Stochastic Thermodynamics of Langevin systems under time-delayed feedback control

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1 Japan-France Joint Seminar (-4 August 25) New Frontiers in Non-equilibrium Physics of Glassy Materials Stochastic Thermodynamics of Langevin systems under time-delayed feedback control M.L. Rosinberg in coll. with T. Munakata (Kyoto), and G. Tarjus (Paris) LPTMC, CNRS and Université. P. et M. Curie, Paris lundi août 5

2 Purpose of Stochastic Thermodynamics: Extend the basic notions of classical thermodynamics (work, heat, entropy production...) to the level of individual trajectories. f(λ) V (x, λ) The observed systems. have only a few degrees of freedom fluctuations play a dominant role and observables are described by probability distributions.. are in contact with one or several heat baths. stay far from equilibrium because of mechanical of chemical «forces». lundi août 5

3 Thermodynamics of feedback control («Maxwell s demon»): Purpose: Extend the second law of thermodynamics and the fluctuation theorems in the presence of information transfer and control Two types of control: ) Feedback is implemented discretely by an external agent through a series of loops initiated at a sequence of predetermined times, e.g. Szilard engines (non-autonomous machines). See recent review in Nature Phys., 3 (25). 2) Feedback is implemented continuously, in real time. Timelags are then unavoidable (or chosen on purpose). Normal operating regime: NESS in which heat and work are permanently exchanged with the environment (autonomous machines). lundi août 5

4 The non-markovian character of the dynamics (which is neither due to coarse-graining nor to the coupling with the heat bath) raises issues that go beyond the current framework of stochastic thermodynamics and that do not occur when dealing with discrete feedback control. Main message: Because of the time-delayed feedback control, the relation between dissipation and timereversibility becomes highly non-trivial (the reverse process is quite unusual). However, in order to understand the behavior of the system (in particular the fluctuations of the observables, e.g. the heat), one must refer to the properties of the reverse process. lundi août 5

5 Time-delayed Langevin equation: m v t = v t + F (x t )+F fb (t)+ p 2 T (t) with F fb (t) =F fb (x t + t ) Inertial effects play an important role in human motor control and in experimental setups involving nano-mechanical resonators (e.g., feedback cooling) Deterministic feedback control: no measurement errors Stochastic Delay Differential Equations (SDDEs) have a rich dynamical behavior (multistability, bifurcations, stochastic resonance, etc.). However, we will only focus on the steadystate regime. lundi août 5

6 Second-law-like inequalities The full description of the time-evolving state of the system in terms of pdf s requires the knowledge of the whole Kolmogorov hierarchy p(x, v, t),p(x,v,t; x 2,v 2,t ), etc. There is an infinite hierarchy of Fokker-Planck (FP) equations that has no close solution in general. There is no unique entropy-balance equation from the FP formalism (and no unique second-law-like inequality in the steady state), but a set of equations and inequalities. Ẇ ext T The definition of the Shannon entropy Z depends on the level of description, e.g. S xv (t) = dx dv p(x, v, t)lnp(x, v, t) apple S xv pump (Ẇext = The «entropy pumping» rate describes the influence of the continuous feedback. One can extract work from the bath if the entropy puming rate is positive For more details, see Phys. Rev. E 9, 424 (25) lundi août 5 Q)

7 Local detailed balance equation: relates the heat exchanged with the bath along a given stochastic trajectory to the conditional probabilities of observing the trajectory and its time-reversed image. q[x, Y] = = Z t Z t ds [ v s p 2 T s ] v s ds [m v s F (x s ) F fb (x s )] v s P[X Y] probability to observe X = {x s } t given the previous path Y = {x s } P[X Y] / J e S[X,Y] S[X, Y] = Onsager-Machlup action functional S[X, Y] = 4 Z t ds mẍ s + ẋ s F (x s ) F fb (x s ) J path-independent Jacobian (contains the factor e 2m t ) lundi août 5

8 By simply reversing time, and taking the logratio of the probabilities, one does not recover the heat because the heat is not odd under time reversal! To recover the heat, one must also reverse the feedback i.e. change into! This defines a conjugate, acausal Langevin dynamics: m v t = v t + F (x t )+F fb (x t+ )+ p 2 T (t) P[X Y] P[X x i, Y ] = J J [X] e Q[X,Y] P[X x i, Y ] / J S[X [X]e,Y ] S[X, Y] = Z t with ds mẍ s + ẋ s F (x s ) F fb (x s+ ) 4 J [X] = non-trivial Jacobian due to the violation of causality in general path dependent lundi août 5

9 From the local detailed balance equation, one can derive another second-law-like inequality in the stationary state Ẇ ext T apple S J where S J := lim t! t hln J J [X] i st This new upper bound to the extracted work is different from the one involving the entropy pumping rate. lundi août 5

10 FLUCTUATIONS To be concrete, we now consider a linear Langevin equation, i.e. a stochastic harmonic oscillator submitted to a linear feedback In reduced units: v t = x t Q v t + g Q x t 3 independent parameters: Q,g, + t Q =! (! = p k/m, = m/ ) Active feedback cooling of the cantilever of an AFM (Liang et al. 2) (Quality factor of the resonator) This equation faithfully describes the dynamics of nano-mechanical resonators (e.g. the cantilever of an AFM) in the vicinity of the resonance frequency. lundi août 5

11 We study the fluctuations of 3 observables: Work: W[X, Y] = 2g Q 2 Z t ds x s v s Heat: Q[X, Y] = W[X, Y] U(x i, x f ) = W[X, Y] (x 2 f x 2 i + vf 2 vi 2 ) Q Pseudo EP [X, Y] = Q[X, Y]+ln p st(x i ) p st (x f ) Quantities of interest: probability distribution functions P A (A, t) =h (A A[X, Y])i st Z Z Z xf = dx f DY P st [Y] DX (A A[X, Y])P[X Y] x i and the characteristic (or moment generating) functions Z A (,t)=he lundi août 5 A[X,Y] i st = Z + da e A P A (A, t)

12 Expected long-time behavior of the pdfs: P A (A = at) e I A(a)t where denotes logarithmic equivalence and I(a) is the LDF Similarly: Z A (,t) g A ( )e µ A( )t where µ A ( )= lim t! t lnhe A[X,Y] i st is the SCGF Scaled Cumulant Generating Function) and the pre-exponential factor g A ( ) typically arises from the average over the initial and final states. Here the initial state is Y The 3 observables only differ by temporal «boundary» terms that are not extensive in time. However, since the potential V(x) is unbounded, these terms may fluctuate to order t! lundi août 5 Pole singularities in the prefactors and exponential tails in the pdf s (e.g. for the heat)

13 r = 8.4). To get a more quantitative picture, the corresponding robability distributions are shown in Figs. 3 and 4. Probability distribution functions: Q = 34.2, g/q =.25 Length of the trajectory: t= τ=8.4 Probability distributions Probability distributions τ= , -.5 w, q, or σ w, q, or σ..2 W, Q, S G. 3: (Color on line) Probability distribution FIG. functions 4: (Color on line) Same as Fig. 3 for = 8.4. W (W = wt), PQ (Q = qt), and P ( = t) for Qdashed = 34.2, blue line on the l.h.s. for q /.42 is the theoret Puzzle: can we of Eq. behavior of Q Main =.25 and = 7.6. How The duration of theexplain trajectory isiq (q)tchange curve e the obtained from (72). =. Points represent numerical data obtained by solvg the Langevin equation (45) for 2.6 realizations of the P ( = t) with PQ(black (Q =circles), qt) Qand? oise: W (blue stars), and (red squares). he solid black line is the theoretical curve e IW (w)ttions obtained (more precisely (/t) ln ZA (, t)) are shown in F om Eq. (66), and the dashed black line is the semi-empirical 5. Again we observe a striking di erence in the behav rge-deviation form given by Eq. (69). The dashedofred linesfunctions for = 7.6 and = 8.4. It is also these lundi août 5 I ( )t

14 Two (related) explanations: ) Existence of exact sum-rules (IFT= integral fluctuation theorems). For the heat: he Q i st = e t/m valid at all times and for any underdamped Langevin dynamics. For the «pseudo» entropy production: where he S J := lim t! t i st e S J t ln J J is a function of valid only asymptotically (somewhat related to Sagawa- Ueda IFT involving the «efficacy» parameter. lundi août 5

15 2) The behavior of the pdf s also depends on whether the conjugate, acausal dynamics reaches or does not reach a stationary state. What does this mean? Although the conjugate dynamics is acausal and therefore cannot be physically implemented, one can still define a response function e(t t )=hx(t) (t )i If e(t)! as t! ± then x(t) = Z Z t dt e(t t ) (t ) dt e + (t t) (t )+ or in the frequency domain: Z t dt e (t t ) (t ) x(!) e(!) (!) In this sense, the acausal dynamics reaches a stationary state that is independent from the initial and final conditions for t! ± lundi août 5

16 Acausal respeonse function Acausal response function ation of the observed trajectory, the boundary terms 8 which are non-extensive in time) are still not negligible. The most striking feature is that they contribute di er6 ntly to the observables depending on the value of : for = 7.6,4 the quantity that exhibits the largest fluctua2 ions is, whereas it is Q for = 8.4. Note that the sysem operates in the feedback cooling regime in both cases Tx /T.42, Tv /T.36, W ext W.9 for -2 = 7.6, and Tx /T.72, Tv /T.84, W ext.5 or = -48.4). To get -6 a more quantitative picture, the corresponding robability distributions are shown in Figs. 3 and t 2 FIG. 2: Acausal response function e(t): Same as Fig. g/q =.45 and =. The poles s ±.257 ± (2),6 and s ±.47 ±.692 i control the behavior of e( t and t!, respectively. Note the cusp behavi,4 t = and the weaker singularities for, 2, etc.,8,2 Note finally that e(t) is not C at t =,, 2, as can be easily seen by di erentiating Eq. (E) t -,2 this is more clearly seen in Fig. 2. -, t + -3 r.h.s., respectively. x e(t) is thus given by an infinite but converging sum of exponent Probability distributions Probability distributions IG. 8: Acausal response function e(t) versus time for Q =, g/q=.55 and =.2. e(s) has no poles on the left-hand FIG. 7: (Color on line) Acausal response function e(t) for Q = 2, g/q =.55 and = τ=7.6 de of the complex plane. The poles s ±.394 ±.847 i τ=8.4 rather a figure for Qthe 43.2.) =behavior nd s (2) on thepresent right-hand side control - f e(t) for t (hence the oscillations and the diverging time ependence) and t! (hence the rapid decay to zero), espectively. The ROC of e(s) is defined by min = Re(s ± ) < -2-2 Re(s) < s (2). the sums in e (t) and e (t) areover max = where the two poles in the l.h.s. of the comp -, -.5 w, q, or σ x e(t) X.5 s2l.h.s. Z -3 A(s) -4 t dt es(t -. t) (t ) + X s2r.h.s.. w, q, or σ B(s).2 Z dt e t and it can be numerically computed for a given noise history (in practice of course, on of on terms the second sum and the quality of the approximation depends on the s IG. 3: (Color line) in Probability distribution functions on line) Same as Fig. 3 for = 8.4. Th we Pcan a representative ensemble of trajectories and obtain th (W = août wt),in PQthis (Q =way, qt), and = t) for Q = FIG. 34.2, 4: (Color Wlundi ( generate 5

17 Z Modified f Crooks FT efor the e work: When the acausal dynamics reaches a stationary state, one can show that P W (W = wt) ep ( W f = wt) e(w+ S J )t,t! Probability distributions,5-2 - W P W (W = wt) P W (W = wt)e wt P W ( W S = wt)e J t In the long-time limit, the atypical trajectories that dominate he W i st are the conjugate twins (Jarzynski 26) of typical realisations of the reverse (acausal) process lundi août 5

18 Alternatively, one can determine the properties of the atypical noise that generates the rare events. Since the conjugate dynamics converges, the solution of the acausal Langevin equation is,8 Acausal respeonse function,6,4,2 -,2 -,4 x(!) e(!) (!) -2 2 t Inserting into the original Langevin equation yields e (!) atyp (!) e(!) (!) And thus: atyp (!) e(!) (!) (!). lundi août 5

19 Hence with h atyp (t) atyp (t )i = (t t ) apple Z + d! e(!) (t) =2 T (t)+ [ 2 (!) 2 ]e i!t Variance of the atypical noise.4 ν(t)-2 γ T δ(t) The «atypical» noise that generates the rares events dominating he is colored! W i st -.4 lundi août t

20 CONCLUSION One can extend the framework of stochastic thermodynamics to treat non-markovian effects induced by a time-delayed feedback. This introduces a new and interesting phenomenology. Experimental tests? Thank you for your attention! lundi août 5

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