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
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∂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
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→ρ
'
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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:
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diS
dt st
= Aγ Jγ
γ =1
c
∑ ≥ 0
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Jγ =1
tjγ (t ')
0
t
∫ dt'
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Aγ =ΔGγ
T=
Gγ − Gγeq
T
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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
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t → +∞
Schnakenberg network theory (Rev. Mod. Phys. 1976): cycles in the graph of the process
BEYOND LINEAR RESPONSE & ONSAGER RECIPROCITY RELATIONS
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q( λ γ ,Aγ{ }) = limt →∞
−1
tln exp − λ γ jγ t '( )dt'
0
t
∫γ
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
noneq.
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Jα =∂q
∂λα λα = 0
= Lαβ Aβ
β
∑ + Mαβγ Aβ Aγ
β ,γ
∑ + Nαβγδ Aβ Aγ
β ,γ ,δ
∑ Aδ +L
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Mαβγ = 12 Rαβ ,γ + Rαγ ,β( )
€
Lαβ = Lβα is totally symmetric
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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:
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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)
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q( λ γ ,Aγ{ }) = q( Aγ − λ γ , Aγ{ })
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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
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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
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deS
€
diS ≥ 0
Entropy production:
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1
kB
diS
dt= hR − h ≥ 0
P. Gaspard, J. Stat. Phys. 117 (2004) 599
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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
thermodynamic entropy production:
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d
dtpt (ω') = pt (ω)Wωω ' − pt (ω')Wω 'ω[ ]
ω
∑
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d
dtp(ω') = 0
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h(τ ) = lne
τ
⎛
⎝ ⎜
⎞
⎠ ⎟ p(ω)Wωω '
ω≠ω '
∑ − p(ω)Wωω '
ω≠ω '
∑ lnWωω ' + O(τ )
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hR (τ ) = lne
τ
⎛
⎝ ⎜
⎞
⎠ ⎟ p(ω)Wωω '
ω≠ω '
∑ − p(ω)Wωω '
ω≠ω '
∑ lnWω 'ω + O(τ )
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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:
thermodynamic entropy production:
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
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limδ →0
(hR − h) = − λ j
j
∑ =1
kB
diS
dt€
limδ →0
hR = − λ j
λ j <0
∑
INTERPRETATION
€
nonequilibrium steady state:
thermodynamic entropy production:
€
1
kB
diS
dt= hR − h ≥ 0
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P(ω) = P(ω0ω1ω2L ωn−1) ≈ exp −n Δt h( )
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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:
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R = α /k = 3.05 10−3s
DRIVEN BROWNIAN MOTION
trap stiffness:
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k = 9.62 10−6 kg s−2
trap velocity:
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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:
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dx
dt= −
x − ut
τ R
+2kBT
αξ t
dissipated heat:
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Qt = −k ˙ x t ' (x t ' − ut') dt'0
t
∫
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Qt = α u2t
driving force:
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F = −k(x − ut)
€
position x
€
friction α
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white noise ξ t
mean dissipated heat:
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relative position z = x − ut
u < 0
u > 0
comoving frame of reference:
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dz
dt= −
z
τ R
− u +2
αβξ t
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z ≡ x − ut
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pst (z) =βk
2πexp −
βk
2(z + uτ R )2 ⎡
⎣ ⎢ ⎤ ⎦ ⎥
PATH PROBABILITIES OF NONEQUILIBRIUM FLUCTUATIONS
thermodynamic entropy production:
stationary probability density:
path probability:
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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:
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diS
dt= lim
t →∞
kB
tDzt P+∫ zt[ ] ln
P+ zt[ ]
P− ztR
[ ]
D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601
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β ≡1
kBT
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Qt = W t − ΔVt heat generated by dissipation:
path:
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Zm = z(mτ ),...,z(mτ + nτ − τ )[ ]
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H(ε,τ ,n) = −1
Mln P+ Zm[ ]
m=1
M
∑
RELATIONSHIP TO DYNAMICAL RANDOMNESS
thermodynamic entropy production:
()-entropy per unit time:
path probability:
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diS
dt= kB lim
ε ,τ →∞hR (ε,τ ) − h(ε,τ )[ ]
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P+ Zm[ ] = P z(mτ ) − z( jτ ) < ε,..., z(mτ + nτ − τ ) − z( jτ + nτ − τ ) < ε[ ]
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H R (ε,τ ,n) = −1
Mln P− Zm
R[ ]
m=1
M
∑
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h(ε,τ ) = limn →∞
1
τH(ε,τ ,n +1) − H(ε,τ ,n)[ ]
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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:
D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601
algorithm of time series analysis by Grassberger & Procaccia (1980’s)
DRIVEN BROWNIAN MOTION
thermodynamic entropy production:
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=k 0.558 nm k =1- 20
sampling frequency: 8192 Hz
resolution:
()-entropy
time-reversed ()-entropy
time series:
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2 107
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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
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=k 0.558 nm k =11- 20
resolution:
dissipated heat along the random path zt
potentiel velocity:
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u = ± 4.24 μm/s
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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:
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R = 9.22 MΩ C = 278 pF τ R = RC = 2.56 ms
DRIVEN ELECTRIC CIRCUIT
thermodynamic entropy production
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diS
dt=
R I2
TJoule law:
D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601
CONCLUSIONSBreaking of time-reversal symmetry in the statistical description
Nonequilibrium work fluctuation theorem: systems driven by an external forcing
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Wdiss
kBT= −ln pR (−W)[ ] − −ln pF(W)[ ]
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Dk2 ≈ −Re sk = λ (DH) −h(DH)
DH
Nonequilibrium modes of diffusion: relaxation rate sk, Pollicott-Ruelle resonance
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D π /L( )2
≈ γ = λ i
λ i >0
∑ − hKS
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟L
Nonequilibrium transients: escape-rate formalism: fractal repeller
diffusion D : (1990)
viscosity : (1995)
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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.
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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:
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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