1. 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

2. BREAKING OF TIME-REVERSAL SYMMETRY (r,p) = (r, p) Newtons 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 T : T Irreversible behavior is obtained by weighting differently the trajectories T and their time-reversal image T with a probability measure. Pollicott-Ruelle resonance (Axiom-A systems): (Pollicott 1985, Ruelle 1986) = generalized eigenvalues s of Liouvilles equation associated with decaying eigenstates singular in the stable directions W s but smooth in the unstable directions W u :

3. POLLICOTT-RUELLE RESONANCES group of time evolution: < t < statistical average of the observable A t = = A( ) p 0 ( t ) d analytic continuation toward complex frequencies: L | > = s | >, < | L = s < | forward semigroup ( 0 < t < ): asymptotic expansion around t = : t = exp(s t) + (Jordan blocks) backward semigroup ( < t < ): asymptotic expansion around t = : t = exp( s t) + (Jordan blocks)

4. 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: (Van Hove, 1954) wavenumber: k s k = lim t (1/t) ln = D k 2 + O(k 4 ) diffusion coefficient: Green-Kubo formula time space concentration wavelength = 2 /k

5. FRACTALITY OF THE NONEQUILIBRIUM MODES OF DIFFUSION The eigenstate k is a distribution which is smooth in W u but singular in W s. cumulative function: fractal curve in complex plane of Hausdorff dimension D H Ruelle topological pressure: Hausdorff dimension: diffusion coefficient: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506.

6. MULTIBAKER MODEL OF DIFFUSION

7. PERIODIC HARD-DISK LORENTZ GAS Hamiltonian: H = p 2 /2m + elastic collisions Deterministic chaotic dynamics Time-reversal symmetric (Bunimovich & Sinai 1980) cumulative functions F k ( ) = 0 k ( ) d

8. PERIODIC YUKAWA-POTENTIAL LORENTZ GAS Hamiltonian: H = p 2 /2m i exp( ar i )/r i Deterministic chaotic dynamics Time-reversal symmetric (Knauf 1989) cumulative functions F k ( ) = 0 k ( ) d

9. DIFFUSION IN A GEODESIC FLOW ON A NEGATIVE CURVATURE SURFACE non-compact manifold in the Poincaré disk D: spatially periodic extension of the octogon infinite number of handles cumulative functions F k ( ) = 0 k ( ) d

10. FRACTALITY OF THE NONEQUILIBRIUM MODES OF DIFFUSION Hausdorff dimension of the diffusive mode: large-deviation dynamical relationship: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506. hard-disk Lorentz gas Yukawa-potential Lorentz gas Re s k

11. DYNAMICAL RANDOMNESS Partition P of the phase space into cells representing the states of the system observed with a certain resolution. Stroboscopic observation: history or path of a system: sequence of states 0 1 2 … n 1 at successive times t = n probability of such a path: (Shannon, McMillan, Breiman) P( 0 1 2 … n 1 ) ~ exp[ h( P ) n ] entropy per unit time: h( P ) 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:

12. DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS nonequilibrium steady state: P ( 0 1 2 … n 1 ) P ( n 1 … 2 1 0 ) If the probability of a typical path decays as P( ) = P( 0 1 2 … n 1 ) ~ exp( h t n ) the probability of the time-reversed path decays as P( R ) = P( n 1 … 2 1 0 ) ~ exp( h R t n ) with h R h entropy per unit time: h = lim n ( 1/n t) P( ) ln P( ) = lim n ( 1/n t) P( R ) ln P( R ) time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 h R = lim n ( 1/n t) P( ) ln P( R ) = lim n ( 1/n t) P( R ) ln P( ) The time-reversed entropy per unit time characterizes the dynamical randomness (temporal disorder) of the time-reversed paths.

13. THERMODYNAMIC ENTROPY PRODUCTION nonequilibrium steady state: thermodynamic entropy production: 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. entropy production dynamical randomness of time-reversed paths h R dynamical randomness of paths h P. Gaspard, J. Stat. Phys. 117 (2004) 599

14. discrete-time Markov chains: ILLUSTRATIVE EXAMPLES time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 Kolmogorov-Sinai entropy per unit time: entropy production: Markov chain with 2 states {0,1}: Markov chain with 3 states {1,2,3}: 123123123123123123123123123122322113333311222112331221 equilibrium

15. 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 Liouvilles equation corresponding to the Pollicott-Ruelle resonances. Escape rate formalism: escape rate, Pollicott-Ruelle resonance diffusion D : D ( / L ) 2 = ( i i + h KS ) L wavenumber k = / L (1990) viscosity : ( / ) 2 = ( i i + h KS ) (1995) Nonequilibrium modes of diffusion: relaxation rate s k, Pollicott-Ruelle resonance D k 2 Re s k = (D H ) h KS (D H )/ D H (2001) Nonequilibrium steady states: The flux boundary conditions explicitly break the time-reversal symmetry. fluctuation theorem: = R( ) R( ) (1993, 1995, 1998) entropy production: ________ = h R ( P ) h( P ) (2004) d i S( P ) k B dt

16. CONCLUSIONS (contd) 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. Boltzmanns interpretation of the second law: Out of equilibrium, the spatial disorder increases in time. http://homepages.ulb.ac.be/ ~ gaspard thermodynamic entropy production = temporal disorder of time-reversed paths temporal disorder of paths = time asymmetry in dynamical randomness ________ = h R ( P ) h( P ) d i S( P ) k B dt