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