TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS Pierre GASPARD Brussels, Belgium J. R. Dorfman, College Park T. Gilbert, Brussels D. Andrieux, Brussels
S. Ciliberto, Lyon N. Garnier, Lyon 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 Θ(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.
∂p = {H, p} = Lˆ p ∂t
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 Θ T :
ΘT≠T Irreversible behavior is obtained by weighting differently the trajectories T and their time-reversal image Θ 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. Gaspard, Physica A 369 (2006) 201-246.
STOCHASTIC DESCRIPTION IN TERMS OF A MASTER EQUATION A trajectory is a solution of Hamilton’s equations of motion: Γ(t;r0,p0) Coarse-graining: cell ω in the phase space stroboscopic observation of the trajectory with sampling time Δt : Γ(nΔt;r0,p0) in cell ωn path or history: ω = ω0ω1ω2…ωn−1 Liouville’s equation of the Hamiltonian dynamics -> reduced description in terms of the coarse-grained states ω -> master equation for the probability to visit the state ω by the time t : Pt(ω)
d Pt (ω ) = ∑ [ Pt (ω ') W ρ (ω '| ω ) − Pt (ω ) W− ρ (ω | ω ')] = 0 dt steady state ρ ,ω '(≠ω ) ρ
W ρ (ω | ω ') €
rate of the transition ω →ω '
due to the elementary process ρ = ±1,...,±r
€ -> € statistical description of the equilibrium and nonequilibrium fluctuations
FLUCTUATION THEOREM FOR THE CURRENTS steady state fluctuation theorem for the currents (2004): fluctuating currents:
1 Jγ = t
t
∫ j (t') dt' γ
0
affinities or thermodynamic forces:
€
ex: • electric currents in a nanoscopic conductor • rates of chemical reactions • velocity of a linear molecular motor • rotation rate of a rotary molecular motor
ΔGγ Gγ − Gγeq Aγ = = T T
Schnakenberg network theory (Rev. Mod. Phys. 1976): cycles in the graph of the process
{Jγ = αγ } ≈ e P{Jγ = −α γ }
€P
t kB
∑ Aγ αγ
t → +∞
γ
-> Onsager reciprocity relations and their generalizations to nonlinear response
€
thermodynamic entropy production:
€
c di S = ∑ Aγ Jγ ≥ 0 dt st γ =1
D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Phys. 127 (2007) 107.
BEYOND LINEAR RESPONSE & ONSAGER RECIPROCITY RELATIONS
1 generating function of the currents: q({λγ , Aγ }) = lim− ln exp−∑ λγ t →∞ t γ (Schnakenberg network theory)
average current:
= ∑ Lαβ Aβ + ∑ Mαβγ Aβ Aγ + €
λα = 0
linear response coefficients: Lαβ (Green-Kubo formulas)
€
Onsager reciprocity relations:
∫ 0
jγ ( t') dt'
noneq.
q({λγ , Aγ }) = q({ Aγ − λγ , Aγ })
fluctuation theorem for the currents:
€ ∂q Jα = ∂λα
t
β ,γ
β
[ jα (t) −
Lαβ = Lβα
higher-order nonequilibrium € coefficients:
Aβ Aγ Aδ + L
jα
][ jβ (0) −
jβ
]
is totally symmetric
Mαβγ = Rαβ ,γ
αβγδ
β ,γ ,δ
1 ∂ 2q 1 +∞ =− (0;0) = ∫ 2 ∂λα∂λβ 2 −∞
€ response: relations for nonlinear
∑N
1 2
(R
αβ ,γ
+ Rαγ ,β )
∂ 3q =− (0;0) ∂λα∂λβ ∂Aγ
D. Andrieux & P. Gaspard, J. Chem. € Phys. 121 (2004) 6167; J. Stat. Mech. (2007) P02006.
dt
FLUCTUATIONS AND MICROREVERSIBILITY Microreversibility: Hamilton’s equations are time-reversal symmetric. If Γ(t;r0,p0) is a solution of Hamilton’s equation, then Γ(−t;r0,−p0) is also a solution. But, typically, Γ(t;r0,p0) ≠ Γ(−t;r0,−p0). Coarse-graining: cell ω in the phase space stroboscopic observation of the trajectory with sampling time Δt : path or history: ω = ω0ω1ω2…ωn−1
Γ(nΔt;r0,p0) in cell ωn
If ω = ω0ω1ω2…ωn−1 is a possible path, then ωR = ωn−1…ω2ω1ω0 is also a possible path. But, again, ω ≠ ωR. €
Statistical description: probability of a path or history: equilibrium steady state: nonequilibrium steady state:
Peq(ω0ω1ω2…ωn−1) = Peq(ωn−1…ω2ω1ω0) Pneq(ω0ω1ω2…ωn−1) ≠ Pneq(ωn−1…ω2ω1ω0)
In a nonequilibrium steady state, ω and ωR have different probability weights. Explicit breaking of the time-reversal symmetry by the nonequilibrium boundary conditions
DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS nonequilibrium steady state:
P (ω0 ω1ω2 … ωn−1) ≠ P (ωn−1 … ω2 ω1 ω0)
If the probability of a typical path decays as P(ω) = P(ω0 ω1 ω2 … ωn−1) ~ exp( −h Δt n ) the probability of the time-reversed path decays as P(ωR) = P(ωn−1 … ω2 ω1 ω0) ~ exp( −hR Δt n )
with hR ≠ h
entropy per unit time: dynamical randomness (temporal disorder) € h = lim n→∞ (−1/nΔt) ∑ω P(ω) ln P(ω)
time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 hR = lim n→∞ (−1/nΔt) ∑ω P(ω) ln P(ωR) The time-reversed entropy per unit time characterizes the dynamical randomness (temporal disorder) of the time-reversed paths.
THERMODYNAMIC ENTROPY PRODUCTION de S entropy flow
Second law of thermodynamics: entropy S
dS de S di S = + dt dt dt
with
di S ≥0 dt
entropy production
di S ≥ 0 €
1 di S € = hR − h ≥ 0 kB dt
€ Entropy production:
P. Gaspard, J. Stat. Phys. 117 (2004) 599
P(ω ) P(ω 0ω1ω 2 Lω n−1 ) nΔt ( h ≈e R = P(ω ) P(ω€ n−1 Lω 2ω1ω 0 )
€
R
−h
)
=e
nΔt d i S kB dt
hR ≥ h
Property:
1 di S 1 P(ω ) = lim P( ω ) ln ∑ R ≥0 n →∞ kB dt nΔt ω P(ω )
€ equality iff
€
P(ω) = P(ωR)
(relative entropy)
(detailed balance) which holds at equilibrium.
PROOF FOR CONTINUOUS-TIME JUMP PROCESSES Pauli-type master equation:
d pt (ω ') = ∑ [ pt (ω )Wωω ' − pt (ω ')Wω 'ω ] dt ω
nonequilibrium steady state:
d p(ω ') = 0 dt
€ τ-entropy per unit time:
P. Gaspard & X.-J. Wang, Phys. Reports 235 (1993) 291
€h(τ ) = ln e ∑ p(ω )Wωω ' − ∑ p(ω )Wωω ' lnWωω ' + O(τ ) τ ω ≠ω ' ω ≠ω ' time-reversed τ-entropy per unit time:
€
P. Gaspard, J. Stat. Phys. 117 (2004) 599
e h (τ ) = ln ∑ p(ω )Wωω ' − ∑ p(ω )Wωω ' lnWω 'ω + O(τ ) τ ω ≠ω ' ω ≠ω ' R
thermodynamic entropy production:
1 p(ω )Wωω ' 1 di S h R (τ ) − h(τ )€= ∑ [ p(ω )Wωω ' − p(ω ')Wω 'ω ] ln + O(τ ) ≈ 2 ω ≠ω ' p(ω ')Wω 'ω k B 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: €
lim h =
δ →0
∑λ
j
= hKS
λ j >0
lim h R = − ∑ λ j
δ →0
λ j 0
€
t
dissipated heat: € Qt = −k ∫ x˙ t' (x t' − ut') dt' 0
mean dissipated heat:
€
u
