hydrodynamic modes and nonequilibrium steady states - Out of

Liouville's equation is also time-reversal symmetric. Equilibrium ... special solutions of Liouville's equation: p(r,v,t) ..... thermostated systems: no Liouville theorem.
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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

ERGODIC PROPERTIES AND BEYOND Ergodicity (Boltzmann 1871, 1884): time average = phase-space average

1 lim T →∞ T

T



A(Φ t Γ0 ) dt =

0

∫ A(Γ) Ψ (Γ) dΓ = 0

A = A Ψ0

Ψ0 stationary probability density representing the equilibrium statistical ensemble Mixing (Gibbs 1902):

A(t)B(0) =







A(Φ t Γ) B(Γ) Ψ0 (Γ) dΓ → A t →∞

B

lim µ(Φ−tA ∩ B ) = µ(A ) µ(B ) t →∞

Spectrum of unitary € time evolution:

ˆ pt = Uˆ t p0 = e iG t p0

Gˆ = iLˆ

Ergodicity: € density is€ unique: Gˆ Ψ = 0 The stationary probability 0 The eigenvalue z = 0 is non-degenerate.



Mixing: The non-degenerate eigenvalue z = 0 is the only one on the real-frequency axis. € The rest of the spectrum is continuous.



Statistical average of a physical observable A(Γ):

At=



A(Φt Γ€ 0 ) p0 (Γ0 ) dΓ0 =



A(Γ) p0 (Φ−t Γ) dΓ ≡



ˆ

A(Γ) e Lt p0 (Γ) dΓ

POLLICOTT-RUELLE RESONANCES group of time evolution: −∞ < t < +∞ t = = ∫ A(x) p0(Φ−t x) dx analytic continuation toward complex frequencies:

L |Ψα> = sα |Ψα> , < Ξα | L = sα < Ξα | s = −i z • forward semigroup ( 0 < t < +∞): asymptotic expansion around t = +∞ :
t = ≈ ∑α exp(sα t) + (Jordan blocks) • backward semigroup (−∞ < t < 0): asymptotic expansion around t = −∞ : t = ≈ ∑α exp(−sα t) + (Jordan blocks)



POLLICOTT-RUELLE RESONANCES Simple example: Hamiltonian of an inverted harmonic potential: Flow:

Φt (x, y) = (e + λt x,e− λt y)

Statistical average of an observable: t → +∞

€A t =



+ λt − λt dx dy A(e x,e y) p0 (x, y) ∫

= e− λt

∫ dx' dy A(x',e

− λt



=

H=λxy

∑e

− λ( l +m +1)t

A Ψlm

€ y) p0 (e− λt x', y) Lˆ t ˜ Ψlm p0 = A e p0 €

l,m= 0

Eigenvalue problem: Lˆ Ψ = s Ψ lm lm lm

˜ lm Lˆ = slm Ψ ˜ lm Ψ

Eigenvalues: Pollicott-Ruelle resonances: slm = − λ (l + m + 1)

1 l x (−∂ y ) m δ (y) Eigenstates: m! € 1 m ˜ Ψlm (x, y) = y € (−∂ x ) l δ (x) l! Ψlm (x, y) =

γ = λ = −s00 hKS = 0

Schwartz-type distributions € breaking of time-reversal symmetry

TIME-REVERSAL SYMMETRY & ITS BREAKING Hamilton’s equations are time-reversal symmetric: t

−t

ΘoΦ oΘ = Φ

If the phase-space curve C = Γ = Φ t (Γ0 ) : t ∈ R

{

}

is solution of Hamilton’s equation, then the time-reversed curve

Θ(C) = {Γ'= Φ€t' (ΘΓ0 ) : t'∈ R} is also solution of Hamilton’s equation. € C ≠ Θ(C)

Typically, the solution breaks the time-reversal symmetry:



Liouville’s equation is also time-reversal symmetric. Equilibrium state:

Ψ0 = Ψ0 o Θ

Relaxation modes:

Ψα ≠ Ψα o Θ

Nonequilibrium steady state: €

µeq (€A ) = µeq (ΘA ) (α ≠ 0)

µ (A ) ≠ µnoneq (ΘA ) € noneq

€Spontaneous or explicit breaking of time-reversal symmetry €

RELAXATION MODES OF DIFFUSION ∂p special solutions of Liouville’s equation: = {H, p} = Lˆ p ∂t

p(r,v,t) = C exp(sk t) Ψk (r,v)

Ψk (r,v) ∝ exp(ik ⋅ r) spatial periodicity

generalized eigenstate of Liouvillian operator:

€ Lˆ Ψk = sk Ψk





eigenvalue = dispersion relation of diffusion:

sk = − D k2 + O(k4) diffusion coefficient: Green-Kubo formula

concentration

wavenumber: € k

space

D=



∫ 0

v x (0)v x (t) dt

wavelength = 2π/k

time

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: l ∈ L lattice Fourier transform: G(Γ,l) = first Brillouin zone of the lattice:



B

1 B

∫ dk e

i k⋅ l

B

initial probability density close to equilibrium: € time evolution of the probability density:



Rt (Γ,l) =

1 B

G˜ (Γ,k)

∫ dk€F

k

B

p0 (Γ,l) = peq [1+ R0 (Γ,l)] pt (Γ,l) = peq [1+ Rt (Γ,l)]

exp{i k ⋅ [l + d(Γ,t)]}

lattice distance travelled by the tracer particle: €

d(Γ,t)

J. R. Dorfman, € P. Gaspard, & T. Gilbert, Entropy production of diffusion in spatially periodic deterministic systems, Phys. Rev. E 66 (2002) 026110



DIFFUSIVE MODES IN SPATIALLY PERIODIC SYSTEMS The Perron-Frobenius operator Pˆ = exp(Lˆ t) is symmetric under the spatial translations {l} of the (crystal) lattice: t

[

Pˆ t , Tˆ l = 0

]

€  Pˆ t Ψ = exp(s t) Ψ k k k common eigenstates:  ˆl  T Ψ = exp(ik ⋅ l) Ψ k



k

eigenstate = hydrodynamic mode of diffusion:

Ψk

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

€ wavenumber: k



sk = lim t→∞ (1/t) ln = − D k2 + O(k4) diffusion coefficient: Green-Kubo formula

D=

concentration

€ (Van Hove, 1954)

space



∫ 0

v x (0)v x (t) dt

wavelength = 2π/k

time

CUMULATIVE FUNCTION OF THE DIFFUSIVE MODES The eigenstate Ψk is a distribution which is smooth in Wu but singular in Ws. = breaking of time-reversal symmetry since Wu = Θ(Ws) but Wu ≠ Ws . θ

cumulative function:

θ

Fk (θ ) =

∫ Ψ (r k

θ'

∫ dθ'

exp[ik ⋅ (rt − r0 )θ ' ]

∫ dθ'

exp[ik ⋅ (rt − r0 )θ ' ]

,vθ ' )dθ ' = lim 20π

0

t →∞

0

fractal curve in complex plane because Ψk is singular in Ws eigenvalue = leading € Pollicott-Ruelle resonance sk = − D k2 + O(k4) = lim t→∞ (1/t) ln

(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. 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.

MULTIBAKER MODEL OF DIFFUSION ...

 y l −1,2x,  ,   2 φ (l, x, y) =   l + 1,2x −1, y + 1,  2 

0≤ x≤

1 2

...

1 < x ≤1 2

dispersion relation of diffusion : sk = lncos k









l-1

l

l+1

...

...

φ

singular diffusive modes : de Rham cumulative functions : α Fk (2y),  1  for 0 < y < ,  2 Fk (y) =  (1− α )Fk (2y −1) + α,  1 for < y < 1.  2 exp(ik) α= 2cos k

top. pressure : P(β ) = (1− β ) ln2 ln2 Hausdorff dimension : DH = ln(2cos k)

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 (θ) = ∫0θ Ψk(xθ’) dθ’

YUKAWA-POTENTIAL LORENTZ GAS • Hamiltonian: H = p2/2m − Σi exp(−ari)/ri • Deterministic chaotic dynamics • Time-reversal symmetric (Knauf 1989) cumulative functions of the diffusive mode: Fk (θ) = ∫0θ Ψk(xθ’) dθ’

HAUSDORFF DIMENSION OF THE DIFFUSIVE MODES

Proof of the formula for the Hausdorff dimension θ

cumulative function:

∫ dθ '

exp[ik ⋅ (rt − r0 )θ ' ]

∫ dθ '

exp[ik ⋅ (rt − r0 )θ ' ]

Fk (θ ) = lim 20π t →∞

0

polygonal approximation of the fractal curve: 1 exp[ik ⋅ (rt − r0 )ω ] Fk (θ ) = lim ∑ = lim ∑ ΔFk (ω ) n →∞ n →∞ € exp(sk nτ ) ω Λ(ω ) ω Hausdorff dimension:

∑ ΔF (ω ) k

DH

∝1

for

n →∞

ω

€ 1 −β P( β ) ≡ lim ln Λ( ω ) Ruelle topological pressure: ∑ n →∞



€ Hausdorff dimension:

ω

P(DH ) = DH Re sk

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.



DIFFUSION COEFFICIENT FROM THE HAUSDORFF DIMENSION Hausdorff dimension:

P(DH ) = DH Re sk µβ (ω ) ≅

probability measure:



Λ(ω )

−β

∑ Λ(ω )

−β

ω

1 ∑ µβ (ω ) ln Λ(ω ) n →∞ nτ ω

average Lyapunov exponent:

λ(β ) ≡ lim

entropy per unit time: €

h(β ) = β λ(β ) + P(β )  P(DH ) h(DH ) 1  −Re sk = − = λ (DH ) − ≈ λ1−  DH DH D  H

Hausdorff dimension: €



dispersion relation of diffusion: low-wavenumber expansion:



diffusion coefficient:



Re sk = −Dk 2 + O(k 4 ) D DH (k) = 1+ k 2 + O(k 4 ) λ

DH (k) −1 k →0 k2

D = λ lim

P. Gaspard, I. Claus,€ T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506.

FRACTALITY OF THE DIFFUSIVE MODES Hausdorff dimension: DH (k) = 1+ D k 2 + O(k 4 )

λ

Yukawa-potential Lorentz gas hard-disk Lorentz gas



−Re sk large-deviation dynamical relationship:

λ(DH )

P(DH ) h(DH ) Dk ≈ −Re sk = − = λ (DH ) − DH DH 2



P. Gaspard, I. Claus, T. Gilbert, & J. R. Dorfman, Phys. Rev. Lett. 86 (2001) 1506.



h(DH ) DH

NONEQUILIBRIUM STEADY STATES Steady state of gradient g=(p+−p−)/L in x: pnoneq(Γ) = (p++ p−)/2 + g [ x(Γ) + ∫0−T(Γ) vx(Φt Γ) dt ] pnoneq(Γ) = (p++ p−)/2 + g [ x(Γ) + x(Φ−T(Γ) Γ) − x(Γ) ]

p+

p−

pnoneq(Γ) = (p++ p−)/2 + (p+−p−) x(Φ−T(Γ) Γ) /L pnoneq(Γ) = p±

for

x(Φ−T(Γ) Γ) = ±L/2

Ψg.(Γ) = g [ x(Γ) + ∫0− ∞ vx(Φt Γ) dt ] = − i g•∂k Ψk(Γ)|k=0

x t

Green-Kubo formula:

D = ∫0∞ eq dt

Fick’s law: neq= g [eq + ∫0− ∞ eqdt ] =

−Dg

SINGULAR CHARACTER OF THE NONEQUILIBRIUM STEADY STATES cumulative functions

hard-disk Lorentz gas

Tg (θ) = ∫0θ

Ψg(Γθ’) dθ’ Yukawa-potential Lorentz gas

(generalized Takagi functions)

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 −sk, Pollicott-Ruelle resonance

h(DH ) Dk ≈ −Re sk = λ (DH ) − DH 2

reminiscent of the escape-rate formula:



D(π /L) ≈ γ = ( λ − hKS ) L 2

(1/2) wavelength = L = π/k



CONCLUSIONS (cont’d) Escape-rate formalism: nonequilibrium transients fractal repeller diffusion D :

viscosity η :

D(π /L)

2

η (π / χ )

2



  ≈ γ =  ∑ λi − hKS   λi >0 L

(1990)

  ≈ γ =  ∑ λi − hKS   λi >0 L 2f

Hamiltonian systems: Liouville theorem:

∑λ

i

(1995)

γ = ∑ λi − hKS = − ∑ λi − hKS

=0

λi >0

i=1

λi