1. Phase-space instability for particle systems in equilibrium and stationary nonequilibrium states Harald A. Posch Institute for Experimental Physics, University of Vienna Ch. Forster, R. Hirschl, J. van Meel, Lj. Milanovic, E.Zabey Ch. Dellago, Wm. G Hoover, J.-P. Eckmann, W. Thirring, H. van Beijeren Dynamical Systems and Statistical Mechanics, LMS Durham Symposium July 3 - 13, 2006

2. Outline Localized and delocalized Lyapunov modes Translational and rotational degrees of freedom Nonlinear response theory and computer thermostats Stationary nonequilibrium states Phase-space fractals for stochastically driven heat flows and Brownian motion Thermodynamic instability: Negative heat capacity in confined geometries

3. Lyapunov instability in phase space

4. Perturbations in tangent space

5. Lyapunov spectra for soft and hard disks Left: 36 soft disks, rho = 1, T = 0.67 Right: 400 disks, rho = 0.4, T = 1

6. Properties of Lyapunov spectra Localization Lyapunov modes

7. Localization

8. 102.400 soft disks Red: Strong particle contribution to the perturbation associated with the maximum Lyaounov exponent, Blue: No particle contribution to the maximum exponent. Wm.G.Hoover, K.Boerker, HAP, Phys.Rev. E 57, 3911 (1998)

9. Localization measure at low density 0.2 T. Taniguchi, G. Morriss

10. N-dependence of localization measure

11. N = 780 hard disks,  = 0.8, A = 0.8, periodic boundaries

12. N = 780

13. Hard disks, N = 780,  = 0.8, A = 0.867 Transverse mode T(1,1) for l = 1546

14. Continuous symmetries and vanishing Lyapunov exponents

15. Hard disks: Generators of symmetry transformations

16. N = 780

17. Classification of modes

18. Classification for hard disks Rectangular box, periodic boundaries

19. Hard disks: Transverse modes, N = 1024,  = 0.7, A = 1

20. Lyapunov modes as vector fields

21. Dispersion relation N = 780 hard disks,  = 0.8, A = 0.867

22. Shape of Lyapunov spectra

23. Time evolution of Fourier spectra

24. Propagation of longitudinal modes N = 200, density = 0.7, L x = 238, L y = 1.2

25. LP(1,0), N = 780 hard disks,  = 0.8, A = 0.867 reflecting boundaries

26. LP(1,1), N=780 hard disks,  =0.8, A=0.867 reflecting boundaries

27. N = 375

28. Soft disks N = 375 WCA particles,  = 0.4; A = 0.6

29. Power spectra of perturbation vectors

30. Density dependence: hard and soft disks

31. Rough Hard Disks and Spheres Hard disks:

32. Rough particles: collision map

33. N = 400,  = 0.7, A = 1

35. Convergence :  = 0.5, A = 1, I = 0.1

37. Rough hard disks N = 400

38. Localization, N = 400, I = 0.1, density = 0.7

39. Summary I: Equilibrium systems with short-range forces Lyapunov modes: formally similar to the modes of fluctuating hydrodynamics Broken continuous symmetries give rise to modes Unbiased mode decomposition Soft potentials require full phase space of a particle Hard dumbbells,...... Applications to phase transitions, particles in narrow channels, translation-rotation coupling,......

40. Response theory

41. Time-reversible thermostats

42. Isokinetic thermostat

43. Stationary States: Externally-driven Lorentz gas

44. B.L.Holian, W.G.Hoover, HAP, Phys.Rev.Lett. 59, 10 (1987), HAP, Wm. G. Hoover, Phys. Rev A38, 473 (1988)

45. Externally-driven Lorentz gas

47. Frenkel-Kontorova conductivity, 1d

48. Stationary nonequilibrium states II: The case for dynamical thermostats qpzx-oscillator

49. Stationary Heat Flow on a Nonlinear Lattice Nose-Hoover Thermostats HAP and Wm.G.Hoover, Physica D187, 281 (2004)

50. Control of 2nd and 4th moment

51. Extensivity of the dimensionality reduction

52. Stochastic  4 lattice model

53. Temperature field, Lyapunov spectrum

54. Projection onto Newtonian subspace

55. Summary II Fractal phase-space probability is fingerprint of Second Law Insensitive to thermostat: dynamical or stochastic Sum of the Lyapunov exponents is related to transport coefficient Kinetic theory for low densities and fields (Dorfman, van Beijeren,..... )

56. Unstable Systems

57. Negative heat capacity

58. Stability of “stars”

59. B: Heating of cluster core; C: Cooling at boundary HAP and W. Thirring, Phys. Rev. Lett 95, 251101 (2005)

60. Jumping board model (PRL 95, 251101 (2005)

61. Jumping board model

65. N = 1000 particles

67. Coupled systems

68. Uncoupled systems

70. Coupled systems, N(P) = N(N) = 1

71. Summary III Systems with c<0: more-than-exponential energy growth of phase volume Jumping-board model: gas of interacting particles in specially-confined gravitational box Problems with ergodicity

72. Self-gravitating system: Sheet model

73. Chaos in the gravitational sheet model

74. Sheet model: non-ergodicity

75. Family of gen. sheet models: Hidden symmetry? Lj. Milanovic, HAP abd W. Thirring, Mol. Phys. 2006

76. Gravitational particles confined to a box Case A: E = const

77. Case B: energy E = const ; angular momentum L = 0 Case C: energy E = const ; linear momentum P = 0

78. 3 particles in external potential

79. 3 particles in reflecting box

80. Summary IV: Gravitational collapse and ergodicity Sheet model: Lack of ergodicity for thirty- particle system Symmetric dependence on parameter Hint of additional integral of the motion Stabilization by additional conserved quantities