Kyoto University Masa-aki SAKAGAMI

1. Non-equilibrium evolution of Long-range Interacting Systems and Polytrope Kyoto University Masa-aki SAKAGAMI Collaboration with T.Kaneyama　（Kyoto U.） A.Taruya(RESCEU, Tokyo U.)

2. Outlines of this talk Two systems with long-range interaction (1) Self-gravitating N-body system Taruya and Sakagami, PRL90(2003)181101, Physica A322(2003)285 A short review of our previous work. Extremal of generalized entropy by Tsallis polytrope generalization of B-G dist. It well describes non-equilibrium evolution of the system. (2) 1-dim. Hamiltonian Mean-Field model (HMF model) We show another example where non-equilibrium evolution of the system moves along a sequence of polytrope. Thermodynamical instability implies a superposition of polytrope. (negative specific heat)

3. Antonov 1962 Lynden-Bell & Wood 1968 Long-term (thermodynamic) instability Self-gravitating N-body System A System with N Particles (stars) (N>>1) Particles interact with Newtonian gravity each other Typical example of a system under long-range force Key word Negative Specific Heat Gravothermal instability

4. 2 l= – reE/GM Boltzmann-Gibbs entropy Extremization Maxwell-Boltzmann distribution = 0.335 Large re = 709 D= rc/re Maxwell-Boltzmann distribution as Equilibrium Adiabatic wall (perfectly reflecting boundary)

5. q-entropy BG limit q→1 A naïve generalization of BG statistics Thermostatistical treatment by generalized entropy Tsallis, J.Stat.Phys.52 (1988) 479 One-particle distribution function identified with escort distribution Power-law distribution

6. Energy-density contrastrelation for stellar polytrope n=6 Polytropic equation of state (e.g., Binney & Tremaine 1987) Polytrope index n→∞ BG limit

7. n=6 stable unstable unstable state appears at n>5 (gravothermal instability) Stellar polytrope as quasi-equilibrium state Energy-density contrast relation for stellar polytrope

8. Survey results of group (A) The evolutionary track keeps the directionincreasing the polytrope index “n”. Once exceeding the critical value “Dcrit“, central density rapidly increases toward the core collapse.

9. Initial cond: Stellar polytrope (n=3,D=10 ) 4 Density profile One-particle distribution function Fitting to stellar polytropes is quite good until t ~ 30 trh,i. Run n3A : N-body simulation

10. Stellar polytropes are not stable in timescale of two-body relaxation. However, focusing on their transients, we found : Quasi-equilibrium property Transient states approximately follow a sequence of stellar polytropes with gradually changing polytrope index “n”. Quasi-attractive behavior Even starting from non-polytropic states, system soon settles into a sequence of stellar polytropes. Overview of the results in sel-gravitating system

11. Application of generalized entropy and polytrope to another long-range interacting system, 1-dimensional Hamiltonian mean-field model 1D-HMF model Antoni and Ruffo, PRE 52(1995)2361

12. Antoni and Ruffo, PRE 52(1995)2361

13. q-entropy BG limit q→1 A naïve generalization of BG statistics Thermostatistical treatment by generalized entropy Tsallis, J.Stat.Phys.52 (1988) 479 One-particle distribution function identified with escort distribution Power-law distribution (polytrope)

14. Physical quantities by 1-particle distribution number energy magnetization potential

15. Polytropic distribution function Chavanis and Campa, Eur.Phys.J. B76(2010)581 Taruya and Sakagami, unpublished polytrope index BG limit q→1 n →∞ For given U and n, this eq. self-consistently determines magnitization M. As for defenition of Tphys, Abe, Phys.Lett. A281(2001)126

16. Antoni and Ruffo, PRE 52(1995)2361 Thermal equilibrium,ＢＧ　Ｌｉｍｉｔ n→∞ inhomogeneous state homogeneous state generalized to seqences of polytrope, describing non-eqilibrium (?) n=1000 n=10 n=4 n=2 n=0.5 n=1000 n=10 n=4 n=2 n=0.5

17. Time evolution of M and Tphys (1) N=10000, 10 samplesof simulations Initial cond. : Water Bag Spatially homogeneous

18. Time evolution of M and Tphys (2) N=10000, 10 samplesof simulations Initial cond. : Water Bag Spatially inhomogeneous

19. Three stages of evolution of M and Tphys N=10000, 10 samplesof simulations nearly equilibrium transient state quasi-stationary state (qss) The evolution over three stages are totally well described by sequences of polytropes.

20. Tphys – U relation at equilibrium N=1000, single sample Long term behavior early stage behavior Theoretical prediction for thermal equilibrium.

21. Tphys – U relation at qss N=1000, single sample Long term behavior early stage behavior Tphys at early stage of qss

22. Tphys – U relation at qss N=1000, single sample Long term behavior polytrope n=0.5 early stage behavior Tphys at early stage of qss are explained by polytrope with n=0.5.

23. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations f by simulation prediction by polytrope

24. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations Even in qss, polytrope index n and distribution f change.

25. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

26. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

27. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

28. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

29. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

30. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

31. Evolutionary track on the polytrope sequence N=10000, 10 samplesof simulations

32. Failure of single polytrope description due to Thermodynamical instability (negative specific heat).

33. n=6 stable unstable unstable state appears at n>5 (gravothermal instability) Stellar polytrope as quasi-equilibrium state Energy-density contrast relation for stellar polytrope

34. self-gravitating system Sel-similar core-collapse in Fokker-Planck eq. Halo could not catch up with core collapse. Heat flow core halo When self-similar core collapse takes place, polytrope could not fit distribution function. H.Cohn Ap.J 242 p.765 (1980)

35. fitting of self-similar sol. with double polytrope Black dots: Numerical Self-Similar sol. by Heggie and Stevenson Magenta lines: fitting by double polytrope

36. Thermodynamical Instability due to negative specific heat After a short time, single polytrope fails to describe the simulated distribution. We prepare the initial state as Polytrope with U=0.6, n=1.

37. A description by double polytropes might work. double polytrope single polytrope

38. A description by double polytropes might work. double polytrope parameters coexistence conditions (preliminary)

39. Summary and Discussion • Polytrope(Extremal of Generalized Entropy) describes evolution along quasi-stationary and transient statesto thermal equilibrium Self-gravitating system, 1D-HMF (Long-range interaction) (2) Break down of single polytrope description (due to negative specific heat) implies superposition of polytropes.

40. How to determine polytropic index n.

43. Polytropeによる準定常状態の記述の限界 self-similar evolution Fokker-Planck eq.によるCore-Collapseの解析 Heat flow core halo self-similar core collapse が 始まると polytrope で fit できない H.Cohn Ap.J 242 p.765 (1980)

44. Self-similar sol. of F-P eq. Heggie and Stevenson, MN 230 p.223 (1988) power law envelope isothermal core

45. fitting of self-similar sol. with double polytrope Black dots: Numerical Self-Similar sol. by Heggie and Stevenson Magenta lines: fitting by double polytrope

46. 2D ＨＭＦモデル Antoni&Torcini PRE 57(1998) R6233 Antoni, Ruffo&Torcini, PRE 66(2003) 025103R Interaction by Mean-field：　Long-range interacting system 2D HMF have the effect of energy transfer due to 2-body scattering process. Negative specific heat in some range of energy

48. T-U curveBoltzmann case Thermal equilibrium T Magnetization Transient st polytrope ? U (energy) t / N Vlasov phase ? dist. func. dist. func.

49. Initial : polytrope U=1.9 Negative specific heat dist. func. Magnetization t (logarithmic)

50. Initial: polytrope U=1.7 positive specific heat dist. func. Magnetization t (logarithmic)