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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
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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
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Lyapunov instability in phase space
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Perturbations in tangent space
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Lyapunov spectra for soft and hard disks Left: 36 soft disks, rho = 1, T = 0.67 Right: 400 disks, rho = 0.4, T = 1
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Properties of Lyapunov spectra Localization Lyapunov modes
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Localization
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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)
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Localization measure at low density 0.2 T. Taniguchi, G. Morriss
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N-dependence of localization measure
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N = 780 hard disks, = 0.8, A = 0.8, periodic boundaries
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N = 780
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Hard disks, N = 780, = 0.8, A = 0.867 Transverse mode T(1,1) for l = 1546
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Continuous symmetries and vanishing Lyapunov exponents
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Hard disks: Generators of symmetry transformations
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N = 780
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Classification of modes
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Classification for hard disks Rectangular box, periodic boundaries
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Hard disks: Transverse modes, N = 1024, = 0.7, A = 1
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Lyapunov modes as vector fields
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Dispersion relation N = 780 hard disks, = 0.8, A = 0.867
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Shape of Lyapunov spectra
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Time evolution of Fourier spectra
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Propagation of longitudinal modes N = 200, density = 0.7, L x = 238, L y = 1.2
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LP(1,0), N = 780 hard disks, = 0.8, A = 0.867 reflecting boundaries
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LP(1,1), N=780 hard disks, =0.8, A=0.867 reflecting boundaries
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N = 375
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Soft disks N = 375 WCA particles, = 0.4; A = 0.6
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Power spectra of perturbation vectors
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Density dependence: hard and soft disks
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Rough Hard Disks and Spheres Hard disks:
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Rough particles: collision map
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N = 400, = 0.7, A = 1
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Convergence : = 0.5, A = 1, I = 0.1
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Rough hard disks N = 400
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Localization, N = 400, I = 0.1, density = 0.7
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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,......
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Response theory
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Time-reversible thermostats
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Isokinetic thermostat
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Stationary States: Externally-driven Lorentz gas
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B.L.Holian, W.G.Hoover, HAP, Phys.Rev.Lett. 59, 10 (1987), HAP, Wm. G. Hoover, Phys. Rev A38, 473 (1988)
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Externally-driven Lorentz gas
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Frenkel-Kontorova conductivity, 1d
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Stationary nonequilibrium states II: The case for dynamical thermostats qpzx-oscillator
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Stationary Heat Flow on a Nonlinear Lattice Nose-Hoover Thermostats HAP and Wm.G.Hoover, Physica D187, 281 (2004)
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Control of 2nd and 4th moment
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Extensivity of the dimensionality reduction
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Stochastic 4 lattice model
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Temperature field, Lyapunov spectrum
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Projection onto Newtonian subspace
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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,..... )
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Unstable Systems
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Negative heat capacity
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Stability of “stars”
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B: Heating of cluster core; C: Cooling at boundary HAP and W. Thirring, Phys. Rev. Lett 95, 251101 (2005)
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Jumping board model (PRL 95, 251101 (2005)
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Jumping board model
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N = 1000 particles
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Coupled systems
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Uncoupled systems
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Coupled systems, N(P) = N(N) = 1
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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
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Self-gravitating system: Sheet model
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Chaos in the gravitational sheet model
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Sheet model: non-ergodicity
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Family of gen. sheet models: Hidden symmetry? Lj. Milanovic, HAP abd W. Thirring, Mol. Phys. 2006
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Gravitational particles confined to a box Case A: E = const
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Case B: energy E = const ; angular momentum L = 0 Case C: energy E = const ; linear momentum P = 0
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3 particles in external potential
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3 particles in reflecting box
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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
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