POLLICOTTRUELLE RESONANCES, FRACTALS, AND NONEQUILIBRIUM MODES OF RELAXATION

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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
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BREAKING OF TIME-REVERSAL SYMMETRY Q(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 Q T Q T ?
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 Liouvilles
equation associated with decaying
eigenstates singular in the stable
directions Ws but smooth in the unstable
directions Wu
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POLLICOTT-RUELLE RESONANCES

• group of time evolution -8 lt t lt 8
  statistical average of the observable A
• ltAgtt ltAexp(L t) p0 gt ? A(G)
  p0(F-t G) dG

analytic continuation toward complex
frequencies L Yagt sa Yagt , lt Xa
L sa lt Xa forward semigroup ( 0 lt t lt
8) asymptotic expansion around t 8
ltAgtt ltAexp(L t) p0gt ?a ltAYagt
exp(sa t) ltXa p0gt (Jordan blocks)
backward semigroup (-8 lt t lt 0) asymptotic
expansion around t -8 ltAgtt
ltAexp(L t) p0gt ?a ltAYaQgt exp(-sa t)
ltXaQ p0gt (Jordan blocks)
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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 sk lim t8
(1/t) ln ltexp i k(rt - r0)gt - D k2
O(k4) diffusion coefficient Green-Kubo formula
concentration
space
wavelength 2p/k
time
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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.
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MULTIBAKER MODEL OF DIFFUSION
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PERIODIC HARD-DISK LORENTZ GAS

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

cumulative functions Fk (q) ?0q Yk(Gq) dq
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PERIODIC YUKAWA-POTENTIAL LORENTZ GAS

• Hamiltonian
• H p2/2m - Si exp(-ari)/ri
• Deterministic chaotic dynamics
• Time-reversal symmetric
• (Knauf 1989)

cumulative functions Fk (q) ?0q Yk(Gq) dq
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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) ?0q Yk(Gq) dq
infinite number of handles
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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.
Yukawa-potential Lorentz gas
hard-disk Lorentz gas
-Re sk
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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 w0 w1 w2 wn-1 at successive
times t n t probability of such a path
(Shannon, McMillan, Breiman)
P(w0 w1 w2 wn-1 ) exp
-h(P) t 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
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DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS
nonequilibrium steady state P (w0
w1w2 wn-1) ? P (wn-1 w2 w1 w0) If the
probability of a typical path decays as
P(w) P(w0 w1 w2
wn-1) exp( -h Dt n ) the probability of
the time-reversed path decays as P(wR)
P(wn-1 w2 w1 w0) exp( -hR Dt n )
with hR ? h entropy per unit time
h lim n8 (-1/nDt) ?w P(w) ln P(w) lim
n8 (-1/nDt) ?w P(wR) ln P(wR)
time-reversed entropy per unit time P. Gaspard,
J. Stat. Phys. 117 (2004) 599 hR lim
n8 (-1/nDt) ?w P(w) ln P(wR) lim n8
(-1/nDt) ?w P(wR) ln P(w) The time-reversed
entropy per unit time characterizes the
dynamical randomness (temporal disorder) of the
time-reversed paths.
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THERMODYNAMIC ENTROPY PRODUCTION
nonequilibrium steady state thermodynamic
entropy production
entropy production
dynamical randomness of time-reversed
paths hR
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.
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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
123123123123123123123123123
122322113333311222112331221
equilibrium
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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 g, Pollicott-Ruelle
resonance diffusion D D (p / L )2 g
(?i li - hKS )L wavenumber k
p / L (1990) viscosity h h
(p / c )2 g (?i li - hKS )c
(1995) Nonequilibrium modes of diffusion
relaxation rate -sk, Pollicott-Ruelle resonance
D k2 - Re sk
l(DH) - hKS(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 ________
hR(P) - h(P)
(2004)

diS(P)
kB dt
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CONCLUSIONS (contd)
diS(P)
________ hR(P) - h(P)
kB 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.
Boltzmanns interpretation of the second
law Out of equilibrium, the spatial disorder
increases in time.
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