POLLICOTTRUELLE RESONANCES FRACTALS AND NONEQUILIBRIUM MODES OF RELAXATION
POLLICOTT-RUELLE RESONANCES, FRACTALS, AND NONEQUILIBRIUM MODES OF RELAXATION Pierre GASPARD Brussels, Belgium J. R. Dorfman, College Park G. Nicolis, Brussels S. Tasaki, Tokyo T. Gilbert, Brussels D. Andrieux, Brussels • INTRODUCTION • TIME-REVERSAL SYMMETRY BREAKING • POLLICOTT-RUELLE RESONANCES • NONEQUILIBRIUM MODES OF RELAXATION: DIFFUSION • ENTROPY PRODUCTION & TIME ASYMMETRY IN DYNAMICAL RANDOMNESS OF NONEQUILIBRIUM FLUCTUATIONS • CONCLUSIONS
BREAKING OF TIME-REVERSAL SYMMETRY Q(r, p) = (r, -p) Newton’s equation of mechanics is time-reversal symmetric if the Hamiltonian H is even in the momenta. Liouville equation of statistical mechanics, ruling the time evolution of the probability density p is also time-reversal symmetric. The solution of an equation may have a lower symmetry than the equation itself (spontaneous symmetry breaking). Typical Newtonian trajectories T are different from their time-reversal image Q T : QT≠T Irreversible behavior is obtained by weighting differently the trajectories T and their time-reversal image Q T with a probability measure. Pollicott-Ruelle resonance (Axiom-A systems): (Pollicott 1985, Ruelle 1986) = generalized eigenvalues sa of Liouville’s equation associated with decaying eigenstates singular in the stable directions Ws but smooth in the unstable directions Wu :
POLLICOTT-RUELLE RESONANCES group of time evolution: -∞ < t < +∞ statistical average of the observable A <A>t = <A|exp(L t)| p 0 > = ∫ A(G) p 0(F-t G) d. G analytic continuation toward complex frequencies: L |Ya> = sa |Ya> , < Xa | L = s a < Xa | • forward semigroup ( 0 < t < +∞): asymptotic expansion around t = +∞ : <A>t = <A|exp(L t)| p 0> ≈ ∑a <A|Ya> exp(sa t) <Xa| p 0> + (Jordan blocks) • backward semigroup (-∞ < t < 0): asymptotic expansion around t = -∞ : <A>t = <A|exp(L t)| p 0> ≈ ∑a <A|Ya°Q> exp(-sa t) <Xa°Q| p 0> + (Jordan blocks)
DIFFUSION IN SPATIALLY PERIODIC SYSTEMS Invariance of the Perron-Frobenius operator under a discrete Abelian subgroup of spatial translations {a}: common eigenstates: eigenstate = nonequilibrium mode of diffusion: eigenvalue = Pollicott-Ruelle resonance = dispersion relation of diffusion: wavenumber: k sk = lim t®∞ (1/t) ln <exp[ i k • (rt - r 0)]> = - D k 2 + O(k 4) diffusion coefficient: Green-Kubo formula concentration (Van Hove, 1954) space wavelength = 2 p/k time
FRACTALITY OF THE NONEQUILIBRIUM MODES OF DIFFUSION The eigenstate Yk is a distribution which is smooth in Wu but singular in Ws. cumulative function: fractal curve in complex plane of Hausdorff dimension DH Ruelle topological pressure: Hausdorff dimension: diffusion coefficient: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506.
MULTIBAKER MODEL OF DIFFUSION
PERIODIC HARD-DISK LORENTZ GAS • Hamiltonian: H = p 2/2 m + elastic collisions • Deterministic chaotic dynamics • Time-reversal symmetric (Bunimovich & Sinai 1980) cumulative functions Fk (q) = ∫ 0 q Yk(Gq’) dq’
PERIODIC YUKAWA-POTENTIAL LORENTZ GAS • Hamiltonian: H = p 2/2 m - Si exp(-ari)/ri • Deterministic chaotic dynamics • Time-reversal symmetric (Knauf 1989) cumulative functions Fk (q) = ∫ 0 q Yk(Gq’) dq’
DIFFUSION IN A GEODESIC FLOW ON A NEGATIVE CURVATURE SURFACE non-compact manifold in the Poincaré disk D: spatially periodic extension of the octogon cumulative functions Fk (q) = ∫ 0 q Yk(Gq’) dq’ infinite number of handles
FRACTALITY OF THE NONEQUILIBRIUM MODES OF DIFFUSION Hausdorff dimension of the diffusive mode: Yukawa-potential Lorentz gas hard-disk Lorentz gas -Re sk large-deviation dynamical relationship: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506.
DYNAMICAL RANDOMNESS Partition P of the phase space into cells w representing the states of the system observed with a certain resolution. Stroboscopic observation: history or path of a system: sequence of states w 0 w 1 w 2 … wn-1 at successive times t = n t probability of such a path: P(w 0 w 1 w 2 … wn-1 ) ~ exp[ -h(P) t n ] entropy per unit time: h(P) (Shannon, Mc. Millan, Breiman) h(P) is a measure of dynamical randomness (temporal disorder) of the process: h(P) = ln 2 for a coin tossing random process. The dynamical randomness of all the different random and stochastic processes can be characterized in terms of their entropy per unit time (Gaspard & Wang, 1993). Deterministic chaotic systems: Kolmogorov-Sinai entropy per unit time: h. KS = Sup. P h(P) Pesin theorem for closed systems:
DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS nonequilibrium steady state: P (w 0 w 1 w 2 … wn-1) ≠ P (wn-1 … w 2 w 1 w 0) If the probability of a typical path decays as P(w) = P(w 0 w 1 w 2 … wn-1) ~ exp( -h Dt n ) the probability of the time-reversed path decays as P(w. R) = P(wn-1 … w 2 w 1 w 0) ~ exp( -h. R Dt n ) with h. R ≠ h entropy per unit time: h = lim n®∞ (-1/n. Dt) ∑w P(w) ln P(w) = lim n®∞ (-1/n. Dt) ∑w P(w. R) ln P(w. R) time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 h. R = lim n®∞ (-1/n. Dt) ∑w P(w) ln P(w. R) = lim n®∞ (-1/n. Dt) ∑w P(w. R) ln P(w) The time-reversed entropy per unit time characterizes the dynamical randomness (temporal disorder) of the time-reversed paths.
THERMODYNAMIC ENTROPY PRODUCTION nonequilibrium steady state: thermodynamic entropy production: dynamical randomness of time-reversed paths h. R entropy production dynamical randomness of paths h P. Gaspard, J. Stat. Phys. 117 (2004) 599 If the probability of a typical path decays as the probability of the corresponding time-reversed path decays faster as The thermodynamic entropy production is due to a time asymmetry in dynamical randomness.
ILLUSTRATIVE EXAMPLES discrete-time Markov chains: Kolmogorov-Sinai entropy per unit time: time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 entropy production: Markov chain with 2 states {0, 1}: Markov chain with 3 states {1, 2, 3}: 123123123123123 122322113333311222112331221 equilibrium
CONCLUSIONS • Breaking of time-reversal symmetry in the statistical description • Large-deviation dynamical relationships Nonequilibrium transients: Spontaneous breaking of time-reversal symmetry for the solutions of Liouville’s equation corresponding to the Pollicott-Ruelle resonances. Escape rate formalism: escape rate g, Pollicott-Ruelle resonance diffusion D : D (p / L )2 ≈ g = (∑i li+ - h. KS )L viscosity h : h (p / c )2 ≈ g = (∑i li+ - h. KS )c wavenumber k = p / L (1990) (1995) Nonequilibrium modes of diffusion: relaxation rate -sk, Pollicott-Ruelle resonance D k 2 ≈ - Re sk = l(DH) - h. KS(DH)/ DH (2001) Nonequilibrium steady states: The flux boundary conditions explicitly break the time-reversal symmetry. fluctuation theorem: z = R(-z) - R(z) (1993, 1995, 1998) entropy production: ____ = h. R(P) - h(P) d S(P) (2004) i k. B dt
CONCLUSIONS (cont’d) d S(P) i ____ = h. R(P) - h(P) k. B dt thermodynamic entropy production = temporal disorder of time-reversed paths - temporal disorder of paths = time asymmetry in dynamical randomness Principle of temporal ordering as a corollary of the second law: In nonequilibrium steady states, the typical paths are more ordered in time than the corresponding time-reversed paths. Boltzmann’s interpretation of the second law: Out of equilibrium, the spatial disorder increases in time. http: //homepages. ulb. ac. be/~gaspard
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