Stochastic Thermodynamics of Langevin systems under time-delayed - PowerPoint PPT Presentation

Japan-France Joint Seminar (11-14 August 2015) 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 10 août 15

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 10 août 15

Demon System Information Feedback 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: 1) 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. 11, 131 (2015). 2) Feedback is implemented continuously , in real time. Time- lags 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 10 août 15

 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 time- reversibility 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 10 août 15

Time-delayed Langevin equation: p m ˙ v t = − γ v t + F ( x t ) + F fb ( t ) + 2 γ T ξ ( t ) F fb ( t ) = F fb ( x t − τ + η t − τ ) with  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 steady- state regime. lundi 10 août 15

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 1 , v 1 , t ; x 2 , v 2 , t − τ ) , etc. There is an infinite hierarchy of Fokker-Planck (FP) equations that has no close solution in general. The definition of the Shannon entropy depends on the Z level of description, e.g. S xv ( t ) = dx dv p ( x, v, t ) ln p ( x, v, t ) 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. The «entropy pumping» rate describes the influence of the ˙ W ext ( ˙ W ext = − ˙ ≤ ˙ continuous feedback. One can S xv Q ) pump T extract work from the bath if the entropy puming rate is positive For more details, see Phys. Rev. E 91 , 042114 (2015) lundi 10 août 15

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. Z t p q [ X , Y ] = ds [ γ v s � 2 γ T ξ s ] � v s 0 Z t = � ds [ m ˙ v s � F ( x s ) � F fb ( x s − τ )] � v s 0 P [ X | Y ] probability to observe X = { x s } t 0 given the previous path Y = { x s } 0 − τ P [ X | Y ] ∝ J e − β S [ X , Y ] S [ X , Y ] = Onsager-Machlup action functional Z t S [ X , Y ] = 1 ⇥ ⇤ m ¨ x s + γ ˙ x s − F ( x s ) − F fb ( x s − τ ) ds 4 γ 0 γ 2 m t ) J path-independent Jacobian (contains the factor e lundi 10 août 15

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 τ into − τ ! This defines a conjugate, acausal Langevin dynamics: p m ˙ v t = − γ v t + F ( x t ) + F fb ( x t + τ ) + 2 γ T ξ ( t ) P [ X | Y ] J e β Q [ X , Y ] = ˜ ˜ P [ X † | x † i , Y † ] J [ X ] J [ X ] e − β ˜ S [ X † , Y † ] P [ X † | x † ˜ i , Y † ] ∝ ˜ Z t S [ X , Y ] = 1 ˜ with ⇥ ⇤ m ¨ x s + γ ˙ x s − F ( x s ) − F fb ( x s + τ ) ds 4 γ 0 ˜ J [ X ] = non-trivial Jacobian due to the violation of causality in general path dependent lundi 10 août 15

From the local detailed balance equation, one can derive another second-law-like inequality in the stationary state ˙ W ext ≤ ˙ S J T 1 J ˙ S J := lim t h ln i st where ˜ J [ X ] t →∞ This new upper bound to the extracted work is different from the one involving the entropy pumping rate. lundi 10 août 15

FLUCTUATIONS To be concrete, we now consider a linear Langevin equation, i.e. a stochastic harmonic oscillator submitted to a linear feedback v t + g v t = − x t − 1 In reduced units: x t − τ + ξ t ˙ Q 0 Q 0 3 independent parameters: Q 0 , g, τ p (Quality factor of the resonator) Q 0 = ω 0 τ 0 ( ω 0 = k/m, τ 0 = m/ γ ) This equation faithfully describes the dynamics of Active feedback n a n o - m e c h a n i c a l cooling of the resonators (e.g. the cantilever of an AFM cantilever of an AFM) in (Liang et al. 2000) t h e v i c i n i t y o f t h e resonance frequency. lundi 10 août 15

We study the fluctuations of 3 observables: Z t β W [ X , Y ] = 2 g Work: ds x s − τ v s Q 2 0 0 Heat: β Q [ X , Y ] = β W [ X , Y ] − ∆ U ( x i , x f ) = β W [ X , Y ] − 1 ( x 2 f − x 2 i + v 2 f − v 2 i ) Q 0 Σ [ X , Y ] = β Q [ X , Y ] + ln p st ( x i ) “Pseudo EP” p st ( x f ) Quantities of interest: probability distribution functions P A ( A, t ) = h δ ( A � β A [ X , Y ]) i st Z x f Z Z = d x f D Y P st [ Y ] D X δ ( A � β A [ X , Y ]) P [ X | Y ] x i and the characteristic (or moment generating) functions Z + ∞ Z A ( λ , t ) = h e − λβ A [ X , Y ] i st = dA e − λ A P A ( A, t ) −∞ lundi 10 août 15

P A ( A = at ) ∼ e − I A ( a ) t Expected long-time behavior of the pdfs: where ∼ denotes logarithmic equivalence and I ( a ) is the LDF Z A ( λ , t ) ≈ g A ( λ ) e µ A ( λ ) t Similarly: 1 t ln h e − λβ A [ X , Y ] i st is the SCGF where µ A ( λ ) = lim t →∞ (Scaled Cumulant Generating Function) and the pre-exponential factor g A ( λ ) 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 ! Pole singularities in the prefactors and exponential tails in the pdf’s (e.g. for the heat) lundi 10 août 15

Probability distribution functions: Q 0 = 34 . 2 , g/Q 0 = 0 . 25 Length of the trajectory: t=100 0 0 10 10 τ =8.4 τ =7.6 Probability distributions Probability distributions -1 -1 10 10 -2 -2 10 10 -3 -3 10 10 -4 -4 10 10 -0.15 -0,1 -0.05 0 0.05 -0.1 0 0.1 0.2 w, q, or σ w, q, or σ °°° W, °°° Q, °°° S Main Puzzle: How can we explain the change of behavior of with ? and P Q ( Q = qt ) P Σ ( Σ = σ t ) τ lundi 10 août 15

Two (related) explanations: 1) Existence of exact sum-rules (IFT= integral fluctuation theorems) h e − β Q i st = e γ t/m . For the heat: valid at all times and for any underdamped Langevin dynamics . For the «pseudo» entropy production: ˙ h e − β Σ i st ⇠ e S J t 1 t ln J where ˙ S J := lim is a function of τ ˜ J t →∞ valid only asymptotically (somewhat related to Sagawa- Ueda IFT involving the «efficacy» parameter. lundi 10 août 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 χ ( t � t 0 ) = h x ( t ) ξ ( t 0 ) i If e χ ( t ) → 0 as t → ± ∞ then Z 1 dt 0 e χ ( t − t 0 ) ξ ( t 0 ) x ( t ) ≈ �1 Z t Z 1 dt 0 e χ + ( t − t ) ξ ( t 0 ) + dt 0 e χ � ( t − t 0 ) ξ ( t 0 ) = t �1 or in the frequency domain: x ( ω ) ≈ e χ ( ω ) ξ ( ω ) In this sense, the acausal dynamics reaches a stationary state that is independent from the initial and final conditions for t → ± ∞ lundi 10 août 15

e 0,8 8 Acausal response function 6 Acausal respeonse function 0,6 4 0,4 2 0,2 0 0 -2 -4 -0,2 -6 -0,4 -8 -20 0 20 -20 -10 0 10 20 30 t t 0 10 0 10 τ =7.6 τ =8.4 Probability distributions Probability distributions -1 10 -1 10 -2 10 -2 10 -3 10 -3 10 -4 10 -0.15 -0,1 -0.05 0 0.05 -4 w, q, or σ 10 -0.1 0 0.1 0.2 w, q, or σ lundi 10 août 15