Phase-space instability for particle systems in equilibrium and stationary nonequilibrium states

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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 disksRectangular 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, Lx 238, Ly 1.2
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LP(1,0), N 780 hard disks, ? 0.8, A
0.867reflecting boundaries
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LP(1,1), N780 hard disks, ?0.8, A0.867
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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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 IIThe case for
dynamical thermostats

• qpzx-oscillator

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Stationary Heat Flow on a Nonlinear
LatticeNose-Hoover ThermostatsHAP 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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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 clt0 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 IVGravitational 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