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