paris, fifem 2012, january 25. udo seifertdavid/fr2012/talk-udo.pdf · 2012. 1. 27. · paris,...

Paris, FIFEM 2012, January 25.

Stochastic thermodynamics of nano-machines

Udo Seifert

II. Institut für Theoretische Physik, Universität Stuttgart

Acknowledgments:

• D. Abreu and T. Schmiedl

1

• Rough classification of micro/nano machines

– autonomous (NESS) vs ext driven (cyclic or feedback)

– isothermal vs “heat engines”

– discrete states vs continuous states

2

• Smallest rotor: F1-ATP-ase

[K. Hayashi, ... H. Noji, PRL 104, 218103 (2010)]

– single steps

– kinetics vs thermodynamics

– first law?

3

• Stochastic th’dynamics of a driven enzym with internal states

[T.Schmiedl, T.Speck and U.S., JSP 128 77, 2007; U.S., EPJ E 34 26, 2011]

w0nm

A1

A2

A3

A1

A2

A3

n

m

w0mn

– A1 + nwnm⇀↽

wmnm+A2 +A3

– mass action law kinetics:

– wnmwmn=

w0nmw0mn

[A1]/[A2][A3]

– Fn = En − TSn

• First law w = ∆E + q for a single step:

– internal energy: ∆E ≡ Em −En +∆Esol

– dissipated heat: qnm ≡ w −∆E = lnwnm/wmn − T∆(Snm + Ssol)

4

• Experiment: Stochastic thermodynamics of F1-ATP-ase

[S. Toyabe et al, PRL 104, 198103 (2010)]

5

• Paradigmatic example: Efficiency of one step molecular motors

[cf. F. Jülicher at al, Rev. Mod. Phys. 1997]

l

F

V (x)δl

w+ = [ATP ] w+0 exp[−δl F ]

w− = [ADP ][P ] w−0 exp[(1− δ)l F ]

– Power output Pout = (w+ − w−) Fl

– Power input Pin = (w+ − w−)∆µ

– efficiency 0 ≤ η ≡ Pout/Pin ≤ 1

6

• Efficiency at maximum power:

[T. Schmiedl and U.S., EPL 83, 30005, 2008]

– Maximum power for fixed ∆µ: P ∗out = [w+(F ∗) − w−(F ∗)] F ∗l

η∗

10.50.20.1

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

∆µ/(kBT )

δ

l

F

V (x)δl

– universal (η∗ ≈ 1/2) close to eq,, i.e.. in lin response

cf [Esposito ... vd Broeck, PRL ’09, PRL ’10]

– η∗ can increase above 1/2 [ 6= Gaveau et al. PRL ’10]

7

• Efficiency of autonomous nano-machines at maximum power

[U.S., PRL 106, 020601, 2011]

– output: wout = fd

– input: win = ∆µ

– mean currents: j± ≡ 1/τ±

– local detailed balance: j−/j+ = exp(−∆S) = exp(wout − win)

– power:

Pout,in = (j+ − j−)× wout,in

= j+(wout, win) [1− exp(−∆S)] wout,in

8

- Efficiency: η ≡ wout/win

- Efficiency at maximum power (EMP)

* case I: fix win, maximize Pout with respect to wout:

* win = w∗out + ln

(

1+w∗out

1+xoutw∗out

)

with xout ≡ d ln j+/dwout

* EMP: η∗ ≡w∗outwin

≈ 1/2+ (xout +1/2)win.

xout=5

xout=1

2

xout=-2

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

Ωin

Η*

9

- Efficiency at maximum power η∗∗

* case III: maximize Pout with respect to wout and win

– three regimes

10

• Multicyclic machines

– strong coupling (SC) vs weak coupling (WC)

– SC: essentially as unicyclic case

– WC: less universal and η∗LR < 1/2

• similarly for autonomous heat engines: η∗ ≈ ηc/2

11

• Generalized Green-Kubo relations [U.S., PRL 104, 138101, 2010]

rates

– show sensitivity

rαmn ≡ ∂ lnwmn/∂hα

– obey local detailed balance

wmn(h)wnm(h)

= wmnwnm

∣∣∣|h=0

exp (hαdαmn/T)

12

– currents

– net probab’ty currents Kmn ≡ pmwmn − pnwnm

– ordinary current jα(t) ≡∑

j δ(t− tj)dαn−j n

+j

– sensitivity-weighted current

να(t) =∑

kKn(t)k rαn(t)k

/ pn(t)

– (differential) mobility µαβ = ∂jα/∂hβ

Dαβ = Tµαβ +∫∞0 dt 〈jα(t)(νβ(0)− jβ)〉

– additive relationship between dispersion and mobility,

better not: T effαβ ≡Dαβ/µαβ

13

• Isothermal machines with feedback: Work from a single heat bath?

– Maxwell’s demon (1867) – Szilard engine (1929)

but: Landauer’s principle (erasure of 1 bit of information costs kBT ln 2)

14

• Stochastic th’dynamics of a colloidal particle

λ(τ )

V (x, λ)

x

��

��

��

��

V (x, λ)

f (λ)

– Langevin dynamics ẋ = µ[−V ′(x, λ) + f(λ)] + ζ

with external driving λ(τ) and 〈ζ1ζ2〉 = 2µkBT︸︷︷︸(≡1)

δ12

– First law [(Sekimoto, 1997)]: dw = du+ dq

∗ applied work: dw = ∂λV (x, λ)dλ+ f dx

∗ internal energy: du = dV

∗ dissipated heat: dq = dw − du = [−∂xV (x, λ) + f ]dx = Tdsm

– stochastic entropy: ds ≡ −d [ln p(x, t)] [U.S., PRL ’05]

15

• Measurement and feedback

– initially thermal equilibrium peq(x)

– measurement yields xm (±ym, precision)

– acquired information: I ≡ lnp(x|xm)peq(x)

similar to stochastic entropy

– subsequent control λ(τ)|xm in order to extract work

16

• Fluctuation theorems with measurement and feedback

fixed λ without with

equilibrium 〈exp[−(w −∆F)]〉 = 1 〈exp[−(w −∆F − I)]〉 = 1

Jarzynski, PRL’97 Sagawa and Ueda, PRL’08

〈exp[−∆stot]〉 = 1 〈exp[−(∆stot + I)]〉 = 1

NESS U.S., PRL’05

��

��

��

��

V (x, λ)

f (λ)

〈exp[−(∆(stot − shk)]〉 = 1 〈exp[−(∆(stot − shk + I)]〉 = 1

Hatano and Sasa, PRL’01 D. Abreu & U.S., PRL 108 030601, 2012

with concise “universal” proof

17

• Fluctuation theorem implies bounds on power

IFT ”2nd law”

equilibrium 〈exp[−(w −∆F + I)]〉 = 1 Wout ≤ −∆F + I

〈exp[−(∆stot + I)]〉 = 1 ∆Stot ≥ −INESS

��

��

��

��

V (x, λ)

f (λ)

〈exp[−(∆(stot − shk + I)]〉 = 1 ∆Stot ≥ ∆Shk − I

⇒ ˙Wout ≤ Ẇ in − Q̇hk + İ

18

• Example: Brownian particle in a harmonic trap

[D. Abreu and U.S., EPL 94, 10001, 2011 ]

* measurement yields

xm (±ym, precision)

– Case 1: One control parameter (position)

0 xmb(0)

V(x,0)=V(x,t)V(x,0 )

V(x,t )-+

jumps

linear displacement

∗ Optimal extracted mean work:

Wout∗(t) = t

2(2+t)1

1+y2m−→ 1

2(1+y2m)

19

• Case 2: Two control parameters (position λ and stiffness k)

– Controlling both degrees of freedom leads to:

Wout∗(t → ∞) =

1

2ln

(

1+1

y2m

)

= −I

– optimal finite-time behaviour

20

• Feedback with genuine non-equilibrium states

[D. Abreu and U.S., PRL 108, 030601 (2012)]

w+1 = e(f−E)/2 = 1/w−1

w+2 = e(f+E)/2 = 1/w−2

- measure position every tm

- adjust energy of state 2 as ±|E|

21

• Results for cyclic operation

- optimal E(f, tm)

- for tm → ∞ : Ẇ in ≈ Q̇hk

- for tm → 0 : İ → ∞

- net gain for tm < t∗m

22

• Heat engines at maximal power

– Carnot (1824)

Th

Tc

−Qh

Qc

−W

– ηc ≡ 1− Tc/Thbut zero power

– Curzon-Ahlborn (1975)

Th

Tc

−W

−Qh = α(Th − Tmh )

Qc = β(Tmc − Tc)

Tmc

Tmh

– efficiency at maximum power

ηca ≡ 1−√

Tc/Th

23

• Brownian heat engine at maximal power with universal bounds

[T. Schmiedl and U.S., EPL 81, 20003, (2008)]

VV

V

Tc

Th

V

pa pb

pbpa

1

3

4

2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

η∗

(Th − Tc)/Tc

ηC

α = 1/2α = αCA

α = 1

– η∗ = ηc2−αηcwith 0 ≤ α ≤ 1 implies ηc2 ≤ η

∗ ≤ ηc2−ηc

– Curzon-Ahlborn neither universal nor a bound

– cf. Esposito, Kawai, Lindenberg, vd Broeck, PRL ’10

– cf. extended classification by Wang & Tu on arxiv

24

• Experimental realization in the Sterling version

[V. Blickle and C. Bechinger, nature physics 2012]

T =22 °Cc

expansion

isothermal

-0.2 0.0 0.2

position [µm]

(4)

-0.2 0.0 0.2

position [µm]

(1)

-0.2 0.0 0.2

position [µm]

(2)

V

(3)

-0.2 0.0 0.2

position [µm]

(4)

(1)

(2)

(3)

H

T =86 °Ch

compression

isothermal

isochoric

isochoric

pmacroscopic Stirling cycle

transitio

n

transitio

n

a

0 1

-1

0

1

p(wn)

b

(3)

(1) (1) (1)(2) (2)(2)

(3) (3)(4)(4) (4)

wn

0 500 1000 1500 2000-100

-80

-60

-40

-20

0

20

40

0 30

-1

0

1

2

W[k

BT

c]

time[s]

W[k

BT

c]

time[s]

0 100

-1

0

1

wn

[kBT

c]

cycle number n

c

a

b0 10 20 30 40 50

-0.2

0.0

0.2

Eff

icie

ncy

ht [s]

h¥

0 10 20 30 40 50

-0.10

-0.05

0.00

0.05

-0.6

-0.4

-0.2

0.0

0.2

P[k

BT

c/s

]

w[k

BT

c]

25

• Conclusions

– classification of stochastic engines

– general theory for EMP of autonomous machines

∗ separate system specific from universal contributions

∗ EMP depends crucially on parameter space

∗ isothermal: no bounds beyond LR regime

– feedback controlled isothermal machines

∗ fluctuation theorems and bounds on power

∗ ex’s: harmonic trap set-up and a genuine NESS machine

– heat engines: universal one-parameter EMP with bounds

26

paris, fifem 2012, january 25. udo seifertdavid/fr2012/talk-udo.pdf · 2012. 1. 27. · paris,...

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