POLLICOTT-RUELLE RESONANCES, FRACTALS, AND NONEQUILIBRIUM MODES OF RELAXATION Pierre GASPARD Brussels, Belgium J. R. Dorfman, College Park G. Nicolis,

POLLICOTT-RUELLE RESONANCES, FRACTALS,AND NONEQUILIBRIUM MODES OF RELAXATION

Pierre GASPARDBrussels, 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 (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 T :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 Liouville’s equation associated with

decaying eigenstates singular in the stable directions Ws

but smooth in the unstable directions Wu :

ˆ L s with s complex

p

t H, p ˆ L p

POLLICOTT-RUELLE RESONANCESgroup of time evolution: ∞ < t < ∞ statistical average of the observable A <A>t =<A|exp(L t)| p0 > = ∫ A() p0(t ) d

analytic continuation toward complex frequencies: L |> = s|> , < | L = s < |

• forward semigroup ( 0 < t < ∞): asymptotic expansion around t = ∞ :

<A>t = <A|exp(L t)| p0> ≈ ∑<A|> exp(s t) <| p0> + (Jordan blocks)

• backward semigroup (∞ < t < ): asymptotic expansion around t = ∞ :

<A>t = <A|exp(L t)| p0> ≈ ∑<A|°> exp(s t) <°| p0> + (Jordan blocks)

DIFFUSION IN SPATIALLY PERIODIC SYSTEMSInvariance of the Perron-Frobenius operator under a discrete Abelian subgroup of spatial translations {a}:

common eigenstates:

eigenstate = nonequilibrium mode of diffusion:

ˆ P t exp( ˆ L t)

ˆ P t , ˆ T a 0

ˆ P t k exp(sk t) k

ˆ T a k exp(ik a) k

k

eigenvalue = Pollicott-Ruelle resonance = dispersion relation of diffusion:

(Van Hove, 1954) wavenumber: k

sk = lim t∞ (1/t) ln <exp[ i k•(rt r0)]>

= D k2 + O(k4)

diffusion coefficient: Green-Kubo formula

time

space

c

once

ntra

tion

wavelength = 2/k

D vx (0)vx (t) 0

dt

FRACTALITY OF THE NONEQUILIBRIUM MODES OF DIFFUSION

The eigenstatek 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.

Fk () limt

d ' exp[ik (rt r0) ']0

d ' exp[ik (rt r0) ']0

2

P(DH) DH Re sk

P() limt

1

t ln t

1 h() ()

DH(k) 1D

k2 O(k4 )

D limk 0

DH(k) 1

k2

Re sk Dk2 O(k4 )

MULTIBAKER MODEL OF DIFFUSION

(l,x,y) l 1,2x,

y

2

, 0 x

1

2

l 1,2x 1,y 1

2

,

1

2 x 1

singular diffusive modes k :

cumulative function

Fk () k ( ')0

d '

(de Rham functions)

. . . . . .

ll-1 l+1. . . . . .

Hausdorff dimension :

DH ln2

ln(2cosk)

PERIODIC HARD-DISK LORENTZ GAS

• Hamiltonian: H = p2/2m + elastic collisions• Deterministic chaotic dynamics• Time-reversal symmetric (Bunimovich & Sinai 1980)

cumulative functions Fk () = ∫0 k(’) d’

PERIODIC YUKAWA-POTENTIAL LORENTZ GAS

• Hamiltonian:

H = p2/2m i exp(ari)/ri

• Deterministic chaotic dynamics• Time-reversal symmetric (Knauf 1989)

cumulative functions Fk () = ∫0 k(’) d’

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 Fk () = ∫0 k(’) d’

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.

Dk2 Re sk P(DH)

DH

(DH) h(DH)

DH

DH(k) 1D

k2 O(k4 )

hard-disk Lorentz gas

Yukawa-potential Lorentz gas

Re sk

h(DH)

DH

(DH)

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 … n1 at successive times t = n

probability of such a path: (Shannon, McMillan, Breiman) P(0 1 2 … n1 ) ~ 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: hKS = SupP h(P)

Pesin theorem for closed systems:

hKS i

i 0

DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS

nonequilibrium steady state: P (0 12 … n1) ≠ P (n1 … 2 1 0)

If the probability of a typical path decays as

P() = P(0 1 2 … n1) ~ exp( h t n )

the probability of the time-reversed path decays as

P(R) = P(n1 … 2 1 0) ~ exp( hR t n ) with hR ≠ h

entropy per unit time:

h = lim n∞ (1/nt) ∑ P() ln P() = lim n∞

(1/nt) ∑ P(R) ln P(R)

time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599

hR = lim n∞ (1/nt) ∑ P() ln P(R) = lim n∞

(1/nt) ∑ P(R) ln P()

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:

1

kB

diS

dt= hR h 0

P() P(012n 1) exp n t h

P(R ) P(n 1210) exp n t hR exp n t h exp n t diS

dt

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 randomnessof time-reversed paths

hR

dynamical randomness of paths

h

P. Gaspard, J. Stat. Phys. 117 (2004) 599

discrete-time Markov chains:

p P ' p ' P '

'

1

h p P '

, '

ln P '

iS hR h

ILLUSTRATIVE EXAMPLES

time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599

Kolmogorov-Sinai entropy per unit time:

entropy production:

hR p P '

, '

ln P '

Markov chain with 2 states {0,1}:

hR h

Markov chain with 3 states {1,2,3}:

h ln2

(1 ) ln(1 )

iS hR h (1 32 ) ln

2(1 )

P

2 1

2

2

2 1

1 2

2

123123123123123123123123123 122322113333311222112331221equilibrium

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 , Pollicott-Ruelle resonance

diffusion D : D( / L )2 ≈ =(∑i i+hKS )L wavenumber k = / L

(1990)

viscosity : ( / )2 ≈ =(∑i i+hKS ) (1995)

Nonequilibrium modes of diffusion: relaxation rate sk, Pollicott-Ruelle resonance

D k2 ≈ Re sk = (DH) hKS(DH)/ DH (2001)

Nonequilibrium steady states:

The flux boundary conditions explicitly break the time-reversal symmetry. fluctuation theorem: = R() R() (1993, 1995, 1998)

entropy production: ________ = hR(P) h(P) (2004)

diS(P)

kB dt

CONCLUSIONS (cont’d)

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

thermodynamic entropy production = temporal disorder of time-reversed paths temporal disorder of paths

= time asymmetry in dynamical randomness

________ = hR(P) h(P) diS(P)

kB dt