AB INITIO DERIVATION OF ENTROPY PRODUCTION

Pierre GASPARDBrussels, Belgium

J. R. Dorfman, College Park

S. Tasaki, Tokyo

T. Gilbert, Brussels

• MIXING & POLLICOTT-RUELLE RESONANCES

• COARSE-GRAINED ENTROPY & ENTROPY PRODUCTION

• DECOMPOSITION INTO HYDRODYNAMIC MEASURES

• AB INITIO DERIVATION OF ENTROPY PRODUCTION

• CONCLUSIONS

MIXING & POLLICOTT-RUELLE RESONANCESCorrelation function between observables A and B:

€

A(t)B(0)eq

= A(Γ∫ ) eˆ L tB(Γ) Ψ0 Γ( ) dΓ

≈ A Ψα

α

∑ esα t ˜ Ψ α BΨ0 →t →∞

Aeq

Beq

Statistical average of a physical observable A:

€

At= A(Γ∫ ) e

ˆ L t p0 Γ( ) dΓ

≈ A Ψα

α

∑ esα t ˜ Ψ α p0 →t →∞

Aeq

DIFFUSIVE MODES: CUMULATIVE FUNCTIONS

φ. . . . . .

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

multibaker map hard-disk Lorentz gas Yukawa-potential Lorentz gas

TIME EVOLUTION OF ENTROPY

coarse-grained entropy: partition of phase-space region Ml into cells A

St(Ml |{A}) = kB A Pt(A) ln[Pt(A)/Peq(A)] + Seq with Pt(A) ≈ p(t)

Gibbs mixing property: Pt(A) Peq(A) for t ∞

time asymptotics for t ∞ : Pt(A) = Peq(A) + C exp(s t) + …

Pollicott-Ruelle resonances s and associated eigenstates fixing the coefficients C

Selection of initial conditions by a larger system including the system of interest: problem of regression.

anti-diffusion

∂tn ≈ D ∂l2n

diffusion

∂tn ≈ D ∂l2n

eigenstates singular

in unstable directions,

smooth in stable directions

eigenstates singular

in stable directions,

smooth in unstable directions

Gibbs (1902)

ENTROPY PRODUCTION

coarse-grained entropy: partition of phase-space region Ml into cells A

St(Ml |{A}) = kB A Pt(A) ln[Pt(A)/Peq(A)] + Seq with Pt(A) ≈ p(t)

time variation over time : S = St(Ml |{A}) St(Ml |{A})

entropy flow: eS = St(Ml |{A}) St(Ml |{A})

entropy production: iS = S e

S = St(Ml |{A}) St(Ml |{ A})

Direct calculation shows that

iS ≈ kB D n1 (grad n)2 with the particle density: n = Pt(Ml)

because of the singular character of the nonequilibrium states

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

MOLECULAR DYNAMICS SIMULATION OF DIFFUSION

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

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 Fourier transform:

€

G(Γ,l) =1

Bdk e i k⋅l ˜ G (Γ,k)

B

∫ first Brillouin zone of the lattice:

€

B

initial probability density close to equilibrium:

€

p0(Γ,l) = peq[1+ R0(Γ,l)]

time evolution of the probability density:

€

pt (Γ,l) = peq[1+ Rt (Γ,l)]

€

Rt (Γ,l) =1

Bdk Fk exp i k ⋅ l + d(Γ, t)[ ]{ }

B

∫

€

d(Γ, t) lattice distance travelled by the tracer particle:

lattice vector:

€

l ∈ L

DECOMPOSITION INTO DIFFUSIVE MODES

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

measure of a cell A at time t:

with

€

C(k, t) esk t ≡

dΓ peq e i k⋅d(Γ,t )

M

∫dΓ peq

M

∫

dispersion relation of diffusion:

hydrodynamic measure at time t:

€

μt (A) = pt (Γ,l)dΓA

∫ = μ eq (A) +1

Bdk Fk e ik⋅l

B

∫ C(k, t) esk tχ k (A, t) ≡ μ eq (A) + δμ t (A)

€

sk = limt →∞

1

t ln

dΓ peq e i k⋅d(Γ,t )

M

∫dΓ peq

M

∫

€

limt →∞

1

t ln C(k, t) = 0

€

χk (A, t) ≡ μ eq (M)

dΓ peq e i k⋅d(Γ,t )

A

∫dΓ peq e i k⋅d(Γ,t )

M

∫

invariance under time evolution: de Rham-type equation:

€

eskτ χ k (A) = e i k⋅d(ΓA ,τ )χ k (Φ−τ A)

HYDRODYNAMIC MEASURE

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

invariant hydrodynamic measure:

€

χk (A) ≡ μ eq (M) limt →∞

dΓ peq e i k⋅d(Γ,t )

A

∫dΓ peq e i k⋅d(Γ,t )

M

∫

sum rules: partition

€

d jμ eq (A j ) = 0j

∑

d jT(A j ) + T(A j )d j + d jd jμ eq (A j )[ ] = 2 D τ μ eq (M) 1j

∑

expansion in powers of the wavenumber k:

€

χk (A) = μ eq (A) + i k ⋅T(A) +O(k2)

€

T(A) = T(Φ -τ A) +μ eq (A) d(ΓA ,τ ) measure of cell A by the nonequilibrium steady state:

(~ Green-Kubo formula)

(no mean drift)

€

∪j

A j = M l

€

d j = d(ΓΦτ A j

,τ ) distance:

AB INITIO DERIVATION OF ENTROPY PRODUCTION

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

entropy production:

€

iτ S(M l ) = St (M l |{A j}) − St (M l |{Φτ A j})

= − μ t (A j )lnμ t (A j )

μ eq (A j )A j ⊂M l

∑ + μ t (Φτ A j )ln

μ t (Φτ A j )

μ eq (A j )Φτ A j ⊂M l

∑

= −1

2

δμ t (A j )[ ]2

μ eq (A j )A j ⊂M l

∑ +1

2

δμ t (Φτ A j )[ ]

2

μ eq (A j )Φτ A j ⊂M l

∑ + O(δμ 3)

€

μt (A) = μ eq (A) + δμ t (A)

€

δμt (A) ≈1

Bdk Fk e ik⋅l

B

∫ C(k, t) esk t μ eq (A) + i k ⋅T(A) +O(k2)[ ]

€

iτ S(M l ) =

1

2B2 dk1dk2Fk1

Fk 2e i(k1 +k 2 )⋅lC(

B

∫∫ k1, t)C(k2, t)e(sk1

+sk 2)t

×1

μ eq (A j ) k1k2 : T(A j )T(A j ) − T(Φτ A j )T(Φτ A j )[ ]

j

∑ ≈ D τ 1

neq

∂n

∂l

⎛

⎝ ⎜

⎞

⎠ ⎟2

wavenumber expansion:

entropy production of nonequilibrium thermodynamics

CONCLUSIONS

In the long-time limit, the approach to equilibrium is controlled by the Pollicott-Ruelle resonances (including the dispersion relation of diffusion) and the associated eigenstates (including the diffusive modes). The same applies to the coarse-grained entropy.

Ab initio derivation of the entropy production expected from nonequilibrium thermodynamics:

iS ≈ kB D n1 (grad n)2 (2002)

because the diffusive modes are singular and break the time-reversal symmetry.

This result is obtained in the limit of long times and low wavenumbers, where the

diffusive mode gives the singular distribution of the nonequilibrium steady state. This

latter appears as part of the Green-Kubo formula giving the diffusion coefficient D.

http://homepages.ulb.ac.be/~gaspard

Singular nonequilibrium steady state: g.() = g [ x() + ∫0 ∞ vx(t ) dt ]

Green-Kubo formula: D = ∫0∞ <vx(0)vx(t)>eq dt

Fick’s law: <vx>neq= g [<vx x>eq + ∫0 ∞ <vx(0)vx(t)>eqdt ] = D g