Fluctuation theorems: where do we go from here?

Transcription

1 Fluctuation theorems: where do we go from here? D. Lacoste Laboratoire Physico-Chimie Théorique, UMR Gulliver, ESPCI, Paris

2 Outline of the talk 1. Fluctuation theorems for systems out of equilibrium 2. Modified fluctuation-dissipation theorems out of equilibrium 3. Fluctuations theorems for systems at equilibrium 4. Non-invasive estimation of dissipation

3 Fluctuation theorems for systems out of equilibrium

4 What is stochastic thermodynamics? 1. Classical Thermodynamics P(W) First and second law Macroscopic systems : fluctuations are gaussian and small Absence of time as a parameter <W> 2. Stochastic thermodynamics First and second law at the trajectory level P(W) Small systems : large non-gaussian fluctuations Time enters as an essential parameter

5 Jarzynski relation Assume system equilibrated by the contact with a reservoir at temperature T at t=0 Then a control parameter l t is applied; final state at time t is not in equilibrium If system is disconnected from heat bath at t=0 he W t i = = Z dx 0 p eq (x 0 )e 1 Z( 0 ) Z dx 0 W t 1 W t = H(x t, t) H(x 0, 0) e H(x t, t ) = Z( t) Z( 0 ) = e F An exponential average over non-equilibrium trajectories leads to equilibrium behavior A method to evaluate free energies from non-equilibrium experiments C. Jarzynski, PRL 78, 2690 (1997)

6 «Violations» of the second law of thermodynamics Using Jensen s inequality hexp(x)i exphxi one obtains the second law (only valid on average) hw i F Second law can be «violated» for particular realizations, but P (W < F ) = apple Z ( F ) 1 Z ( F ) 1 dw P (W ) P (W )e ( F W ) apple e DF Such «violations» are required for the JE to hold but they are exponentially rare. «Violations» are pratically unobservable for macroscopic systems

7 Systems in contact with a thermostat Interaction with thermostat now described by a markovian evolution dp t dt = M P t Accumulated work up to time t: W t = Z t 0 (x t )d Laplace distribution of joint distribution of work and of the fluctuating variable ˆP t (x, )= Z dw P t (x, W )e W satisfies The solution ˆP = M ˆP t ˆP t (x, )=h (x ˆP t W t i = 1 Z A e H (x) Jarzynski relation proved by integration over all x: More generally, for any observable A(x): W he t i = e F W ha(x t )e t i = A eq (x t )

8 Hatano-Sasa relation Generalization when initial condition is in a non-equilibrium steady state (NESS) Work like functional Y t = Z t 0 (x, ) (x, )= ln P stat (x, ) Average over non-equilibrium trajectories leads to steady-state behavior he Y t i =1 T. Hatano and S. Sasa, (2001) Now hy t i 0 where the equality holds for a quasi-stationary process

9 Crooks relation Let us define forward and reverse processes which both start in an equilibrium with fixed value of l Forward trajectory g F ={x F (t), 0<t<t f } Reverse trajectory g R ={x R (t), 0<t<t f } with x R (t) =x F (t f t) If the reservoir is removed during the process P F ( F ) P R ( R ) = P F (x F (0)) P R (x R (0)) = e H(xF (0), i) Z( i ) Z( f ) e H(x F (t f ), f ) = e (W F F ) As a particular case, at equilibrium, fluctuations are symmetric under time-reversal P eq ( F )=P eq ( R )

10 Same result holds for systems obeying markovian stochastic dynamics provided M (x! x 0 ) M (x 0! x) = e (H(x 0, ) H(x, )) Rk: this detailed balance condtion holds only if the heat bath is ideal Through integration of trajectories of given value of W=W F P F (W ) P R ( W ) = e (W F ) G. Crooks, PRE 61,2361 (1999) Experiments : using RNA hairpin pulled by optical tweezers D. Collin, Nature (2005)

11 Introducing KL divergence D(p q) = X c p(c)ln p(c) q(c) 0 - asymmetric character of the measure is crutial P F ( F ) P R ( R ) = e (W F F ) From by taking the average D(P F P R )= hw i k B T F = hw diss i k B T W diss = W F The functional leads to the same estimate of dissipation as the KL divergence of path probabilities Non-equilibrium fluctuations created by irreversible processes (such as systems presenting hysteresis) are asymmetric with respect to time-reversal

12 Evans-Searles and Gallavotti-Cohen relation 1. Transient fluctuation theorem of Evans-Searles The system is initially at equilibrium and evolves towards a NESS P ( S) P ( S) = e S/k B Evans DJ, Searles DJ, (1994) NESS can be created from multiple reservoirs or from time-symmetric driving Also holds separetely for parts of entropy production under conditions 2. Fluctuation theorem of Gallavotti-Cohen = S/ The asymptotic distribution of entropy production rate in a NESS lim!1 1 ln P ( ) P ( ) = k B T Gallavotti G., Cohen EGD, (1995) Implies relations for distribution of currents in a NESS

13 Trajectory dependent entropy for a particle or system 1. Within Fokker-Planck equation (markovian) Definition : stochastic t p t (x t )= s t = ln p t (x t x j t (x) Seifert U., (2005) 2. Second law of thermodynamics Heat dumped into heat bath assumed ideal according to Sekimoto : q ln P F [x t x 0 ] P R [x R t x R 0 ] = s m = q T Then s =lnp 0 (x 0 ) ln p t (x t ) difference of Shanon entropy s tot =ln P F [x t ] P R [x R t ] = s m + s which satisfies an integral fluctuation theorem by construction Rk: condtions for a detailed FT for s tot are more restrictive he s tot i =1

14 Fluctuation theorems for molecular motors Discrete ratchet model Velocity data of kinesin K. Vissher et al., (1999) Average ATP consumption rate: r = s Strong coupling l = v r 0.97 A.W.C. Lau et al., PRL 99, (2007); D. L. et al., PRE 78, (2008).

15 Minimal ratchet: thermodynamics Statistics of the displacement n(t) and of the number of ATP molecules consumed y(t) as function of external and chemical loads? Velocity (mechanical current) ATP consumption rate (chemical current) v( f,δµ) = lim t d n(t) dt d y() t r( f, Δ µ ) = lim t dt At thermodynamic equilibrium: f=0 and Dµ=0, fluctuations of n(t) and y(t) are gaussian, characterized by two diffusion coefficients D 1 and D 2 Near equilibrium: for small f and Dµ=0, linear response theory holds, v= L f + L Δµ r = L f + L Δµ What happens far from equilibrium? Einstein relations: L 11 =D 1 and L 22 =D 2 Onsager relations: L 12 =L 21

16 Large deviations of the currents Long time properties of n(t) and y(t) are embodied in the large deviation function G(v,r) Prob( n(t) t = v, y(t) t = r)! e tg(v,r) v = v Equivalently, one has the cumulant generating function E(l,g) e λn+γ y! e te(λ,γ ) E v( f, Δ µ ) = (0,0) λ In particular and E r( f, Δ µ ) = (0,0) γ The functions G(v,r) and E(l,g) are related by Legendre transforms

17 Gallavotti-Cohen relation for a discrete ratchet model The function E(l,g) satisfies the Gallavotti-Cohen symmetry : E( λγ, ) = E( λ f, γ Δµ ) which implies the Fluctuation theorem in the long time limit: Prob( n(t) t Prob( n(t) t = v, y(t) t = v, y(t) t = r) = r) t( fv+rδµ)! e Valid arbitrary far from equilibrium Linear expansion near equilibrium leads to Einstein and Onsager relations Results from local thermodynamic constraints : Generalized Detailed Balance

18 Phase diagram and thermodynamic efficiency 4 regions of mechano-transduction : Thermodynamic efficiency A: ATP in excess -> mechanical work B: mechanical work -> ATP C: ADP in excess -> mechanical work D: mechanical work -> ADP A: f v < 0 and r Δ µ > 0 B: f v > 0 and r Δ µ < 0 C: f v < 0 and r Δ µ > 0 D: f v < 0 and r Δ µ < 0 fv η = r Δµ is maximum far from equilibrium (which is reached on a point in (Dµ,f) plane)

19 Gallavotti-Cohen relation for a continuous ratchet model Flashing ratchet model φ ( x) 1 φ ( x) 2 ω ( x) 2 ω ( x) 1 x also obeys the GC symmetry provided both mechanical and chemical variables are included E( f λ, Δµ γ) = E( λ, γ) but in general e( f λ) e( λ) D. L. et al., PRE 80, (2009); Séminaires Poincaré, biological physics (2009)

20 Modified fluctuation-dissipation theorems out of equilibrium

21 Standard fluctuation-dissipation theorem (FDT) Only holds for systems which are close to an equilibrium state H0 > H0 h t ' O A perturbation : is applied at time t : Response function (for t>t ) At () d Rtt (, ') = = β AtOt () ( ') h dt' t ' h= 0 eq Einstein (1905), Nyquist (1928); H. Callen and T. Welton (1951), Kubo (1966) Many attempts to extend the result to non-equilibrium systems

22 From Jarzynski equality to fluctuation-dissipation theorem (FDT) More general form of Jarzynski equality If dissipated work is small with respect to kt Take functional derivative with respect to perturbation

23 Modified fluctuation-dissipation theorem (MFDT) near NESS Variant of Hatano-Sasa (2001) : Response function near a non-equilibrium steady state for a general observable A Particular case J. Prost et al. (2009) General case U. Seifert et al. (2010) G. Verley, K. Mallick, D. L., EPL, 93, (2011)

24 Perturbation near an arbitrary non-stationary state Most general MFDT obtained from an Hatano-Sasa like relation: Several alternate formulations: G. Verley et al., J. Stat. Mech. (2011) - In terms of an additive correction (the asymmetry) which vanishes at equilibrium M. Baiesi et al. (2009); E. Lippiello et al. (2005) - In terms of a local velocity/current R. Chétrite et al. (2008); U. Seifert et al. (2006) Rk: in all these formulations, markovian dynamics is assumed

25 Origin of the additive structure of MFDT Stochastic trajectory entropy s t (c t, [h]) = ln t (c t, [h]) o Distinct but similar to s t (c t, [h]) = ln p t (c t, [h]) U. Seifert, (2005) o It can be split into Reservoir entropy + Total entropy production [ h] s ( c [ h] + s c,[ h] Δ s ( c, ) = Δ, ) Δ ( ) t t r t tot t Consequence of this decomposition for MFDT: d r Req (, t t') = hδst '( ct ', h) A( ) ' h 0 t ct dt = j ( c ) A( c ) t' t' t t Additive structure of the MFDT involving local currents: Rtt (, ') = ( j ( c ) ν ( c )) A( c ) t' t' t ' t d tot Rneq (, t t') = hδst ' ( ct ', h) A( ) ' h 0 t ct dt = ν ( c ) A( c ) t t' t' t t

26 The 1D Ising model with Glauber dynamics Classical model of coarsening : L Ising spins in 1D described by the hamiltonian L i i+ 1 m m i= 1 H({ σ}) = J σσ H σ, T = System intially at equilibrium at is quenched at time t=0 to a final temperature T. At the time t >0, a magnetic field H m is turned on: H () t = H θ( t t'), m The dynamics is controlled by time-dependent (via H m ) Glauber rates m α H m i w ({ σ},{ σ} ) = ( 1 σi tanh ( βj( σi 1+ σi+ 1) + βhmδim) ), 2

27 Analytical verification : - MFDT can be verified although the distributions p t ({s},h m ) even for a zero magnetic field are not analytically calculable - Analytical form of the response is known Numerical verification : the distributions p t ({s},h m ) can be obtained numerically for a small system size (L=14); and the MFDT verified: Integrated response function t χn m(, tt') = dτrn m(, tτ) t ' G. Verley et al., J. Stat. Mech. (2011)

28 Fluctuation theorems for equilibrium systems

29 1. For non-equilibrium systems: Extension of Gallavotti-Cohen lim!1 1 ln P (J) P (J 0 ) = (J J0 ) P. Hurtado et al. (2011) J = J 0 for any pairs of isometric current vectors and is related to entropy production 2. For equilibrium systems with discrete broken symmetry For an ensemble of N Ising spins in a magnetic field P B (M) =P B ( M)e 2 BM P. Gaspard (2012) Equilibrium fluctuations in finite systems are in general non-gaussian Non-gaussianity particularly significant near critical points

30 Consider N Heisenberg spins in a magnetic field H N ( ; B) =H N ( ; 0) B M N ( ) with M N ( )= Fluctuations of the magnetization M obey the general relation: P B (M) =P B 0(M) e (B B0 ) M, NX i=1 i M 0 = Rg 1 M M = M 0 where thus (isometric fluctuation theorems) This fluctuation theorem expresses a symmetry of the large deviation function defined by P B (M) =A N (m)e N B(m), for N!1 These fluctuation theorems quantify the breaking of a symmetry different from time reversal D. L. and P. Gaspard, under review (2014)

31 Illustrative examples Curie-Weiss model in a magnetic field: exact evaluation of the large deviation function XY model in a magnetic field numerical verification at low and high temperatures More complex symmetry breaking with tensorial order parameter in a liquid crystal mean-field model

32 Non-invasive estimation of dissipation from trajectory information

33 Irreversibility as time-reversal symmetry breaking Direct determination of work or heat is difficult for most complex systems Non-equilibrium fluctuations created by irreversible processes have a welldefined arrow of time This arrow of time can be «measured» by comparing the statistics of fluctuations forward and backward in time using the KL divergence This amounts to enforce fluctuation theorems and exploit them to extract a measurement rather than trying to «verify» them

34 Estimation of dissipation in a NESS Mesure of dynamical randomness associated with direct and reverse path where in the reverse path the driving is reversed d i S dt = lim t!1 k B t h P +[z t z 0 ] P [z R t z R 0 ]i D. Andrieux et al. (2008) In other types of NESS, the driving if present is constant and does not need to be reversed -> simpler implementation with a single data series

35 Simpler implementation with no reversal of the driving Probability to observe a block of length m, (x 1..x m ) within a trajectory of total length n in the forward direction p F =p(x 1..x m ) Connection between thermodynamics and information-theoretic estimation E. Roldan et al. (2010)

36 Dissipation in chemical reactions networks A k 1 k 3 k -1 B k -2 k 2 k -3 C Equilibrium condition (detailed balance): Conservation of the total number of particles: linear dynamics Can one detect that the system is out of equilibrium using only information contained in the fluctuations of {n A,n B } or n A?

37 with full information {n A,n B } trajectory with partial information {n A } trajectory Possibility to distinguish equilibrium from non-equilibrium fluctuations in a non-invasive way, even when only partial information is available Method works for arbitrary number of non-linear chemical reactions The quality of the estimate depends primarily on the resolution, i.e. on the degree of coarse-graining of the input data S. Muy et al., J. Chem. Phys. (2013)

38 Estimating dissipation in systems with time-dependent driving S. Tush et al., PRL, (2014) Stochastic work and heat on a cycle:

39 Energy vs. information based estimation of dissipation For information based estimation: a) compare forward/backward probability distributions b) compare equilibrium/non-equilibrium probability distributions R. Kawai et al. (2007) S. Vaikuntanathan et al. (2009)

40 Acknowledgements 1. Fluctuation theorems for systems out of equilibrium K. Mallick, A. Lau 2. Fluctuations theorems for systems at equilibrium P. Gaspard 3. Modified fluctuation-dissipation theorems out of equilibrium R. Chétrite, G. Verley 4. Non-invasive estimation of dissipation from trajectory information Theory: A. Kundu, G. Verley Experiments with manipulated colloids: S. Tush, J. Baudry