Stochastic thermodynamics
London, July 2013
Stochastic thermodynamics and the role of hidden slow degrees of freedoms
Udo Seifert II. Institut f¨ ur Theoretische Physik, Universit¨ at Stuttgart
Review: U.S., Rep. Prog. Phys. 75 126001, 2012.
1
• Intro on stoch th’dynamics • Hidden slow degrees of freedom – ... and the fluctuation theorem – ... in a molecular motor – ... in cellular sensing (i.e., in a bipartite Markov process)
2
• Stochastic thermodynamics for small systems
W λ0
T, p
λt
driving: mechanical
(electro)chemical
(bio)chemical
– First law: how to define work, internal energy and exchanged heat? – fluctuations imply distributions:
p(W ; λ(τ )) ...
– entropy: distribution as well?
3
• Stochastic thermodynamics
applies to such systems where
– non-equilibrium is caused by mechanical, hydrodyn. or chemical forces – ambient solution/reservoirs provides thermal bath(s) of well-defined Ti and µi – “all” slow variables are observed
• Main idea:
Energy conservation (1st law) and entropy production (2nd law) along a single stochastic trajectory
• Precursors: – notion “stoch th’dyn” by Nicolis, van den Broeck mid ‘80s ( on ensemble level) – fluct’theorem: Evans, Cohen, Galavotti, Kurchan, Lebowitz & Spohn ’90s – stochastic energetics (1st law) by Sekimoto late ‘90s – non-eq work theorems by Jarzynski and Crooks late ’90’s – quantities like stochastic entropy by Crooks, Qian, Gaspard in early ’00s – ...
4
• Paradigm for mechanical driving: V (x, λ) x1
f (λ)
x4 x6 x3
x0
λ(τ )
x5 x2
x
λ(τ )
– Langevin dyn’s
x˙ = µ [−V ′(x, λ) + f (λ)] +ζ
– external protocol λ(τ ) • First law [(Sekimoto, 1997)]: – applied work:
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
|
{z F (x,λ)
}
hζζi = 2µ kB T δ(...) | {z } (≡1)
dw = du + dq
– internal energy:
dw = ∂λV (x, λ)dλ + f (λ)dx du = dV
– dissipated heat:
dq = dw − du = F (x, λ)dx = T dsm
5
• Experimental illustration: Colloidal particle in V (x, λ(τ )) [V. Blickle, T. Speck, L. Helden, U.S., C. Bechinger, PRL 96, 070603, 2006]
– work distribution – non-Gaussian distribution ⇒ – Langevin valid beyond lin response [T. Speck and U.S., PRE 70, 066112, 2004]
6
• Stochastic entropy
[U.S., PRL 95, 040602, 2005]
– Fokker-Planck equation ∂τ p(x, τ ) = −∂xj(x, τ ) = −∂x (µF (x, λ) − µ∂x) p(x, τ ) – Common non-eq ensemble entropy
[kB ≡ 1]
R
S(τ ) ≡ − dx p(x, τ ) ln p(x, τ ) – Stochastic entropy for an individual trajectory x(τ ) s(τ ) ≡ − ln p(x(τ ), τ )
with hs(τ )i = S(τ )
Rt – ∆stot ≡ ∆sm+∆s = 0 dτ x(τ ˙ )ν(x(τ )) [with ν(x) ≡ hx|xi ˙ = j(x)/p(x)]
– hexp[−∆stot]i = 1 ⇒ h∆stot i ≥ 0 ∗ arbitrary initial state, driving, length of trajectory ∗ cf. Jarzynski relation (PRL ’97)
hexp[−W ]i = exp[−∆F ]
7
• NESSs: Examples and common characteristics
f (λ)
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
– Time-independent driving beyond linear response regime – Broken detailed-balance – Persistent “currents” with permanent dissipation
8
• Fluctuation theorem p(−∆stot)/p(∆stot ) = exp(−∆stot) long-time limit: Evans et al (1993), Gallavotti & Cohen (1995), Kurchan (1998), Lebowitz & Spohn (1999) ...
finite times: U.S., PRL’05
– experimental data [Speck, Blickle, Bechinger, U.S., EPL 79 30002 (2007)] f (λ)
~ a)
0 -50 -100 -150 -200 -250
t=2s
0
t=20s
200
b)
600
0 -50 -100 -150 -200 -250
t=2s
0
400
200
800
t=20s
400
600
800
1000
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
t=21.5
1000
total entropy production ∆stot [kB]
9
• F’theorem and slow hidden degrees of freedom [J. Mehl, B. Lander, C. Bechinger, V. Blickle and U.S., PRL 108, 220601, 2012]
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111
dτ [x˙1ν1(x1, x2) + x˙2ν2(x1, x2)]
f (λ)
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111
+1
(a)
with ν1(x1, x2) ≡ hx˙1|x1, x2i obeys FT
p(∆stot )/p(−∆stot ) = exp ∆stot
+-
min.
f1
R
pR
0,1500 0,1650 0,1800 0,1950 0,2100 0,2250 0,2400 0,2550 0,2700 0,2850 0,3000
max.
0
0
x1 B
(b)
G=0 -1 +1
G x2
f2 0
– suppose x2 is hidden:
+-
pR
0
R
ν ˜1(x1) ≡
ν(x1, x2)p(x2|x1)dx2
– apparent entropy production
potential (units of kBT)
(c)
R
G=300
600
U1
300
U2
0
-300 -600 0
x1, x2 (pR)
+1
(d)
G=1150 -1
-1
0
x1 (pR)
Rt ∆˜ stot ≡ 0 dτ x˙1ν ˜1(x1)
-1 +1
0
-1
x (pR)
0
x2
f (λ)
x (pR)
∆stot ≡
Rt
x1
obeys FT ?? 10
+1
x (pR)
– total entropy production in the NESS
• Experimental data – with and without coupling
6 4 2 0 -3
-2
-1
0
1
2
3
3 2
3
(c)
~ ln[p(Ds )/p(-Ds~tot)] tot
ln[p(Ds~tot)/p(-Ds~tot)]
-2
p(Ds~tot) ( 10 )
8
[rarely:]
1 0 -1 -2 -3 -3
-2
-1
0
1
2
3
0 -1 -2
(a) -2
-1
0
1
2
3
~
s
Dstot
1
-3 -3
~ D tot
~
2
Dstot
– slope α 1,1
slope a
1,0 0,9 0,8 0,7 0,6
(a) 0
(b) 100
200
G
300
0,5
1,0
1,5
2,0
t (s)
11
• Theory for
f (σ) ≡ ln[p(σ)/p(−σ)]
– for small t :
with
f (σ) ≈ σ + t1/2g(σ) + O(t):
σ ≡ ∆˜ stot FT universal
– for any t is f (σ) asymmetric by construction ∗ σ≪1: ∗ σ≫1:
f (σ) ≈ α(t)σ + γ(t)σ 3 + ... (i)
! R
1=
dσ p(σ) exp[−f (σ)]
(ii) if p(σ) ≥ (i)+(ii)
=⇒
Gaussian for |σ| >> 1
linear slope expected, but typically α(σ → 0) 6= α(σ → ∞)
12
• F1-ATPase and the fluctuation theorem [K. Hayashi, ... H. Noji, PRL 104, 218103 (2010)]
– kinetics vs thermodynamics – first law? – efficiency(ies)?
13
• F1-ATPase and the fluctuation theorem
– Γθ˙ = N + ζ
hζ1ζ2) = 2ΓkB T δ(τ1 − τ2)
⇒ ln[p(∆θ)/p(−∆θ)] = N ∆θ/kB T independent of friction coefficient
cf our two-ring system: 1,1
slope a
– cf f’theorem
1,0 0,9 0,8 0,7
ln[p(∆stot )/p(−∆stot )] = ∆stot/kB
0,6
(a) 0
(b) 100
200
G
300
0,5
1,0
1,5
t (s)
14
2,0
• Hybrid model
[E. Zimmermann and U.S., New J. Phys. 14, 103023, 2012]
– probe particle ∗ x˙ = µ(−∂y V (y) + f ex) + ζ
with
y(τ ) ≡ n(τ ) − x(τ )
– motor ∗ w+/w− = exp[∆µ − V (n + d, x) − V (n, x)] ∗ local detailed balance condition
15
• First law (i) probe
−f ex∆x
=
∆qp + ∆V|p
Sekimoto ’97
(ii) motor
0
=
∆qm + ∆V|m + ∆Esol
U.S., EPJE 34 26, 2011
mean
−f exv
=
˙p + Q ˙ m + ∆E ˙ sol Q
∆Esol = − ∆µ + T ∆Ssol
→
˙ − f exv ∆µ
=
˙p + Q
˙ m + T S˙ sol Q
| {z } not distinguishable
16
• Efficiencies – Thermodynamic efficiency
[Parmeggiani et al, PRE 1999]
˙ η ≡ f exv/∆µ – for f ex = 0:
pseudo efficiency
[Toyabe et al, PRL 2010]
˙ p/∆µ ˙ ηQ ≡ Q
4 4 4 4
40
∆µ
∆µ
17
• Experimental determination of ηQ [S. Toyabe et al, PRL 104, 198103 (2010)]
– Harada-Sasa relation ˙ P = v2 + µQ
Z
[PRL 2006]
dω[Cx˙ (ω) − 2kB T ReRx˙ (ω)]
from violation of fluc-diss-theorem
18
• Comparison to experiment
Θ+=0.1, Θ+=0.01, Exp.
– quite reasonable agreement for small θ+ – future: substeps of 90 + 30 degrees
19
• Measuring dissipation in a NESS [B. Lander, J. Mehl, V. Blickle, C. Bechinger and U.S. PRE 86, 030401, 2012]
– Harada Sasa: R −1 2 −1 dω[Cx˙ (ω) − 2kB T ReRx˙ (ω)] hqi ˙ =µ v +µ
∗ Rx˙ difficult to measure – heat production along a trajectory: D
E
hqi ˙ = µ−1 ν 2(x(t))
∗ trajectories x(t) from experiments ∗ ν(x) ≡ hx|xi ˙ from x(t)
20
• Classical non-autonomous information machines:
– Maxwell’s demon (1867)
– Szilard engine (1929)
but: Landauer’s principle (erasure of 1 bit of information costs kBT ln 2) recent theoretical developments: [Sagawa and Ueda PRL 2010, Horowitz and Parrondo EPL+NJP 2011, Berut et al Nature 2012, Mandal and Jarzynski PNAS 2012, Schaller and Esposito, EPL 2012, ....]
21
• Efficency of an information machine [M. Bauer, D. Abreu and U.S.; EPL 94 10001, 2011, J. Phys. A 45, 162001, 2012, PRL 108 030601, 2012 ]
* efficiency
(i) measurement (with error ym ) yields information: I
m) ≡ ln p(x|x eq p (x)
(ii) subsequent optimization of V (x, λ(τ )) extracts work W out
0 ≤ η ≡ W out /hIi ≤ 1
cf. hexp[wout − I )]i = 1 [Sagawa and Ueda, PRL’08]:
22
• Information and entropy production in cellular sensing [AC Barato, D Hartich and U.S., PRE 87, 042104, 2013, and 1306.1698]
– Berg & Purcell, – groups headed by Wingreen, Bialek, Rein ten Wolde, Vergassola ... – Lan et al, Nature Phys. 2012; Mehta and Schwab PNAS 2012]
Information about an external process {X(t)} (conc’ of a nutrient) is recorded by an internal variable {Y (t)} (conc’ of phosph’d protein) Q: Is the rate of information I acquired about {X(t)} related to the thermodynamic cost σ of maintaining the sensory network?
23
• Toy model γ XA YA
γ k−
k+
XB YA
XA YB
k+ γ γ
k−
XB YB
– external time series X(t) alternates between XA and XB – internal “record” Y (t) alternates between YA and YB – discretize time in steps τ – entropy rates
∗ hX,Y
P
P αW αβ (τ ) ln W αβ (τ ) P ≡ limN →∞ S(X 1, Y 1 , X 2, Y 2....X n, Y n)/N = −(1/τ ) αβij Piα Wijαβ (τ ) ln Wijαβ (τ )
∗ hX ≡ limN →∞ S(X 1 , X 2, ....X n)/N = −(1/τ )
αβ
∗ hY =??
– rate of mutual information I≡hX +hY −hX,Y
24
• Bounds on entropy rate – S(Y n+1|Y n, ..., Y 1, X 1) ≤ hY ≤ S(Y n+1|Y n, ...Y 1) – in limit τ → 0 upper and lower bounds for different n converge: –
P u I =
P α α ¯ ij αi Pi j6=i wij /w
P β with w ¯ij ≡ β P (β|i)wij
γ XA YA
XA YB
4
k+ γ γ
5
XB YA
γ k−
k+
I l =0
k−
First bounds Fifth bounds Ninth bounds Simulation
σ (u) I I
3
3
2
1
XB YB
1 0 0
0.01
τ
0.03
γ = 1, k− = 5, k+ = 25
0 0
5
10
15
20
k-
σ I possible
25
25
• A thermodynamically consistent model σ (u) I I
4
k on
off Y
on Y
k off
ω+ ω −
ω+ ω − κ+ κ−
a κ + aκ − k on
off Y*
2
on Y*
k off
0
1
10
a
100
I≡hX −hY −hX,Y (a)κ+
ω+
(a)κ−
ω−
⇀ − ⇀ Y + AT P − ↽ − Y + ADP + Pi ↽ −− − − Y ∗ + ADP − enzymatic enhancement a ∆µ = kB T ln(κ+ ω+ /κ− ω−)
rate of mutual information σ = th’dynamic entr’ prod’n I > σ
possible
26
• Intro on stoch th’dynamics
T. Speck
• Hidden slow degrees of freedom – ... and the fluctuation theorem B. Lander, J. Mehl, V. Blickle, C. Bechinger
– ... in a molecular motor
E. Zimmermann
– ... in cellular sensing (i.e., in a bipartite Markov process) AC Barato, D. Abreu, D. Hartich
27
Stochastic thermodynamics and the role of hidden slow degrees of freedoms
Udo Seifert II. Institut f¨ ur Theoretische Physik, Universit¨ at Stuttgart
Review: U.S., Rep. Prog. Phys. 75 126001, 2012.
1
• Intro on stoch th’dynamics • Hidden slow degrees of freedom – ... and the fluctuation theorem – ... in a molecular motor – ... in cellular sensing (i.e., in a bipartite Markov process)
2
• Stochastic thermodynamics for small systems
W λ0
T, p
λt
driving: mechanical
(electro)chemical
(bio)chemical
– First law: how to define work, internal energy and exchanged heat? – fluctuations imply distributions:
p(W ; λ(τ )) ...
– entropy: distribution as well?
3
• Stochastic thermodynamics
applies to such systems where
– non-equilibrium is caused by mechanical, hydrodyn. or chemical forces – ambient solution/reservoirs provides thermal bath(s) of well-defined Ti and µi – “all” slow variables are observed
• Main idea:
Energy conservation (1st law) and entropy production (2nd law) along a single stochastic trajectory
• Precursors: – notion “stoch th’dyn” by Nicolis, van den Broeck mid ‘80s ( on ensemble level) – fluct’theorem: Evans, Cohen, Galavotti, Kurchan, Lebowitz & Spohn ’90s – stochastic energetics (1st law) by Sekimoto late ‘90s – non-eq work theorems by Jarzynski and Crooks late ’90’s – quantities like stochastic entropy by Crooks, Qian, Gaspard in early ’00s – ...
4
• Paradigm for mechanical driving: V (x, λ) x1
f (λ)
x4 x6 x3
x0
λ(τ )
x5 x2
x
λ(τ )
– Langevin dyn’s
x˙ = µ [−V ′(x, λ) + f (λ)] +ζ
– external protocol λ(τ ) • First law [(Sekimoto, 1997)]: – applied work:
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
|
{z F (x,λ)
}
hζζi = 2µ kB T δ(...) | {z } (≡1)
dw = du + dq
– internal energy:
dw = ∂λV (x, λ)dλ + f (λ)dx du = dV
– dissipated heat:
dq = dw − du = F (x, λ)dx = T dsm
5
• Experimental illustration: Colloidal particle in V (x, λ(τ )) [V. Blickle, T. Speck, L. Helden, U.S., C. Bechinger, PRL 96, 070603, 2006]
– work distribution – non-Gaussian distribution ⇒ – Langevin valid beyond lin response [T. Speck and U.S., PRE 70, 066112, 2004]
6
• Stochastic entropy
[U.S., PRL 95, 040602, 2005]
– Fokker-Planck equation ∂τ p(x, τ ) = −∂xj(x, τ ) = −∂x (µF (x, λ) − µ∂x) p(x, τ ) – Common non-eq ensemble entropy
[kB ≡ 1]
R
S(τ ) ≡ − dx p(x, τ ) ln p(x, τ ) – Stochastic entropy for an individual trajectory x(τ ) s(τ ) ≡ − ln p(x(τ ), τ )
with hs(τ )i = S(τ )
Rt – ∆stot ≡ ∆sm+∆s = 0 dτ x(τ ˙ )ν(x(τ )) [with ν(x) ≡ hx|xi ˙ = j(x)/p(x)]
– hexp[−∆stot]i = 1 ⇒ h∆stot i ≥ 0 ∗ arbitrary initial state, driving, length of trajectory ∗ cf. Jarzynski relation (PRL ’97)
hexp[−W ]i = exp[−∆F ]
7
• NESSs: Examples and common characteristics
f (λ)
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
– Time-independent driving beyond linear response regime – Broken detailed-balance – Persistent “currents” with permanent dissipation
8
• Fluctuation theorem p(−∆stot)/p(∆stot ) = exp(−∆stot) long-time limit: Evans et al (1993), Gallavotti & Cohen (1995), Kurchan (1998), Lebowitz & Spohn (1999) ...
finite times: U.S., PRL’05
– experimental data [Speck, Blickle, Bechinger, U.S., EPL 79 30002 (2007)] f (λ)
~ a)
0 -50 -100 -150 -200 -250
t=2s
0
t=20s
200
b)
600
0 -50 -100 -150 -200 -250
t=2s
0
400
200
800
t=20s
400
600
800
1000
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000 111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111
t=21.5
1000
total entropy production ∆stot [kB]
9
• F’theorem and slow hidden degrees of freedom [J. Mehl, B. Lander, C. Bechinger, V. Blickle and U.S., PRL 108, 220601, 2012]
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111
dτ [x˙1ν1(x1, x2) + x˙2ν2(x1, x2)]
f (λ)
111111111111111 000000000000000 000000000000000 111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 V (x, λ) 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 000000000000000 111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111
+1
(a)
with ν1(x1, x2) ≡ hx˙1|x1, x2i obeys FT
p(∆stot )/p(−∆stot ) = exp ∆stot
+-
min.
f1
R
pR
0,1500 0,1650 0,1800 0,1950 0,2100 0,2250 0,2400 0,2550 0,2700 0,2850 0,3000
max.
0
0
x1 B
(b)
G=0 -1 +1
G x2
f2 0
– suppose x2 is hidden:
+-
pR
0
R
ν ˜1(x1) ≡
ν(x1, x2)p(x2|x1)dx2
– apparent entropy production
potential (units of kBT)
(c)
R
G=300
600
U1
300
U2
0
-300 -600 0
x1, x2 (pR)
+1
(d)
G=1150 -1
-1
0
x1 (pR)
Rt ∆˜ stot ≡ 0 dτ x˙1ν ˜1(x1)
-1 +1
0
-1
x (pR)
0
x2
f (λ)
x (pR)
∆stot ≡
Rt
x1
obeys FT ?? 10
+1
x (pR)
– total entropy production in the NESS
• Experimental data – with and without coupling
6 4 2 0 -3
-2
-1
0
1
2
3
3 2
3
(c)
~ ln[p(Ds )/p(-Ds~tot)] tot
ln[p(Ds~tot)/p(-Ds~tot)]
-2
p(Ds~tot) ( 10 )
8
[rarely:]
1 0 -1 -2 -3 -3
-2
-1
0
1
2
3
0 -1 -2
(a) -2
-1
0
1
2
3
~
s
Dstot
1
-3 -3
~ D tot
~
2
Dstot
– slope α 1,1
slope a
1,0 0,9 0,8 0,7 0,6
(a) 0
(b) 100
200
G
300
0,5
1,0
1,5
2,0
t (s)
11
• Theory for
f (σ) ≡ ln[p(σ)/p(−σ)]
– for small t :
with
f (σ) ≈ σ + t1/2g(σ) + O(t):
σ ≡ ∆˜ stot FT universal
– for any t is f (σ) asymmetric by construction ∗ σ≪1: ∗ σ≫1:
f (σ) ≈ α(t)σ + γ(t)σ 3 + ... (i)
! R
1=
dσ p(σ) exp[−f (σ)]
(ii) if p(σ) ≥ (i)+(ii)
=⇒
Gaussian for |σ| >> 1
linear slope expected, but typically α(σ → 0) 6= α(σ → ∞)
12
• F1-ATPase and the fluctuation theorem [K. Hayashi, ... H. Noji, PRL 104, 218103 (2010)]
– kinetics vs thermodynamics – first law? – efficiency(ies)?
13
• F1-ATPase and the fluctuation theorem
– Γθ˙ = N + ζ
hζ1ζ2) = 2ΓkB T δ(τ1 − τ2)
⇒ ln[p(∆θ)/p(−∆θ)] = N ∆θ/kB T independent of friction coefficient
cf our two-ring system: 1,1
slope a
– cf f’theorem
1,0 0,9 0,8 0,7
ln[p(∆stot )/p(−∆stot )] = ∆stot/kB
0,6
(a) 0
(b) 100
200
G
300
0,5
1,0
1,5
t (s)
14
2,0
• Hybrid model
[E. Zimmermann and U.S., New J. Phys. 14, 103023, 2012]
– probe particle ∗ x˙ = µ(−∂y V (y) + f ex) + ζ
with
y(τ ) ≡ n(τ ) − x(τ )
– motor ∗ w+/w− = exp[∆µ − V (n + d, x) − V (n, x)] ∗ local detailed balance condition
15
• First law (i) probe
−f ex∆x
=
∆qp + ∆V|p
Sekimoto ’97
(ii) motor
0
=
∆qm + ∆V|m + ∆Esol
U.S., EPJE 34 26, 2011
mean
−f exv
=
˙p + Q ˙ m + ∆E ˙ sol Q
∆Esol = − ∆µ + T ∆Ssol
→
˙ − f exv ∆µ
=
˙p + Q
˙ m + T S˙ sol Q
| {z } not distinguishable
16
• Efficiencies – Thermodynamic efficiency
[Parmeggiani et al, PRE 1999]
˙ η ≡ f exv/∆µ – for f ex = 0:
pseudo efficiency
[Toyabe et al, PRL 2010]
˙ p/∆µ ˙ ηQ ≡ Q
4 4 4 4
40
∆µ
∆µ
17
• Experimental determination of ηQ [S. Toyabe et al, PRL 104, 198103 (2010)]
– Harada-Sasa relation ˙ P = v2 + µQ
Z
[PRL 2006]
dω[Cx˙ (ω) − 2kB T ReRx˙ (ω)]
from violation of fluc-diss-theorem
18
• Comparison to experiment
Θ+=0.1, Θ+=0.01, Exp.
– quite reasonable agreement for small θ+ – future: substeps of 90 + 30 degrees
19
• Measuring dissipation in a NESS [B. Lander, J. Mehl, V. Blickle, C. Bechinger and U.S. PRE 86, 030401, 2012]
– Harada Sasa: R −1 2 −1 dω[Cx˙ (ω) − 2kB T ReRx˙ (ω)] hqi ˙ =µ v +µ
∗ Rx˙ difficult to measure – heat production along a trajectory: D
E
hqi ˙ = µ−1 ν 2(x(t))
∗ trajectories x(t) from experiments ∗ ν(x) ≡ hx|xi ˙ from x(t)
20
• Classical non-autonomous information machines:
– Maxwell’s demon (1867)
– Szilard engine (1929)
but: Landauer’s principle (erasure of 1 bit of information costs kBT ln 2) recent theoretical developments: [Sagawa and Ueda PRL 2010, Horowitz and Parrondo EPL+NJP 2011, Berut et al Nature 2012, Mandal and Jarzynski PNAS 2012, Schaller and Esposito, EPL 2012, ....]
21
• Efficency of an information machine [M. Bauer, D. Abreu and U.S.; EPL 94 10001, 2011, J. Phys. A 45, 162001, 2012, PRL 108 030601, 2012 ]
* efficiency
(i) measurement (with error ym ) yields information: I
m) ≡ ln p(x|x eq p (x)
(ii) subsequent optimization of V (x, λ(τ )) extracts work W out
0 ≤ η ≡ W out /hIi ≤ 1
cf. hexp[wout − I )]i = 1 [Sagawa and Ueda, PRL’08]:
22
• Information and entropy production in cellular sensing [AC Barato, D Hartich and U.S., PRE 87, 042104, 2013, and 1306.1698]
– Berg & Purcell, – groups headed by Wingreen, Bialek, Rein ten Wolde, Vergassola ... – Lan et al, Nature Phys. 2012; Mehta and Schwab PNAS 2012]
Information about an external process {X(t)} (conc’ of a nutrient) is recorded by an internal variable {Y (t)} (conc’ of phosph’d protein) Q: Is the rate of information I acquired about {X(t)} related to the thermodynamic cost σ of maintaining the sensory network?
23
• Toy model γ XA YA
γ k−
k+
XB YA
XA YB
k+ γ γ
k−
XB YB
– external time series X(t) alternates between XA and XB – internal “record” Y (t) alternates between YA and YB – discretize time in steps τ – entropy rates
∗ hX,Y
P
P αW αβ (τ ) ln W αβ (τ ) P ≡ limN →∞ S(X 1, Y 1 , X 2, Y 2....X n, Y n)/N = −(1/τ ) αβij Piα Wijαβ (τ ) ln Wijαβ (τ )
∗ hX ≡ limN →∞ S(X 1 , X 2, ....X n)/N = −(1/τ )
αβ
∗ hY =??
– rate of mutual information I≡hX +hY −hX,Y
24
• Bounds on entropy rate – S(Y n+1|Y n, ..., Y 1, X 1) ≤ hY ≤ S(Y n+1|Y n, ...Y 1) – in limit τ → 0 upper and lower bounds for different n converge: –
P u I =
P α α ¯ ij αi Pi j6=i wij /w
P β with w ¯ij ≡ β P (β|i)wij
γ XA YA
XA YB
4
k+ γ γ
5
XB YA
γ k−
k+
I l =0
k−
First bounds Fifth bounds Ninth bounds Simulation
σ (u) I I
3
3
2
1
XB YB
1 0 0
0.01
τ
0.03
γ = 1, k− = 5, k+ = 25
0 0
5
10
15
20
k-
σ I possible
25
25
• A thermodynamically consistent model σ (u) I I
4
k on
off Y
on Y
k off
ω+ ω −
ω+ ω − κ+ κ−
a κ + aκ − k on
off Y*
2
on Y*
k off
0
1
10
a
100
I≡hX −hY −hX,Y (a)κ+
ω+
(a)κ−
ω−
⇀ − ⇀ Y + AT P − ↽ − Y + ADP + Pi ↽ −− − − Y ∗ + ADP − enzymatic enhancement a ∆µ = kB T ln(κ+ ω+ /κ− ω−)
rate of mutual information σ = th’dynamic entr’ prod’n I > σ
possible
26
• Intro on stoch th’dynamics
T. Speck
• Hidden slow degrees of freedom – ... and the fluctuation theorem B. Lander, J. Mehl, V. Blickle, C. Bechinger
– ... in a molecular motor
E. Zimmermann
– ... in cellular sensing (i.e., in a bipartite Markov process) AC Barato, D. Abreu, D. Hartich
27
