Stochastic thermodynamics

Transcript

1 Boulder lectures, July 2009 Stochastic thermodynamics Udo Seifert II. Institut für Theoretische Physik, Universität Stuttgart thanks to: F. Berger, J. Mehl, T. Schmiedl and Th. Speck (theory) V. Blickle and C. Bechinger (expt s on colloidal particles) C. Tietz, S. Schuler and J. Wrachtrup (expt s on single atoms) Review: U.S., Eur. Phys. J. B, 64 : ,

2 LECTURE I Classical vs. stochastic thermodynamics First law General fluctuation theorem and Jarzynski relation Optimization 2

3 Perspective classical thermodynamics dw = du + dq ds eq stat phys p i = exp[ (E i F)/k B T] non-eq: linear response Onsager Green-Kubo, FDT 1993 non-eq: beyond linear response Fluctuation theorem stochastic thermodynamics Jarzynski relation 3

4 Thermodynamics of macroscopic systems [19 th cent] λ 0 W T λ t First law energy balance: W = E + Q = E + T S M Second law: S tot S + S M > 0 W > E T S F W diss W F > 0 4

5 Macroscopic vs mesoscopic vs molecular machines [Bustamante et al, Physics Today, July 2005] 5

6 Stochastic thermodynamics for small systems W A 1 n A 1 λ 0 T, p λ t A 2 A 3 w 0 nm m wmn 0 A 2 A 3 First law: how to define work, internal energy and exchanged heat? fluctuations imply distributions: p(w; λ(τ))... entropy: distribution as well? 6

7 Nano-world Experiment: Stretching RNA [Liphardt et al, Science , 2002.] distributions of W diss : 7

8 Mechanically driven systems x 0 x 4 x 1 x 6 x 3 x 5 x 2 f(λ) V (x,λ) λ(τ) Pulling a biomolecule Colloidal partical in a laser trap 8

9 Flow driven systems Stretching a polymer (dumbbell) Tank-treading vesicle in shear flow 9

10 (Bio)chemically driven systems F1-ATPase 10

11 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 (1 st law) and entropy production (2 nd law) along a single stochastic trajectory Precursors: notion stoch th dyn by Nicolis, van den Broeck mid 80s ( ensemble level) stochastic energetics (1 st law) by Sekimoto late 90s work theorem(s): Jarzynski, Crooks late 90s fluct theorem: Evans, Cohen, Galavotti, Kurchan, Lebowitz & Spohn 90s quantities like stochastic entropy by Crooks, Qian, Gaspard in early 00s... 11

12 Paradigm for mechanical driving: x 0 x 4 x 1 x 6 x 3 x 5 x 2 λ(τ) V (x, λ) λ(τ) x f(λ) V (x,λ) Langevin dynamics ẋ = µ[ V (x, λ) + f(λ)] }{{} F(x,λ) external protocol λ(τ) +ζ ζζ = 2µ k B T δ(...) }{{} ( 1) First law [(Sekimoto, 1997)]: dw = du + dq applied work: dw = λ V (x, λ)dλ + f(λ)dx internal energy: du = dv dissipated heat: dq = dw du = F(x, λ)dx = Tds m 12

13 Experimental illustration: Colloidal particle in V (x, λ(τ)) [V. Blickle, T. Speck, L. Helden, U.S., C. Bechinger, PRL 96, , 2006] work distribution non-gaussian distribution Langevin valid beyond lin response [T. Speck and U.S., PRE 70, , 2004] 13

14 Role of external flow and frame invariance [T. Speck, J. Mehl and U.S., PRL , 2008] Lab frame V (x, τ) = k(vτ x) 2 /2 ẋ = µk(vτ x) + ζ ẇ = τ V = kv(vτ x(τ)) 0 comoving frame: y x(τ) vτ V (y) = ky 2 /2 ẏ = µky v + ζ ẇ = τ V = 0?? det balance satisfied: equilibrium?? 14

15 Correct definitions of th dynamic quantities no flow flow u(r) 0 dw = t V + fdr dw D t V + f[dr udt] (D t t + u ) dq = ( V + f)dr dq ( V + f)(dr udt) 15

16 Path integral representation Boltzmann factor for a whole trajectory p[ζ(τ)] exp [ p[x(τ) x 0 ] exp [ x x 0 x 0 0 x(τ) τ x(τ) t 0 dτ ζ2 (τ)/4d] t 0 dτ (ẋ µf)2 /4D] time reversal x(τ) x(t τ) and λ(τ) λ(t τ) Ratio of forward to reversed path t x t x t λ λ t λ 0 0 λ(τ) τ λ(τ) t p[x(τ) x 0 ] p[ x(τ) x 0 ] = exp [ t 0 dτ (ẋ µf) 2 /4D] exp [ t 0 dτ ( x µ F) 2 /4D] = exp β t 0 dτ ẋf = exp βq[x(τ)] = exp s m 16

17 General fluctuation theorem [U.S., PRL 95, , 2005; generalizing Jarzynski, Crooks, Maes] 1 = x(τ), x 0 p[ x(τ) x 0 ] p 1 ( x 0 ) = p[x(τ) x 0 ] p 0 (x ) p[ x(τ) x 0] p 1 ( x 0 ) 0 p[x(τ) x x(τ),x 0 ] p 0 (x 0 ) 0 = exp[ βq[x(τ)] }{{} s m +ln p 1 (x t )/p 0 (x 0 )] for arbitrary initial condition p 0 (x) for arbitrary (normalized) function p 1 (x t ) 17

18 Jarzynski relation (PRL, 1997) λ 0 W T, p λ t 2 nd law: W λ(τ) F F(λ t )) F(λ 0 ) exp[ W] = exp[ F] (k B T = 1) Short proof: 1 = exp[ q[x(τ)] }{{} s m +ln p 1 (x t )/p 0 (x 0 )] p 0 (x 0 ) exp[ (V (x 0, λ 0 ) F(λ 0 )] p 1 (x t ) exp[ (V (x t, λ t ) F(λ t )] within stochastic dynamics an identity! 18

19 Jarzynski (cont d) (k B T = 1) W λ 0 T, p λ t e W λ(τ)! = e F start with initial thermal distribution valid for any protocol λ(τ) valid beyond linear response allows to extract free energy differences from non-eq data implies a variant of the second law e x e x W F 19

20 Dissipated work W d W G exp[ W d ] + dw d p(w d )exp[ W d ] = 1 p(w d ) W d = W G red events violate the second law (??) Special case: Gaussian distribution p(w d ) exp[ (W d W d ) 2 /2σ 2 ] with W d = σ 2 /2 scenario 1: slow driving of any process [T. Speck and U.S., Phys. Rev E 70, , 2004] 20

21 Scenario 2: linear equations of motion and arbitrary driving Ex: Stretching of Rouse polymer [T.S. and U.S., EPJ B 43, 521, 2005] λ(τ) cantilever x L λ different protocols 0 t τ linear: λ(τ) = τl/t W d = (Nγ/3)L 2 /t periodic: λ(τ) = Lsin πτ/2t W d = [π 2 /8](Nγ/3)L 2 /t 21

22 Optimal finite-time processes in stochastic thermodynamics [T. Schmiedl and U.S., PRL 98, , 2007] W λ i T λ f optimal protocol λ (τ) minimizes W for given λ i, λ f and finite t 22

23 Ex 1: Moving a laser trap V (x, λ) = (x λ(τ)) 2 /2 V (x, 0) V (x,t) λ f λ lin (τ) λ λ (τ) λ 0 λ f 0 t λ (τ) requires jumps at beginning and end λ = λ f /(t + 2) gain 1 W (t)/w lin (t)

24 Ex 2: Stiffening trap V (x, λ) = λ(τ)x 2 /2 (a) 5 4 λ f /λ i = 2, λ i t = 0.1 λ f /λ i = 2, λ i t = 1 λ f /λ i = 2, λ i t = 10 λ f /λ i = 5, λ i t = 0.1 λ f /λ i = 5, λ i t = 1 λ f /λ i = 5, λ i t = 10 λ f λ i λ /λi τ/t jumps are generic should help to improve convergence of exp( W) generalization: underdamped dynamics delta-peaks [A. Gomez-Marin, T.Schmiedl, U.S., J. Chem. Phys., 129 : , 2008] 24

25 Heat engines at maximal power Carnot (1824) Curzon-Ahlborn (1975) T h T h Q h = α(t h T m h ) Q h T m h W W T m c Qc = β(t m c Tc) Qc η c 1 T c /T h but zero power Tc efficiency at maximum power η ca 1 T c /T h recent claims for universality(?) what about fluctuations? Tc 25

26 Brownian heat engine at maximal power [T. Schmiedl and U.S., EPL 81, 20003, (2008)] V 1 V p a p b 1 V 4 p a T h T c 3 V 2 p b η α = 1/2 α = α CA α = (T h T c )/T c η C η = η c 2 αη c with α = 1/2 for temp-independent mobility Curzon-Ahlborn neither universal nor a bound 26

27 Optimizing potentials for temperature ratchets [F. Berger, T. Schmiedl, U.S., PRE 79, , 2009] T c T h T h T c T c T h V(x) T(x) 2 f V(x) V(x) T(x) L/2 L x sin(2πx)+1 d=1 d=0.1 d= j(d) T(x)=0.5sin(2πx)+1 d=1 d=0.1 d=0.005 T(x) 1 V(x) x x 27