TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS Pierre GASPARD Brussels, 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
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TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS Pierre GASPARD Brussels, Belgium J. R. Dorfman, College ParkS. Ciliberto, Lyon T. Gilbert, BrusselsN.
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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
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:
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
thermodynamic entropy production:
€
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
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