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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