Fluctuation relations and nonequilibrium
thermodynamics – VII
Alberto Imparato† and Luca Peliti‡
†
Dipartimento di Fisica, Unità CNISM and Sezione INFN
Politecnico di Torino, Torino (Italy)
‡
Dipartimento di Scienze Fisiche, Unità CNISM and Sezione INFN
Università “Federico II”, Napoli (Italy)
Fluctuations VII – p.1/36
Stochastic non–equilibrium systems
Fluctuations VII – p.2/36
Motivations
“Simple” lattice systems
Steady states depends on initial state, boundary
conditions, and internal parameters
Bulk and boundary perturbations may induce phase
transition in steady state and dynamic behaviour
Heat conduction, diffusion, sand pile models,
avalanches, ....
Fluctuations VII – p.3/36
Random walk on 1d-lattice
W−
W+
Pl (t) probability that the particle is at site l at time t
dPl (t)
= W+ Pl−1 + W− Pl+1 − (W+ + W− )Pl = Jl−1/2 − Jl+1/2
dt
Jl+1/2 = W+ Pl − W− Pl+1 ;
Jl−1/2 = W+ Pl−1 − W− Pl
Fluctuations VII – p.4/36
ASEP
the asymmetric simple exclusion process (ASEP)
β
α
W+
W−
ρA
γ
nl = 0, 1 occupation number, ρl (t) ≡ hnl (t)i
Jl+1/2
Jl−1/2
ρB
δ
dρl (t) = Jl−1/2 − Jl+1/2
dt
= W+ nl (1 − nl+1 ) − W− nl+1 (1 − nl )
= W+ nl−1 (1 − nl ) − W− nl (1 − nl−1 )
Fluctuations VII – p.5/36
ASEP: mean field approximation
hnl (t)nl±1 (t)i = hnl (t)i hnl±1 (t)i = ρl (t)ρl±1 (t)
dρl (t)
dt
dρ0 (t)
dt
dρN (t)
dt
=
W+ ρl−1 (1 − ρl ) + W− ρl+1 (1 − ρl ) − W+ ρl (1 − ρl+1 ) − W− ρl (1 − ρl−1 )
=
α(1 − ρ0 ) + W− ρ1 (1 − ρ0 ) − γρ0 − W+ ρ0 (1 − ρ1 )
=
δ(1 − ρN ) + W+ ρN −1 (1 − ρN ) − βρN − W− ρN (1 − ρN −1 )
a = L/N, x = la, ν = a(W+ − W− ), µ = a2 (W+ + W− )/2
∂ρ
∂ρ
∂
µ
=
− νρ(1 − ρ)
∂t
∂x
∂x
Fluctuations VII – p.6/36
ASEP: mean field approximation II
Steady state: ∂t ρ = 0
Boundary conditions
0 = α(1 − ρ0 ) + W− ρ1 (1 − ρ0 ) − γρ0 − W+ ρ0 (1 − ρ1 )
0 = δ(1 − ρN ) + W+ ρN −1 (1 − ρN ) − βρN − W− ρN (1 − ρN −1 )
p
W+ − W− + α + γ − (α − γ − W+ + W− )2 + 4αγ
ρ(0) =
2(W+ − W− )
p
W+ − W− − β − δ + (β − δ − W+ + W− )2 + 4βδ
ρ(L) =
2(W+ − W− )
Fluctuations VII – p.7/36
ASEP: mean field approximation III
N = 100, W+ = 1, W− = 0.75, ρA = 0.75, ρB = 0.25,
0.8
0.07
J
0.7
0.06
0.05
0.6
0.04
ρ
0 0.2 0.4 0.6 0.8 1
0.5
0.4
0.3
0.2
0
0.2
0.4
x
0.6
0.8
1
Fluctuations VII – p.8/36
ASEP: mean field approximation IV
Phase diagram
1
C
1 − ρ(L)
A
B
0
0
ρ(0)
1
A: low density phase, B: high density phase, C: high
current phase
Schütz and Domany, 1993
Fluctuations VII – p.9/36
Heat Fluctuation: Motivations
The concept of entropy is usually associated with
probability distributions (ensembles) via Gibbs’
formula.
Fluctuations VII – p.10/36
Heat Fluctuation: Motivations
The concept of entropy is usually associated with
probability distributions (ensembles) via Gibbs’
formula.
In nonequilibrium systems one can consistently define
the entropy production along a single trajectory, Crooks
PRE 1999, 2000; Qian PRE 2001; Seifert PRL 2005.
Fluctuations VII – p.10/36
Heat Fluctuation: Motivations
The concept of entropy is usually associated with
probability distributions (ensembles) via Gibbs’
formula.
In nonequilibrium systems one can consistently define
the entropy production along a single trajectory, Crooks
PRE 1999, 2000; Qian PRE 2001; Seifert PRL 2005.
Fluctuation theorems can be interpreted as giving
connections between the probability of
entropy-generating trajectories with respect to that of
entropy-annihilating trajectories.
Fluctuations VII – p.10/36
Heat Fluctuation: Motivations
The concept of entropy is usually associated with
probability distributions (ensembles) via Gibbs’
formula.
In nonequilibrium systems one can consistently define
the entropy production along a single trajectory, Crooks
PRE 1999, 2000; Qian PRE 2001; Seifert PRL 2005.
Fluctuation theorems can be interpreted as giving
connections between the probability of
entropy-generating trajectories with respect to that of
entropy-annihilating trajectories.
The entropy flow along a given trajectory is
experimentally accessible, Tietz et al. PRL 2006
Fluctuations VII – p.10/36
Stochastic systems
A stochastic system is described by a Markovian dynamics
X
dpi (t)
=
[Wij (t)pj (t) − Wji (t)pi (t)]
dt
j (6=i)
Path ω: ω(t) = ik if tk ≤ t < tk+1 , k = 0, 1, . . . , M ,
tM +1 = tf
Time-reversed path ω
e: ω
e (t) = ik if t̃k+1 ≤ t < t̃k ),
t̃ = t0 + tf − t
e ω ): probability of the forward and of the
P(ω), P(e
time-reversed path
by their initial states)
" (conditioned
#
M
X
Wik ik−1 (tk )
P(ω)
Q(ω) = − ln
.
ln
=−
e ω)
Wik−1 ik (tk )
P(e
Fluctuations VII – p.11/36
Entropy flow and entropy production
P
Gibbs entropy of the system S(t) = − i pi (t) ln pi (t),
(kB = 1).
X
X
dS
Wij
Wji pi
−
Wij pj ln
.
=−
Wij pj ln
dt
Wij pj
Wji
i6=j
i6=j
The first sum is non-negative (ln x ≤ x − 1): entropy
production rate
The second sum defines the entropy Sf flowing into the
reservoir
hQit : average of Q(ω), over all possible paths up to
time t
d hQit /dt = dSf /dt
Fluctuations VII – p.12/36
Q(ω) as heat exchange
"
#
P(ω)
Q(ω) = − ln
=−
e ω)
P(e
M
X
Wik ik−1 (tk )
.
ln
Wik−1 ik (tk )
k=1
if the detailed balance conditions holds for the transition
rates Wij (t), and an energy Ei (t) is associated to the
system states
Wji (t)/Wij (t) = exp {[Ei (t) − Ej (t)] /T }
T ln [Wji (t)/Wij (t)] represents the heat exchanged with the
reservoir in the jump from state j to state i.
Q(ω) is the heat exchanged with the reservoir along the trajectory ω
Fluctuations VII – p.13/36
Entropy probability distribution
function
Joint probability distribution φi (Q, t), that the system is
found at time t in state i, having exchanged a total entropy
flow Q
∆sij = log [Wji (t)/Wij (t)]: entropy which flows into the
reservoir in the jump j → i
P
φi (Q, t + τ ) ≃ φi (Q, t) + τ j (6=i) Wij φj (Q − ∆sij , t) − Wji φi (Q, t)
n
hP
i
o
n n
P
∞ (−∆sij ) ∂ φj (Q,t)
= φi (Q, t) + τ j (6=i) Wij
− Wji φi (Q, t) ,
n=0
n!
∂Qn
Differential equation for φi (Q, t):
#
)
(
"∞
X
X (−∆sij )n ∂ n φj (Q, t)
∂φi (Q, t)
=
− Wji φi (Q, t) .
Wij
n
∂t
n!
∂Q
n=0
j (6=i)
Fluctuations VII – p.14/36
Generating Function
ψi (λ, t) =
Z
dQ eλQ φi (Q, t),
λ
P
∂ψi (λ, t)
W
= j (6=i) Wij Wji
ψj (λ, t) − Wji ψi (λ, t)
ij
∂t
P
= j Hij (λ) ψj (λ, t).
Lebowitz and Spohn, J. Stat. Phys. 1999.
g(λ) M.E. of Hij (λ), in the limit t → ∞, ψ(λ, t) = exp [tg(λ)]
Z
dλ −λQ
t g(λ∗ )−λ∗ Q
φ(Q, t) =
e
ψ(λ, t) ∝ e
2πi
λ∗ saddle point value implicitly defined by ∂g/∂λ|λ∗ = Q/t.
Fluctuations VII – p.15/36
Perron–Frobenius theorem
M, n × n matrix with positive entries mij > 0. Then the
following statements hold:
There is a positive real eigenvalue α∗ of M such that
|αi | < α∗ .
the eigenvalue α∗ is simple
vR right eigenvector: M · vR = α∗ vR , with vRi > 0
P
P
∗
eigenvalue estimate mini j Mij ≤ α ≤ maxi j Mij
Fluctuations VII – p.16/36
Maximum eigenvalue
ψ̇(λ, t) = H(λ)ψ(λ, t) ⇒ ψ(λ, t) = eH(λ)t ψ(λ, t = 0)
X
X
ψ(λ, t = 0) =
ci ψ i ⇒ ψ(λ, t) =
ci ψ i eαi (λ)t
i
i
ψ(λ, t → ∞) ∝ ψ max eg(λ)t
Z
XZ
ψ(λ, t) =
dQeλQ Φ(Q) =
dQeλQ φj (Q, t)
=
X
j
j
ψj (λ, t) −−−→ eg(λ)t
t→∞
Fluctuations VII – p.17/36
Gallavotti–Cohen relation
Hij (λ) = Wij
Wji
Wij
λ
;
Hij (1 − λ) = Wji
Wij
Wji
λ
= Hji (λ)
H(1 − λ) = H T (λ)
Z
dλ −λQ
t g(λ∗ )−λ∗ Q
φ(Q, t) =
e
ψ(λ, t) ∝ e
; ∂g/∂λ|λ∗ = Q/t
2πi
Z
Z
′
dλ
dλ
λQ
−λ′ Q
−Q
−Q
e ψ(λ, t) = −
e
ψ(1 − λ′ , t)
e φ(−Q, t) = e
2πi
2πi
∝ e
t g(λ∗ )−λ∗ Q
λ′ = 1 − λ
φ(Q, t)/φ(−Q, t) = e−Q , for, t → ∞
Fluctuations VII – p.18/36
Comparison with experiments
C. Tietz , S. Schuler , T. Speck , U. Seifert and J. Wrachtrup , PRL 2006
Two-state system: an optically driven defect center in
diamond
If excited by a red light laser, the defect exhibits
fluorescence.
By superimposing a green light laser, the rate of
transition from the non-fluorescent to the fluorescent
state (W+ ) and from the fluorescent to the
non-fluorescent state (W− ), turn out to depend linearly
on the lasers intensity
Fluctuations VII – p.19/36
Comparison with experiments
the experimental set-up
W− = (21.8 ms)−1 , W+ (t) = W0 [1 + γ sin(2πt/tm )],
W0 = (15.6 ms)−1 , γ = 0.46, tm = 50 ms.
entropy flux measured, over 2000 trajectories of
time-length 20 tm .
400
Solve the equation for ψ+ , ψ− ,
and compute
φ(q, t) ∝ exp [tg(λ∗ ) − λ∗ Q],
where
g(λ) = 1t log [ψ+ (λ, t) + ψ− (λ, t)]
in the limit t → ∞
300
200
100
0
-4
-2
0
2
4
6
8
−q
Fluctuations VII – p.20/36
Large systems
N equations for the φi (Q) or the ψi (λ): the direct
approach becomes rapidly impracticable, as the
system phase space size increases
Evaluation of φ(Q) by direct simulation of the
stochastic process described by the master equation:
highly difficult task, since one is interested in the tails
of the distribution (rare events)
Solution: sample trajectories in the weighted ensemble
P(ωt ) exp [λQ(ωt )] rather than successions of single
states in the unbiased ensemble.
Fluctuations VII – p.21/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
Fluctuations VII – p.22/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
We haveR
∂ψ(λ,t)
λQ(ωt )
=
Dω
Q(ω
)P(ω
)
e
= hQiλ ψ(λ, t)
t
t
t
∂λ
Fluctuations VII – p.22/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
We haveR
∂ψ(λ,t)
λQ(ωt )
=
Dω
Q(ω
)P(ω
)
e
= hQiλ ψ(λ, t)
t
t
t
∂λ
h. . .iλ is the average in the weighted ensemble
P(ωt ) exp [λQ(ωt )] ψ(λ, t)
Fluctuations VII – p.22/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
We haveR
∂ψ(λ,t)
λQ(ωt )
=
Dω
Q(ω
)P(ω
)
e
= hQiλ ψ(λ, t)
t
t
t
∂λ
h. . .iλ is the average in the weighted ensemble
P(ωt ) exp [λQ(ωt )] ψ(λ, t)
ψ(λ, t) = Zλ is the “partition function” of this ensemble,
Fluctuations VII – p.22/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
We haveR
∂ψ(λ,t)
λQ(ωt )
=
Dω
Q(ω
)P(ω
)
e
= hQiλ ψ(λ, t)
t
t
t
∂λ
h. . .iλ is the average in the weighted ensemble
P(ωt ) exp [λQ(ωt )] ψ(λ, t)
ψ(λ, t) = Zλ is the “partition function” of this ensemble,
hR
i
λ
The solution reads: ψ(λ, t) = exp 0 dλ′ hQiλ′
Fluctuations VII – p.22/36
Biased trajectories
Since ψ(λ, t) =
R
Dωt P(ωt ) eλQ(ωt )
We haveR
∂ψ(λ,t)
λQ(ωt )
=
Dω
Q(ω
)P(ω
)
e
= hQiλ ψ(λ, t)
t
t
t
∂λ
h. . .iλ is the average in the weighted ensemble
P(ωt ) exp [λQ(ωt )] ψ(λ, t)
ψ(λ, t) = Zλ is the “partition function” of this ensemble,
hR
i
λ
The solution reads: ψ(λ, t) = exp 0 dλ′ hQiλ′
hQiλ =
R
Dωt (Q(ωt )/Π(ωt ))Π(ωt )P(ωt )eλQ(ωt )
R
Dωt (1/Π(ωt ))Π(ωt )P(ωt )eλQ(ωt )
=
hQ/Πiλ,Π
h1/Πiλ,Π
h. . .iλ,Π is the average in the P(ωt )Π(ωt ) exp [λQ(ωt )]
ensemble
Fluctuations VII – p.22/36
Biased trajectories II
Probability of a given path ω : P(ω) = KiN ,iN −1 KiN ,iN −1 . . . Ki1 ,i0 p0i0
P
transition probabilities Kij = τ Wij , and Kii = 1 − j(6=i) Kji
Define the new transition probabilities
P
λ
e ji
e
e
Kij = τ Wij (Wji /Wij ) , and Kii = 1 − j(6=i) K
Q
Π(ω) = M
k=1 Πik ,ik−1 (tk ) with
8
<1,
if i 6= j;
Πij (t) =
:K
e jj (t)/Kjj (t), if i = j.
e i ,i
e i ,i
e i ,i p0 ,
P(ω)Π(ω) exp [λQ(ω)] = K
K
...K
1 0 i0
N N −1
N N −1
Evaluate hQiλ =
hQ/Πiλ,Π
h1/Πiλ,Π
Fluctuations VII – p.23/36
ASEP
the asymmetric simple exclusion process (ASEP)
β
α
W+
W−
ρA
ρB
γ
δ
By taking ρA > ρB and W+ > W− , one observes a net
particle current from the left to the right reservoir.
L = 100, W+ = 1, W− = 0.75, ρA = 0.75, ρB = 0.25,
maximum current phase
G. Schütz and E. Domany , J. Stat. Phys. 1993
Fluctuations VII – p.24/36
large deviation function
12
10
8
g
6
4
2
0
-2
-2
ψ(λ, t) = exp
hR
λ
0
-1
dλ′ hQiλ′
0
i
1
2
3
λ
g(λ) = 1t log [ψ(λ, t)] in the long t limit.
Fluctuations VII – p.25/36
Comparison with simulations
entropy flow per time unit q = Q/t, corresponding to 1000
unbiased trajectories.
0.7
f
0.6
0
-10
0.5
-20
-10 -5
0
5
10
φ
0.4
0.3
0.2
0.1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
q
f (q) = log [φ(q)] and f (q) + q exhibit the symmetry required
by the Gallavotti-Cohen relation φ(Q)/φ(−Q) = exp(Q)
Gallavotti and Cohen, J. Stat. Phys. 1995
Fluctuations VII – p.26/36
Typical trajectories
current J(λ) of particles that, in the steady state of
weighted ensembles, jump to the right (positive current) or
to the left (negative current), in the unit time.
Measured as biased trajectories are generated
0.5
0.4
0.3
0.2
J
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-4
-3
-2
-1
0
1
2
3
4
5
λ
Fluctuations VII – p.27/36
Trajectory thermodynamics
the parameter λ, which is the thermodynamic conjugate of
q, selects dynamical trajectories in the same way as an
external field selects states in an ordinary statistical
ensemble
1
f (q) ≡ g(λ ) − λ q = lim log φ(Q, t).
t→∞ t
∗
∗
the functions g(λ) and f (q) are Legendre transform of each
other, and can be then interpreted in terms of path
thermodynamics: g(λ) can be viewed as a path Gibbs free
energy, while f (q) is the corresponding Helmholtz free
energy.
Ritort J. Stat. Mech. 2004; Imparato and Peliti , Phys. Rev. E 2005
Fluctuations VII – p.28/36
The Totally Asymmetric Exclusion
Process (TASEP)
At any given time step t, a given particle moves to the right
with probability α if the target site is empty
Configuration C = (ni ), ni ∈ {0, 1}, i = 1, L, periodic b.c.
Current J:
1, if one particle jumps to the right;
JC ′ C =
0, if nothing happens.
We wish to evaluate
e
Λ(λ)
=
*
exp λ
X
t
JCt+1 Ct
!+
Fluctuations VII – p.29/36
The large-deviation function
Prob[C0 , C1 , . . . , CT ] = UCT CT −1 · · · UC2 C1 · UC1 C0
Xh i
X
Ũ T
ŨCT CT −1 · · · ŨC1 C0 =
eΛ(λ) =
CT
C1 ,...,CT
CT C0
where
ŨC ′ C := eλJC′ C UC ′ C
Define
KC :=
X
ŨC ′ C ,
UC′ ′ C ≡ ŨC ′ C KC−1
C′
e
Λ(λ)
=
X
UC′ T CT −1 KCT −1 · · · UC′ 1 C0 KC0
C2 ,...,CT
Fluctuations VII – p.30/36
The simulation steps
A cloning step:
PC (t + 1/2) = KC PC (t)
G clones of C : G =
[KC ] + 1,
[KC ],
with probability KC − [KC ];
otherwise
Fluctuations VII – p.31/36
The simulation steps
A cloning step:
PC (t + 1/2) = KC PC (t)
G clones of C : G =
[KC ] + 1,
[KC ],
with probability KC − [KC ];
otherwise
A shift step:
PC ′ (t + 1) =
X
UC′ ′ C PC (t + 1/2)
C
Fluctuations VII – p.31/36
The simulation steps
A cloning step:
PC (t + 1/2) = KC PC (t)
G clones of C : G =
[KC ] + 1,
[KC ],
with probability KC − [KC ];
otherwise
A shift step:
PC ′ (t + 1) =
X
UC′ ′ C PC (t + 1/2)
C
Overall cloning step with an adjustable rate
Mt = N/(N + G) (the same for all configurations)
Fluctuations VII – p.31/36
Results
0
-0.1
-0.2
-0.3
σ(λ)
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-1.1
-10
-8
-6
λ
-4
-2
0
For long times
1
Λ(λ)
− lim ln[MT · · · M2 · M1 ] = lim
= σ(λ)
t→∞ t
t→∞
t
Fluctuations VII – p.32/36
The configurations
Space-time diagram for a ring of N = 100 sites, λ = −50 and density 0.5
Note the logarithmic scale on the y-axis
Fluctuations VII – p.33/36
Moving shock waves
Space-time diagram for a ring of N = 100 sites, λ = −30 and density 0.3
Fluctuations VII – p.34/36
Discussion
Biased dynamics is effective to reconstruct entropy
flow distributions of systems with many states.
Fluctuations VII – p.35/36
Discussion
Biased dynamics is effective to reconstruct entropy
flow distributions of systems with many states.
entropy distributions can be used to reconstruct
thermodynamic equilibrium quantities.
Fluctuations VII – p.35/36
Discussion
Biased dynamics is effective to reconstruct entropy
flow distributions of systems with many states.
entropy distributions can be used to reconstruct
thermodynamic equilibrium quantities.
Gene regulation networks, signal processing networks,
molecular motors with many states.
Fluctuations VII – p.35/36
Bibliography
R. Stinchcombe Adv. in Pys. 5: 431–496 (2001).
G. Schütz and E. Domany, J. Stat. Phys. 72: 277, (1993).
C. Enaudand B. Derrida, J. Stat. Phys., 114: 537, (2004).
C. Tietz, S. Schuler, T. Speck, U. Seifert and J. Wrachtrup, Phys. Rev. Lett. 97:
050602, (2006).
S.X. Sun, J. Chem. Phys. 118: 5769, (2003).
H. Oberhofer, C. Dellago and P. L. Geissler, J. Phys. Chem. B 109: 6902, (2005).
A. Imparato, L. Peliti, J. Stat. Mech., L02001, (2007).
C. Giardinà, J. Kurchan, L. Peliti., Phys. Rev. Lett. 96, 120603 (2006).
Fluctuations VII – p.36/36