HYDRODYNAMIC MODES AND NONEQUILIBRIUM STEADY STATES

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HYDRODYNAMIC MODES ANDNONEQUILIBRIUM 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
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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(G)
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POLLICOTT-RUELLE RESONANCES

• group of time evolution -8 lt t lt 8
• ltAgtt ltAexp(L t) p0 gt ? A(x)
  p0(F-t x) dx

analytic continuation toward complex
frequencies L Yagt sa Yagt , lt Xa
L sa lt Xa

s -i z 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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POLLICOTT-RUELLE RESONANCES
Simple example Hamiltonian of an inverted
harmonic potential
Flow
Statistical average of an observable
Eigenvalue problem
Eigenvalues Pollicott-Ruelle resonances
Eigenstates
Schwartz-type distributions
breaking of time-reversal symmetry
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TIME-REVERSAL SYMMETRY ITS BREAKING
Hamiltons equations are time-reversal symmetric
If the phase-space curve is solution of
Hamiltons equation, then the time-reversed
curve
is also solution of Hamiltons
equation. Typically, the solution breaks the
time-reversal symmetry Liouvilles equation is
also time-reversal symmetric. Equilibrium
state Relaxation modes Nonequilibrium steady
state Spontaneous or explicit
breaking of time-reversal symmetry
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RELAXATION MODES OF DIFFUSION
special solutions of Liouvilles equation
spatial periodicity
generalized eigenstate of Liouvillian operator
eigenvalue dispersion relation of diffusion
wavenumber k sk - D k2
O(k4) diffusion coefficient Green-Kubo formula
concentration
space
time
wavelength 2p/k
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MOLECULAR DYNAMICS SIMULATION OF DIFFUSION
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 vector
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
J. R. Dorfman, P. Gaspard, T. Gilbert, Entropy
production of diffusion in spatially periodic
deterministic systems, Phys. Rev. E 66 (2002)
026110
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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 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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CUMULATIVE FUNCTION OF THE DIFFUSIVE MODES
The eigenstate Yk is a distribution which is
smooth in Wu but singular in Ws. breaking
of time-reversal symmetry since Wu Q(Ws) but
Wu ? Ws . cumulative function
fractal curve in complex plane because Yk is
singular in Ws 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
sk - D k2 O(k4) lim t8 (1/t) ln ltexp
i k(rt - r0)gt (Van Hove, 1954) sk
is the continuation of the eigenvalue s0 0 of
the microcanonical equilibrium state and is
not the next-to-leading Pollicott-Ruelle
resonance.
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MULTIBAKER MODEL OF DIFFUSION
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HARD-DISK LORENTZ GAS

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

cumulative functions of the diffusive mode
Fk (q) ?0q Yk(xq) dq
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YUKAWA-POTENTIAL LORENTZ GAS

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

cumulative functions of the diffusive mode
Fk (q) ?0q Yk(xq) dq
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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
Hausdorff dimension
Generalization of the Bowen-Ruelle formula for
the Hausdorff dimension of Julia sets.
P. Gaspard, I. Claus, T. Gilbert, J. R.
Dorfman, Phys. Rev. Lett. 86 (2001) 1506.
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DIFFUSION COEFFICIENT FROM THE HAUSDORFF DIMENSION
Hausdorff dimension
probability measure
average Lyapunov exponent
entropy per unit time
Hausdorff dimension
dispersion relation of diffusion
low-wavenumber expansion
diffusion coefficient
P. Gaspard, I. Claus, T. Gilbert, J. R.
Dorfman, Phys. Rev. Lett. 86 (2001) 1506.
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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.
Yukawa-potential Lorentz gas
hard-disk Lorentz gas
-Re sk
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NONEQUILIBRIUM STEADY STATES

• Steady state of gradient g(p-p-)/L in x
• pnoneq(G) (p p-)/2 g x(G) ?0-T(G) vx(Ft
  G) dt
• pnoneq(G) (p p-)/2 g x(G) x(F-T(G) G) -
  x(G)
• pnoneq(G) (p p-)/2 (p-p-) x(F-T(G) G) /L
• pnoneq(G) p for x(F-T(G) G) L/2
• Yg.(G) g x(G) ?0- 8 vx(Ft G) dt
• - i g?k Yk(G)k0
• Green-Kubo formula D ?08 ltvx(0)vx(t)gteq dt
• Ficks law ltvxgtneq g ltvx xgteq ?0- 8
  ltvx(0)vx(t)gteqdt - D g

p
p-
x
t
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SINGULAR CHARACTER OF THENONEQUILIBRIUM STEADY
STATES
cumulative functions Tg (q) ?0q Yg(Gq)
dq hard-disk Lorentz gas
Yukawa-potential Lorentz gas
(generalized Takagi functions)
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CONCLUSIONS
Breaking of time-reversal symmetry in the
statistical description Nonequilibrium
transients Spontaneous breaking of
time-reversal symmetry for the solutions of
Liouvilles equation corresponding to the
Pollicott-Ruelle resonances. The associated
eigenstates are singular distributions with
fractal cumulative functions. Nonequilibrium
modes of diffusion relaxation rate -sk,
Pollicott-Ruelle resonance
reminiscent of the
escape-rate formula

(1/2) wavelength L p/k
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CONCLUSIONS (contd)

Escape-rate formalism nonequilibrium
transients fractal repeller
diffusion D (1990)
viscosity h (1995)
Hamiltonian systems Liouville theorem
thermostated systems no Liouville theorem
volume contraction rate
Pesins identity on the attractor
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CONCLUSIONS (contd)
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 Liouvilles 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.
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