistic Maps on a lattice, i.e. f( x) = sign( x) mod {2 [ α g ( x) + (1 - α) w], 1} (1) µ ε x = f( x ( i)) + [ f ( x ( i+ 1)) 2 f( x ( i)) + f( x ( i 1))] (2) t+ 1 t 2 t t t g = 1 µ µ Here w is a white noise term controlled by the parameter x 2 α [0,1] See the poster by S. RIZZO

132 Phase diagram of CML FRP= frozen random patterns BD = brownian motion of defect ε DT = defect turbulence PCI= pattern competition intermittency µ FDT= fully developed turbulence K. Kaneko, Simulating Physics with Coupled Map Lattices, World Scientific (1990)

133 103 Coupled Maps on a lattice We fix the control parameter at the edge of chaos <µ <1.8323

134 103 Dynamics of Coupled Maps on a lattice Time evolution with no noise (α=1) µ= ε=0.7

135 104 Dynamics of Coupled Maps on a lattice Time evolution with moderate noise (α=0.73)

136 105 Dynamics of Coupled Maps on a lattice Time evolution with strong noise (α=3)

137 105 Autocorrelation for Coupled Maps on a lattice

138 106 Fluctuations of Coupled Maps on a lattice One can consider the difference between two maps and study the fluctuations of this new variable. We considered for example ut ( ) = x(45) x(55) (a) The time behavior of u(t) can be very intermittent and one can apply a superstatistic approach (more details in the poster by S. RIZZO) u(t) t x 10 4

139 106 Coupled Maps on a lattice 1.5 (a) 50 (b) β(t) u(t) u(t) 0 u(t), β(t) And w is a white noise term regulated by the parameter t x t (a.u.) x 10 4

140 107 Pdfs of Coupled Maps on a lattice With no noise one gets pdfs very similar to the HMF case (black dots). Adding some noise one gets q-gaussians which can be explained by applying the superstatistics approach proposed by Beck and Cohen P(U) (b) CLNL (α=0.73) Gaussian q Gaussian q=1.22 CLNL (α=1.0) CLNL (α=0.3) U=(u <u>)/σ)

141 108 Coupled Maps on a lattice The pdf of β, the inverse variance of u(t), obeys a Lognormal distribution (a) CLNL data "χ 2 " "Lognormal" But a Gamma distribution is very closetoitand represents a good approximation P(β) β

142 SOC model with long-range interactions for earthquakes dynamics

143 Conclusions Summarizing, long-range interacting systems present several dynamical anomalies which pose severe problems to standard statistical mechanics and which can find a natural description in term of q-statistics

144 ALL THE THE TRUTHS PASS THROUGH THREE STAGES: FIRST, THEY ARE ARE CONSIDERED RIDICULOUS, SECOND, THEY ARE ARE VIOLENTLY ADVERSED, THIRD, THEY ARE ARE ACCEPTED AND CONSIDERED SELF- EVIDENT. A. A. SCHOPENHAUER SCHOPENHAUER