1. HYDRODYNAMIC MODES AND NONEQUILIBRIUM STEADY STATES Pierre GASPARD Brussels, Belgium J. R. Dorfman, College Park S. Tasaki, Tokyo T. Gilbert, Brussels INTRODUCTION: POLLICOTT-RUELLE RESONANCES DIFFUSION IN SPATIALLY PERIODIC SYSTEMS FRACTALITY OF THE RELAXATION MODES OF DIFFUSION NONEQUILIBRIUM STEADY STATES CONCLUSIONS

2. ERGODIC PROPERTIES AND BEYOND Ergodicity (Boltzmann 1871, 1884): time average = phase-space average stationary probability density representing the equilibrium statistical ensemble Mixing (Gibbs 1902): Spectrum of unitary time evolution: Ergodicity: The stationary probability density is unique: The eigenvalue is non-degenerate. Mixing: The non-degenerate eigenvalue is the only one on the real-frequency axis. The rest of the spectrum is continuous. Statistical average of a physical observable A(  ):

3. POLLICOTT-RUELLE RESONANCES group of time evolution:  ∞ < t <  ∞ t =  = ∫ A(x) p 0 (   t x) dx analytic continuation toward complex frequencies: L |   > = s  |   >, <   | L = s  <   | s =  i z 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. POLLICOTT-RUELLE RESONANCES Simple example: Hamiltonian of an inverted harmonic potential: Flow: Statistical average of an observable: Eigenvalue problem: Eigenvalues: Pollicott-Ruelle resonances: Eigenstates: breaking of time-reversal symmetry Schwartz-type distributions

5. TIME-REVERSAL SYMMETRY & ITS BREAKING Hamilton’s equations are time-reversal symmetric: If the phase-space curve is solution of Hamilton’s equation, then the time-reversed curve is also solution of Hamilton’s equation. Typically, the solution breaks the time-reversal symmetry: Liouville’s equation is also time-reversal symmetric. Equilibrium state: Relaxation modes: Nonequilibrium steady state: Spontaneous or explicit breaking of time-reversal symmetry

6. RELAXATION MODES OF DIFFUSION special solutions of Liouville’s equation: eigenvalue = dispersion relation of diffusion: wavenumber: k s k =  D k 2 + O(k 4 ) diffusion coefficient: Green-Kubo formula time space concentration wavelength = 2  /k generalized eigenstate of Liouvillian operator: spatial periodicity

7. MOLECULAR DYNAMICS SIMULATION OF DIFFUSION J. R. Dorfman, P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) 026110 Hamiltonian dynamics with periodic boundary conditions. N particles with a tracer particle moving on the whole lattice. The probability distribution of the tracer particle thus extends non-periodically over the whole lattice. lattice Fourier transform: first Brillouin zone of the lattice: initial probability density close to equilibrium: time evolution of the probability density: lattice distance travelled by the tracer particle: lattice vector:

8. DIFFUSIVE MODES IN SPATIALLY PERIODIC SYSTEMS The Perron-Frobenius operator is symmetric under the spatial translations {l} of the (crystal) lattice: common eigenstates: eigenstate = hydrodynamic 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

9. CUMULATIVE FUNCTION OF THE DIFFUSIVE MODES The eigenstate  k is a distribution which is smooth in W u but singular in W s. = breaking of time-reversal symmetry since W u =  (W s ) but W u ≠ W s. cumulative function: fractal curve in complex plane because  k is singular in W s S. Tasaki & P. Gaspard, J. Stat. Phys 81 (1995 935. P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506. eigenvalue = leading Pollicott-Ruelle resonance s k =  D k 2 + O(k 4 ) = lim t  ∞ (1/t) ln (Van Hove, 1954) s k is the continuation of the eigenvalue s 0 = 0 of the microcanonical equilibrium state and is not the next-to-leading Pollicott-Ruelle resonance.

10. MULTIBAKER MODEL OF DIFFUSION

11. HARD-DISK LORENTZ GAS Hamiltonian: H = p 2 /2m + elastic collisions Deterministic chaotic dynamics Time-reversal symmetric (Bunimovich & Sinai 1980) cumulative functions of the diffusive mode: F k (  ) = ∫ 0   k (x  ’ ) d  ’

12. 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 of the diffusive mode: F k (  ) = ∫ 0   k (x  ’ ) d  ’

13. HAUSDORFF DIMENSION OF THE DIFFUSIVE MODES Proof of the formula for the Hausdorff dimension cumulative function: polygonal approximation of the fractal curve: Hausdorff dimension: Ruelle topological pressure: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506. Hausdorff dimension: Generalization of the Bowen-Ruelle formula for the Hausdorff dimension of Julia sets.

14. DIFFUSION COEFFICIENT FROM THE HAUSDORFF DIMENSION low-wavenumber expansion: Hausdorff dimension: P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506. probability measure: average Lyapunov exponent: entropy per unit time: Hausdorff dimension: dispersion relation of diffusion: diffusion coefficient:

15. FRACTALITY OF THE DIFFUSIVE MODES Hausdorff dimension: 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

16. NONEQUILIBRIUM STEADY STATES Steady state of gradient g=(p +  p  )/L in x: p noneq (  ) = (p + + p  )/2 + g [ x(  ) + ∫ 0  T(  ) v x (  t  ) dt ] p noneq (  ) = (p + + p  )/2 + g [ x(  ) + x(   T(  )  )  x(  ) ] p noneq (  ) = (p + + p  )/2 + (p +  p  ) x(   T(  )  ) /L p noneq (  ) = p ± for x(   T(  )  )  ±L/2  g. (  ) = g [ x(  ) + ∫ 0  ∞ v x (  t  ) dt ] =  i g ∂ k  k (  )| k=0 Green-Kubo formula: D = ∫ 0 ∞ eq dt Fick’s law: neq = g [ eq + ∫ 0  ∞ eq dt ] =  D g p+ p+ p p t x

17. SINGULAR CHARACTER OF THE NONEQUILIBRIUM STEADY STATES cumulative functions T g (  ) = ∫ 0   g (   ’ ) d  ’ hard-disk Lorentz gas Yukawa-potential Lorentz gas (generalized Takagi functions)

18. CONCLUSIONS Breaking of time-reversal symmetry in the statistical description Nonequilibrium transients: Spontaneous breaking of time-reversal symmetry for the solutions of Liouville’s equation corresponding to the Pollicott-Ruelle resonances. The associated eigenstates are singular distributions with fractal cumulative functions. Nonequilibrium modes of diffusion: relaxation rate  s k, Pollicott-Ruelle resonance reminiscent of the escape-rate formula: (1/2) wavelength = L =  /k

19. CONCLUSIONS (cont’d) Escape-rate formalism: nonequilibrium transients fractal repeller http://homepages.ulb.ac.be/ ~ gaspard diffusion D :  (1990) viscosity  :  (1995) Hamiltonian systems: Liouville theorem: thermostated systems: no Liouville theorem volume contraction rate: Pesin’s identity on the attractor:

20. CONCLUSIONS (cont’d) http://homepages.ulb.ac.be/ ~ gaspard Nonequilibrium steady states: Explicit breaking of the time-reversal symmetry by the nonequilibrium boundary conditions imposing net currents through the system. For reservoirs at finite distance of each other, the invariant probability measure is still continuous with respect to the Lebesgue measure, but it differs considerably from an equilibrium measure. Considering the invariant probability measure as the solution of Liouville’s equation with nonequilibrium boundary conditions imposed at the contacts with the reservoirs, the invariant probability density takes its value at the reservoir from which the trajectory is coming. In the large-system limit at constant gradient, the phase-space regions coming from either one reservoir or the other alternate more and more densely so that the invariant probability measure soon becomes singular with respect to the Lebesgue measure.