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 : 423-431, 2008. 1
LECTURE I Classical vs. stochastic thermodynamics First law General fluctuation theorem and Jarzynski relation Optimization 2
Perspective 1820 1850 classical thermodynamics dw = du + dq ds 0 1900 eq stat phys p i = exp[ (E i F)/k B T] 1930 1960 non-eq: linear response Onsager Green-Kubo, FDT 1993 non-eq: beyond linear response Fluctuation theorem stochastic thermodynamics Jarzynski relation 3
Thermodynamics of macroscopic systems [19 th cent] 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 λ 0 W T 000 111 000 111 000 111 000 111 000 111 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
Macroscopic vs mesoscopic vs molecular machines [Bustamante et al, Physics Today, July 2005] 5
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
Nano-world Experiment: Stretching RNA [Liphardt et al, Science 296 1832, 2002.] distributions of W diss : 7
Mechanically driven systems x 0 x 4 x 1 x 6 x 3 x 5 x 2 f(λ) 0000 1111 01 01 000000 111111 000 111 0000000 1111111 0000 1111 000000000000000 111111111111111 01 000000000000000 111111111111111 01 000000000000000 111111111111111 01 000000000000000 111111111111111 0000000000000000 1111111111111111 0000000000 1111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000 111111111111111111111111 V (x,λ) λ(τ) Pulling a biomolecule Colloidal partical in a laser trap 8
Flow driven systems Stretching a polymer (dumbbell) Tank-treading vesicle in shear flow 9
(Bio)chemically driven systems F1-ATPase 10
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
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) 0000 1111 01 01 000000 111111 000 111 0000000 1111111 0000 1111 000000000000000 111111111111111 01 000000000000000 111111111111111 01 000000000000000 111111111111111 01 000000000000000 111111111111111 0000000000000000 1111111111111111 0000000000 1111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 000000000000000000000000000000 111111111111111111111111111111 00000000000000000000000 11111111111111111111111 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
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] 13
Role of external flow and frame invariance [T. Speck, J. Mehl and U.S., PRL 100 178302, 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
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
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
General fluctuation theorem [U.S., PRL 95, 040602, 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
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
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
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, 066112, 2004] 20
Scenario 2: linear equations of motion and arbitrary driving Ex: Stretching of Rouse polymer [T.S. and U.S., EPJ B 43, 521, 2005] 01 01 01 01 01 01 01 01 01 0 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
Optimal finite-time processes in stochastic thermodynamics [T. Schmiedl and U.S., PRL 98, 108301, 2007] W λ i T λ f optimal protocol λ (τ) minimizes W for given λ i, λ f and finite t 22
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) 0.88 23
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 3 2 1 0-0.2 0 0.2 0.4 0.6 0.8 1 1.2 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 : 024114, 2008] 24
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
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 0.8 0.6 0.4 0.2 α = 1/2 α = α CA α = 1 0 0 1 2 3 4 5 (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
Optimizing potentials for temperature ratchets [F. Berger, T. Schmiedl, U.S., PRE 79, 031118, 2009] T c T h T h T c T c T h 2.5 2 V(x) T(x) 2 f V(x) V(x) 1.5 1 1 T(x) 0.5 0 0 0 L/2 L -0.5 0 0.2 0.4 0.6 0.8 1 x 1.4 1.2 0.5sin(2πx)+1 d=1 d=0.1 d=0.005 7 6 5 j(d) 2.5 2 1.5 T(x)=0.5sin(2πx)+1 d=1 d=0.1 d=0.005 T(x) 1 V(x) 4 3 0.01 0.1 1 1 0.8 2 0.6 1 0 0 0.2 0.4 0.6 0.8 1 x 0 0.2 0.4 0.6 0.8 1 x 27