Optimal Processes within Stochastic Thermodynamics and beyond Udo - - PowerPoint PPT Presentation

San Diego meeting, July 2009

Optimal Processes within Stochastic Thermodynamics and beyond

Udo Seifert

• II. Institut f¨

ur Theoretische Physik, Universit¨ at Stuttgart Thanks to Tim Schmiedl (PhD thesis work)

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• Intro: Classical vs Stochastic thermodynamics
• Optimization

– directed processes – cyclic processes ∗ heat engines ∗ temperature ratchets ∗ biochemical machines: motor proteins

• beyond

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• Thermodynamics of macroscopic systems [19th cent]
• λ0

λt W T

– First law energy balance: W = ∆E + Q = ∆E + T∆SM – Second law: ∆Stot ≡ ∆S + ∆SM > 0 W > ∆E − T∆S ≡ ∆F Wdiss ≡ W − ∆F > 0

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• Macroscopic vs mesoscopic vs molecular

machines

[Bustamante et al, Physics Today, July 2005]

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• Stochastic thermodynamics for small systems

λt λ0 W T, p

driving: mechanical hydrodynamical (bio)chemical – First law: how to define work, internal energy and exchanged heat? – fluctuations imply distributions: p(W; λ(τ)) ... – entropy: distribution as well?

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• Nano-world Experiment: Stretching RNA

[Liphardt et al, Science 296 1832, 2002.]

– distributions of Wdiss:

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• Stochastic thermodynamics

applies to such systems where – non-equilibrium is caused by mechanical or chemical forces – ambient solution provides a thermal bath of well-defined T – fluctuations are relevant due to small numbers of involved molecules

• Main idea:

Energy conservation (1st law) and entropy production (2nd law) along a single stochastic trajectory

• Review: U.S., Eur. Phys. J. B 64, 423, 2008
• Precursors:

– notion “stoch th’dyn” by Nicolis, van den Broeck mid ‘80s ( on ensemble level) – stochastic energetics (1st law) by Sekimoto late ‘90s – ....

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• Paradigm for mechanical driving:

λ(τ) x1 x6 x5 x2 x0 x4 x3

λ(τ) V (x, λ) x

• V (x, λ)

f(λ)

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

• F(x,λ)

+ζ ζζ = 2µ kBT

(≡1)

– external protocol λ(τ)

• First law [(Sekimoto, 1997)]:

dw = du + dq – applied work: dw = ∂λV (x, λ)dλ + f(λ)dx – internal energy: du = dV – dissipated heat: dq = dw − du = F(x, λ)dx = Tdsm

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• 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]

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• Stochastic entropy

[U.S., PRL 95, 040602, 2005]

– Fokker-Planck equation ∂τp(x, τ) = −∂xj(x, τ) = −∂x (µF(x, λ) − D∂x) p(x, τ) [D = µkBT] – Common non-eq ensemble entropy [kB ≡ 1] S(τ) ≡ −

dx p(x, τ) ln p(x, τ)

– Stochastic entropy for a single trajectory x(τ) s(τ) ≡ − ln p(x(τ), τ) with s(τ) = S(τ) – ∆stot ≡ ∆sm + ∆s – exp[−∆stot] = 1 ⇒ ∆stot ≥ 0 ∗ integral fluctuation theorem for total entropy production ∗ arbitrary initial state, driving, length of trajectory

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• General integral fluctuation theorem

1 = exp[−q[x(τ)]

• −∆sm

+ ln p1(xt)/p0(x0)] for any (normalized ) p1(xt)

• Jarzynski relation (1997)

λt λ0 W T, p

2nd law: W|λ(τ) ≥ ∆F ≡ F(λt)) − F(λ0) – exp[−W] = exp[−∆F]

• r

exp[−Wd] = 1 ∗ p0(x0) ≡ exp[−(V (x0, λ0) − F(λ0)] ∗ p1(xt) ≡ exp[−(V (xt, λt) − F(λt)]

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• Optimal finite-time processes in stochastic thermodynamics

[T. Schmiedl and U.S., PRL 98, 108301, 2007]

λf λi W T

– optimal protocol λ∗(τ) minimizes W for given λi, λf and finite t

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• Ex 1: Moving a laser trap V (x, λ) = (x − λ(τ))2/2

λf V (x, 0) V (x, t)

λlin(τ) λ∗(τ) λf t ∆λ ∆λ

– λ∗(τ) requires jumps at beginning and end ∆λ = λf/(t + 2) – gain 1 ≥ W ∗(t)/W lin(t) ≥ 0.88

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• Ex 2: Stiffening trap

V (x, λ) = λ(τ)x2/2

λf λi

1 2 3 4 5

• 0.2

0.2 0.4 0.6 0.8 1 1.2

(a)

λf/λi = 5, λit = 1 λf/λi = 5, λit = 10 λf/λi = 2, λit = 0.1 λf/λi = 2, λit = 1 λf/λi = 2, λit = 10 λf/λi = 5, λit = 0.1

τ/t λ∗/λi

– jumps are generic – typical size of the jump

∆λ/λf (W ∗−W qs)/W jp

0.5 0.6 0.7 0.8 0.9 1 0.9 0.7 0.6 0.55 10 100 1000 10000 0.01 0.1 1 10 100

(d)

λf/λi λit

∆λ/λf (W ∗−W qs)/W jp

– might help to improve convergence of exp(−W)

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• Underdamped dynamics: role of inertia

[A. Gomez-Marin, T.Schmiedl , U.S., J Chem Phys 129 024114 (2008)]

m¨ x + γx + V ′(x, λ) = ξ ∗ jumps and delta-functions at the boundaries ∗ W ∗/W lin >> 1 possible

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• Heat engines at maximal power

– Carnot (1824)

Th Tc −Qh Qc −W

– ηc ≡ 1 − Tc/Th but zero power – Curzon-Ahlborn (1975)

Th Tc −W −Qh = α(Th − Tm h ) Qc = β(Tm c − Tc) Tm c Tm h

– efficiency at maximum power ηca ≡ 1 −

• Tc/Th

– universality(?)

[cf van den Broeck, PRL 2005]

– what about fluctuations?

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• Brownian heat engine at maximal power

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

V V V

Tc Th

V

pa pb pb pa

1 3 4 2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 η∗ (Th − Tc)/Tc ηC α = 1/2 α = αCA α = 1

– η∗ =

ηc 2−αηc

with α = 1/2 for temp-independent mobility – Curzon-Ahlborn neither universal nor a bound

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• Optimizing potentials for temperature ratchets

[F. Berger, T. Schmiedl, U.S., PRE 79, 031118, 2009]

L/2 L Tc Th Th Tc Tc Th b f V (x)

• 0.5

0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1 1 2 V(x) T(x) x V(x) T(x)

0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 T(x) x 0.5sin(2πx)+1 d=1 d=0.1 d=0.005 1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 V(x) x T(x)=0.5sin(2πx)+1 d=1 d=0.1 d=0.005 0.01 0.1 1 1 1.5 2 2.5 j(d)

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

[T.Schmiedl, T.Speck and U.S., J. Stat. Phys. 128, 77 (2007)]

w0

nm

A1 A2 A3 A1 A2 A3

n m

w0

mn

– A1+n

wnm

⇀ ↽ wmn

m+A2+A3 – mass action law kinetics: – wnm

wmn = w0

nm

w0

mn[A1]/[A2][A3]

• First law along a trajectory w = ∆E + q

for a single reaction step ? – chemical work: wnm

chem ≡ µ1 − µ2 − µ3

– internal energy: ∆Enm ≡ Em − En – dissipated heat: qnm ≡ wnm

chem − ∆Enm = ln [A1] [A2][A3] w0

nm

w0

mn = ln wnm/wmn 19

• Efficiency of molecular motors at maximum power

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

l F V (x) δl

w+ = [ATP]k+ exp[−δl F] w− = [ADP][P]k− exp[(1 − δ)l F]

δ 20 10 5 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 δ 20 10 5 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 ∆µ/(kBT ) ˙ W ∗ ∆µ/(kBT ) δ η∗ 20 10 5 2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 δ η∗ 20 10 5 2 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 η∗ 1 0.5 0.2 0.1 5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 ∆µ/(kBT ) δ

– “Power stroke” (δ ≃ 0) highest efficiency at max power – η∗ can increase beyond lin response regime (η∗ = 1/2)

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• beyond stochastic dynamics

[T. Schmiedl, E. Dieterich, P.S. Dieterich, U.S., J Stat Mech, P07013 (2009)]

λf V (x, 0) V (x, t)

– Hamiltonian dynamics – Quantum dynamics

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• Hamiltonian dynamics

λf V (x, 0) V (x, t)

– ∂τρ(x, p, τ) =

• ρ, p2/2m + V (x, λ(τ))
• PB

– ρ(x, p, τ = 0) = exp[−β(H − F)] – λi → λf in finite t – adiabatic=quasistatic work W ad = ∆F

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• W = [ p2

2m + k 2 (x − λ)2 ]t

⇒ W = 0 if p(t) = 0 and x(t) = λ

• only two conditions on λ(τ)

⇒ optimal protocol highly degenerate

• adiabatic work can be reached in 0 + ǫ time

(price: extreme λ-values)

• Hamiltonian dynamics beats Langevin evolution

( W ∗ → W jp = kλ2

f/2 for t → 0)

λf V (x, 0) V (x, t)

• qualitatively similar for case II:

λf λi

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• Anharmonic potential

V (x, λ) = λx4/4 λ(0) = 1 → λ(t) = 2 0.2 0.22 0.24 0.5 1 1.5 2 W t W ∗, n = 1 2 3 5 W lin W ad W jp 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 λ τ/t t = 2 t = 1 – Fourier protocol better than linear – W ∗(0 + ǫ) < W jp = 0.25 – W ad reached in finite time ??

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• Improvement for Jarzynski estimate:

V (x, λ) = λx4/4 λ = 1 → 2

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• Schr¨
• dinger dynamics

λf V (x, 0) V (x, t)

−i¯ h ∂tρ =

• ρ, p2/2m + V (x, λ(t))
• −

ρ(t = 0) = exp[−β(H − F)] λi = 0 → λf in finite t – Talkner et al PRE 2008: p(W) depends only on z ≡

t

0 ˙

λ(t′)eiωt′dt′ – z = 0 for an adiabatic transition ⇒ W ∗ = W ad for any t > 0 possible ! – case II similarly – general case: open

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• Conclusions and perspectives

– Optimal protocols in stoch th’dynamics: ∗ directed processes: remarkable singularities ∗ cyclic processes: efficiency at max power ∗ optimization wrt other quantities like ∆Stot ? ∗ ... – Hamiltonian and quantum dynamics ∗ systematics beyond case studies? ∗ open quantum systems? – Efficient algorithms for finding optimal protocols?

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