time asymmetry in nonequilibrium statistical mechanics pierre gaspard brussels, belgium j. r....

TIME ASYMMETRY INNONEQUILIBRIUM STATISTICAL MECHANICS

Pierre GASPARDBrussels, Belgium

J. R. Dorfman, College Park S. Ciliberto, Lyon

T. Gilbert, Brussels N. Garnier, Lyon

D. Andrieux, Brussels S. Joubaud, Lyon

A. Petrosyan, Lyon

• INTRODUCTION: THE BREAKING OF TIME-REVERSAL SYMMETRY

• FLUCTUATION THEOREMS FOR CURRENTS & NONLINEAR RESPONSE

• ENTROPY PRODUCTION &

TIME ASYMMETRY OF NONEQUILIBRIUM FLUCTUATIONS

• CONCLUSIONS

BREAKING OF TIME-REVERSAL SYMMETRY (r,v) = (r,v)

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.

Spontaneous symmetry breaking: relaxation modes of an autonomous system

Explicit symmetry breaking: nonequilibrium steady state by the boundary conditions

€

∂p

∂t= H, p{ } = ˆ L p

P. Gaspard, Physica A 369 (2006) 201-246.

STOCHASTIC DESCRIPTION IN TERMS OF A MASTER EQUATION

€

d

dtPt (ω) = Pt (ω') Wρ (ω' |ω) − Pt (ω) W−ρ (ω |ω')[ ]

ρ ,ω '(≠ω )

∑

€

Wρ (ω |ω')

Liouville’s equation of the Hamiltonian dynamics -> reduced description in terms of the coarse-grained states -> master equation for the probability to visit the state by the time t : Pt()

rate of the transition

€

→ρ

'

€

ρ =±1,...,±r due to the elementary process

A trajectory is a solution of Hamilton’s equations of motion: (t;r0,p0)

Coarse-graining: cell in the phase space stroboscopic observation of the trajectory with sampling time t : (nt;r0,p0) in cell n

path or history: 012…n1

-> statistical description of the equilibrium and nonequilibrium fluctuations

= 0 steady state

FLUCTUATION THEOREM FOR THE CURRENTS steady state fluctuation theorem for the currents (2004):

affinities or thermodynamic forces:

fluctuating currents:

thermodynamic entropy production:

€

diS

dt st

= Aγ Jγ

γ =1

c

∑ ≥ 0

€

Jγ =1

tjγ (t ')

0

t

∫ dt'

€

Aγ =ΔGγ

T=

Gγ − Gγeq

T

€

P Jγ = α γ{ }

P Jγ = −α γ{ }≈ e

t

kB

Aγ α γ

γ

∑

-> Onsager reciprocity relations and their generalizations to nonlinear response

D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Phys. 127 (2007) 107.

ex: • electric currents in a nanoscopic conductor • rates of chemical reactions • velocity of a linear molecular motor • rotation rate of a rotary molecular motor

€

t → +∞

Schnakenberg network theory (Rev. Mod. Phys. 1976): cycles in the graph of the process

BEYOND LINEAR RESPONSE & ONSAGER RECIPROCITY RELATIONS

€

q( λ γ ,Aγ{ }) = limt →∞

−1

tln exp − λ γ jγ t '( )dt'

0

t

∫γ

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

noneq.

€

Jα =∂q

∂λα λα = 0

= Lαβ Aβ

β

∑ + Mαβγ Aβ Aγ

β ,γ

∑ + Nαβγδ Aβ Aγ

β ,γ ,δ

∑ Aδ +L

€

Mαβγ = 12 Rαβ ,γ + Rαγ ,β( )

€

Lαβ = Lβα is totally symmetric

€

Rαβ ,γ = −∂ 3q

∂λα∂λ β∂Aγ

(0;0)

average current:

fluctuation theorem for the currents:

Onsager reciprocity relations:

relations for nonlinear response:

higher-order nonequilibrium coefficients:

generating function of the currents:

linear response coefficients:

€

Lαβ = −1

2

∂ 2q

∂λα∂λ β

(0;0) =1

2 jα (t) − jα[ ] jβ (0) − jβ[ ]

−∞

+∞

∫ dt

(Schnakenberg network theory)

D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Mech. (2007) P02006.

linear response coefficients: (Green-Kubo formulas)

€

q( λ γ ,Aγ{ }) = q( Aγ − λ γ , Aγ{ })

€

Microreversibility: Hamilton’s equations are time-reversal symmetric. If (t;r0,p0) is a solution of Hamilton’s equation, then (t;r0,p0) is also a solution. But, typically, (t;r0,p0) ≠ (t;r0,p0).

Coarse-graining: cell in the phase space stroboscopic observation of the trajectory with sampling time t : (nt;r0,p0) in cell n

path or history: 012…n1

If 012…n1 is a possible path, then Rn1…210 is also a possible path. But, again, ≠ R.

Statistical description: probability of a path or history:

equilibrium steady state: Peq(012…n1) = Peq(n1…210) nonequilibrium steady state: Pneq(012…n1) ≠ Pneq(n1…210)

In a nonequilibrium steady state, and R have different probability weights. Explicit breaking of the time-reversal symmetry by the nonequilibrium boundary conditions

FLUCTUATIONS AND MICROREVERSIBILITY

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: dynamical randomness (temporal disorder)

h = lim n∞ (1/nt) ∑ P() ln P()

time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599

hR = lim n∞ (1/nt) ∑ P() ln P(R)

The time-reversed entropy per unit time characterizes

the dynamical randomness (temporal disorder) of the time-reversed paths.

THERMODYNAMIC ENTROPY PRODUCTION

€

Property: hR ≥ h

(relative entropy)

equality iff P() = P(R) (detailed balance) which holds at equilibrium.

Second law of thermodynamics: entropy S

€

dS

dt=

deS

dt+

diS

dt with

diS

dt≥ 0

€

deS

€

diS ≥ 0

Entropy production:

€

1

kB

diS

dt= hR − h ≥ 0

P. Gaspard, J. Stat. Phys. 117 (2004) 599

€

P(ω)

P(ωR )=

P(ω0ω1ω2L ωn−1)

P(ωn−1L ω2ω1ω0)≈ e

nΔt h R −h( ) = enΔt

kB

d i S

dt

€

1

kB

diS

dt= lim

n →∞

1

nΔtP(ω)

ω

∑ lnP(ω)

P(ωR )≥ 0

entropy flow

entropy production

PROOF FOR CONTINUOUS-TIME JUMP PROCESSES

Pauli-type master equation:

nonequilibrium steady state:

-entropy per unit time: P. Gaspard & X.-J. Wang, Phys. Reports 235 (1993) 291

time-reversed -entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599


€

d

dtpt (ω') = pt (ω)Wωω ' − pt (ω')Wω 'ω[ ]

ω

∑

€

d

dtp(ω') = 0

€

h(τ ) = lne

τ

⎛

⎝ ⎜

⎞

⎠ ⎟ p(ω)Wωω '

ω≠ω '

∑ − p(ω)Wωω '

ω≠ω '

∑ lnWωω ' + O(τ )

€

hR (τ ) = lne

τ

⎛

⎝ ⎜

⎞

⎠ ⎟ p(ω)Wωω '

ω≠ω '

∑ − p(ω)Wωω '

ω≠ω '

∑ lnWω 'ω + O(τ )

€

hR (τ ) − h(τ ) =1

2p(ω)Wωω ' − p(ω')Wω 'ω[ ]

ω≠ω '

∑ lnp(ω)Wωω '

p(ω')Wω 'ω

+ O(τ ) ≈1

kB

diS

dt (τ → 0)

Luo Jiu-li, C. Van den Broeck, and G. Nicolis, Z. Phys. B- Cond. Mat. 56 (1984) 165

J. Schnakenberg, Rev. Mod. Phys. 48 (1976) 571

PROOF FOR THERMOSTATED DYNAMICAL SYSTEMS

entropy per unit time:

time-reversed entropy per unit time:


is the diameter of the phase-space cells

T. Gilbert, P. Gaspard, and J. R. Dorfman (2007)

€

limδ →0

h = λ j

λ j >0

∑ = hKS

€

limδ →0

(hR − h) = − λ j

j

∑ =1

kB

diS

dt€

limδ →0

hR = − λ j

λ j <0

∑

INTERPRETATION

€

nonequilibrium steady state:


€

1

kB

diS

dt= hR − h ≥ 0

€

P(ω) = P(ω0ω1ω2L ωn−1) ≈ exp −n Δt h( )

€

P(ωR ) = P(ωn−1L ω2ω1ω0) ≈ exp −n Δt hR( ) = exp −n Δt h( ) exp −n Δt

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

relaxation time:

€

R = α /k = 3.05 10−3s

DRIVEN BROWNIAN MOTION

trap stiffness:

€

k = 9.62 10−6 kg s−2

trap velocity:

€

u = ± 4.24 μm/s

Polystyrene particle of 2 m diameter

in a 20% glycerol-water solution at temperature 298 K, driven by an optical tweezer.

D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

Langevin equation:

€

dx

dt= −

x − ut

τ R

+2kBT

αξ t

dissipated heat:

€

Qt = −k ˙ x t ' (x t ' − ut') dt'0

t

∫

€

Qt = α u2t

driving force:

€

F = −k(x − ut)

€

position x

€

friction α

€

white noise ξ t

mean dissipated heat:

€

relative position z = x − ut

u < 0

u > 0

comoving frame of reference:

€

dz

dt= −

z

τ R

− u +2

αβξ t

€

z ≡ x − ut

€

pst (z) =βk

2πexp −

βk

2(z + uτ R )2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥

PATH PROBABILITIES OF NONEQUILIBRIUM FLUCTUATIONS


stationary probability density:

path probability:

€

Psign u zt[ ]∝ exp −αβ

4dt ' ˙ z t ' + kzt ' + u( )

2

0

t

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

€

lnP+ zt[ ]

P− ztR

[ ]= −

βk

2(zt

2 − z02) − βk u zt ' dt'

0

t

∫ = βQt

ratio of probabilities for u>0 and u<0:

€

diS

dt= lim

t →∞

kB

tDzt P+∫ zt[ ] ln

P+ zt[ ]

P− ztR

[ ]


€

β ≡1

kBT

€

Qt = W t − ΔVt heat generated by dissipation:

path:

€

Zm = z(mτ ),...,z(mτ + nτ − τ )[ ]

€

H(ε,τ ,n) = −1

Mln P+ Zm[ ]

m=1

M

∑

RELATIONSHIP TO DYNAMICAL RANDOMNESS


()-entropy per unit time:

path probability:

€

diS

dt= kB lim

ε ,τ →∞hR (ε,τ ) − h(ε,τ )[ ]

€

P+ Zm[ ] = P z(mτ ) − z( jτ ) < ε,..., z(mτ + nτ − τ ) − z( jτ + nτ − τ ) < ε[ ]

€

H R (ε,τ ,n) = −1

Mln P− Zm

R[ ]

m=1

M

∑

€

h(ε,τ ) = limn →∞

1

τH(ε,τ ,n +1) − H(ε,τ ,n)[ ]

€

hR (ε,τ ) = limn →∞

1

τH R (ε,τ ,n +1) − H R (ε,τ ,n)[ ]time-reversed ()-entropy per unit time:

time-reversed ()-entropy:

()-entropy:


algorithm of time series analysis by Grassberger & Procaccia (1980’s)

DRIVEN BROWNIAN MOTION


€

=k 0.558 nm k =1- 20

sampling frequency: 8192 Hz

resolution:

()-entropy

time-reversed ()-entropy

time series:

€

2 107

€

diS

dt=

1

T

d Qt

dt=

α u2

T

D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud,

and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

€

diS

dt= kB lim

ε ,τ →∞hR (ε,τ ) − h(ε,τ )[ ]

DRIVEN BROWNIAN PARTICLE

€

=k 0.558 nm k =11- 20

resolution:

dissipated heat along the random path zt

potentiel velocity:

€

u = ± 4.24 μm/s

€

lnP+ zt[ ]

P− ztR

[ ]= βQt

ratio of probabilities for u>0 and u<0:

D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud,

and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

RC circuit:

€

R = 9.22 MΩ C = 278 pF τ R = RC = 2.56 ms

DRIVEN ELECTRIC CIRCUIT

thermodynamic entropy production

€

diS

dt=

R I2

TJoule law:


CONCLUSIONSBreaking of time-reversal symmetry in the statistical description

Nonequilibrium work fluctuation theorem: systems driven by an external forcing

€

Wdiss

kBT= −ln pR (−W)[ ] − −ln pF(W)[ ]

€

Dk2 ≈ −Re sk = λ (DH) −h(DH)

DH

Nonequilibrium modes of diffusion: relaxation rate sk, Pollicott-Ruelle resonance

€

D π /L( )2

≈ γ = λ i

λ i >0

∑ − hKS

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟L

Nonequilibrium transients: escape-rate formalism: fractal repeller

diffusion D : (1990)

viscosity : (1995)

€

π /χ( )2

≈ γ = λ i

λ i >0

∑ − hKS

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟L

CONCLUSIONS (cont’d)

thermodynamic entropy production = temporal disorder of time-reversed paths hR temporal disorder of paths h= time asymmetry in dynamical randomness

Theorem of nonequilibrium 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.

€

1

kB

diS

dt= hR − h ≥ 0

Nonequilibrium steady states:

Explicit breaking of time-reversal symmetry by the nonequilibrium conditions.

Fluctuation theorem for the currents:

€

1

kB

Aγα γγ∑ = −lim

t →∞

1

tlnP Jγ = −α γ{ }

⎡ ⎣ ⎢

⎤ ⎦ ⎥− −lim

t →∞

1

tln P Jγ = α γ{ }

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Entropy production and temporal disorder:

Toward a statistical thermodynamics for out-of-equilibrium nanosystems

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

time asymmetry in nonequilibrium statistical mechanics pierre gaspard brussels, belgium j. r....

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