New Frontiers in Non-equilibrium Physics 2015 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 mercredi 29 juillet 15
Both practical and theoretical interest: Time-delayed feedback processes are ubiquitous in biological regulatory networks and engineering. These systems are typically «autonomous» machines that operate in a nonequilibrium steady state (NESS) where work is permanently extracted from the environment. The non-Markovian character of the dynamics raises issues that go beyond the current framework of stochastic thermodynamics and that do not exist when dealing with a discrete (non-autonomous) feedback control. Main theme of the talk: Because of the delay, the time- reversal operation becomes highly non-trivial. However, one cannot understand the behavior of the system (in particular the fluctuations) without referring to the unusual properties of the reverse process. mercredi 29 juillet 15
TALK ROADMAP A. SECOND LAW-LIKE INEQUALITIES: (bounds for the average extracted work) For more details, see PRL 112 , 180601 (2014) and Phys. Rev. E 91 , 042114 (2015). B. FLUCTUATIONS (work, heat, entropy production): large-deviation functions and fluctuation relations For more details, see cond-mat. arXiv soon... mercredi 29 juillet 15
A. SECOND-LAW-LIKE INEQUALITIES 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 mechanical or electromechanical systems. 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. mercredi 29 juillet 15
Consequences of non-Markovianity 1) 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 1 ; x 2 , v 2 , t 2 ) , 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 level of description. There is no unique entropy-balance equation from the FP formalism (nor unique second-law-like inequality), but a set of equations and inequalities. 2) The time-reversal operation is non-trivial and leads to another second-law-like inequality (in this sense, one looses the nice consistency of stochastic thermodynamics). mercredi 29 juillet 15
3) Preparation effects are crucial due to the memory of the dynamics. We will only focus on the steady-state regime and on the asymptotic behavior in the long-time limit (we will not consider transients). mercredi 29 juillet 15
Second-law-like inequalities obtained from the FP description FP equation for the one-time pdf: ∂ t p ( x, v, t ) = − ∂ x J x ( x, v ) − ∂ J v v ( x, v ) where J x ( x, v, t ) = vp ( x, v, t ) J v ( x, v, t ) = 1 F fb ( x, v, t )] p ( x, v, t ) − γ T m [ − γ v + F ( x ) + ¯ m 2 ∂ v p ( x, v, t ) and Z ∞ 1 ¯ F fb ( x, v, t )] := dyF fb ( y ) p ( x, v, t ; y, t − τ ) p ( x, v, t ) −∞ is an effective time-dependent force obtained by formally integrating out the dependence on the variable y t := x t − τ mercredi 29 juillet 15
Corresponding Shannon entropy Z S xv ( t ) = dx dv p ( x, v, t ) ln p ( x, v, t ) d/dt +FP equation => Entropy balance equation: ˙ Q ( t ) d dtS xv ( t ) = ˙ S xv − S xv i ( t ) − pump ( t ) T Q ( t ) = γ ˙ heat exchanged with the bath m ( m h v 2 t i � T ) where i ( t ) = m 2 dxdv [ J v irr ( x, v )] 2 Z ˙ S xv non-negative «EP» rate ≥ 0 p ( x, v, t ) γ T pump ( t ) = � 1 ˙ m h ∂ v ¯ and S xv F fb ( x, v, t ) i «Entropy pumping» rate that describes the influence of the continuous feedback. The effective force contributes to the balance equation because it is velocity - dependent (i.e., it contains a piece which is antisymmetric under time-reversal). mercredi 29 juillet 15
In the steady state regime, one then obtains a second- law-like inequality ˙ W ext ( ˙ W ext = − ˙ ≤ ˙ Q ) S xv pump T • one can extract work from the heat bath if ˙ S xv pump > 0 • (this depends on the delay, among other things) Similarly, by working in momentum space only, and defining the Shannon entropy as Z S v ( t ) = dx dv p ( v, t ) ln p ( v, t ) one obtains another inequality ˙ W ext ≤ ˙ pump ≤ ˙ S v S xv pump T mercredi 29 juillet 15
The entropy pumping rates have no direct interpretation in terms of information-theoretic measures, but one can also consider information flows that reveal how the exchange of information between the system and the controller is affected by the time delay, e.g. dx dv dy ∂ v J v ( x, v, t ; y, t − τ ) ln p ( x, v, t ; y, t − τ ) Z I xv ; y ˙ flow,v ( t ) := p ( x, v, t ) p ( y, t − τ ) For more details, see Phys. Rev. E 91 , 042114 (2015) mercredi 29 juillet 15
Second-law-like inequality obtained from time reversal In the case of non-autonomous feedback control with measurements and actions performed step by step at regular time intervals (e.g. Szilard engines), one can record the measurement outcomes and define a reverse process that does not involve any measurement nor feedback (see recent review in Nature Phys. 11 , 131, 2015). This is not possible when the feedback is implemented continuously. One must also reverse the feedback The feedback force then depends on the future ! The «conjugate» dynamics is acausal. p m ˙ v t = − γ v t + F ( x t ) + F fb ( x t + τ ) + 2 γ T ξ ( t ) mercredi 29 juillet 15
Generalized local detailed balance equation: 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 Z t Fluctuating heat: 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 The heat is odd under time reversal if τ is changed into − τ mercredi 29 juillet 15
Local detailed balance with continuous time-delayed feedback control: 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 ˜ ⇥ ⇤ m ¨ x s + γ ˙ x s − F ( x s ) − F fb ( x s + τ ) ds with 4 γ 0 ˜ J [ X ] = non-trivial Jacobian due to the violation of causality in general path dependent mercredi 29 juillet 15
One can then define a generalized «entropy production» (Kullback-Leibler divergence): D X P [ X ] ln P [ X ] Z h R cg [ X ] i = ˜ P [ X † ] h e R cg [ X ] i = 1 which satisfies an integral fluctuation theorem In the steady state, this leads to another second-law-like inequality: ˙ W ext ≤ ˙ S J T 1 J ˙ where S J := lim t h ln i st i ! � R ˜ J [ X ] t →∞ ? ? � + ? • ( this quantity can be computed exactly in a linear ? ? c � system but this requires a careful analysis of the ? • «response function» associated to the acausal ? � − ? conjugate Langevin equation in Laplace space. ) ? mercredi 29 juillet 15
Example for a linear system: (a) 0.5 Solid black line: extracted work Rates in the NESS red and blue lines: various bounds. 0 -0.5 -1 0 2 4 6 8 10 τ mercredi 29 juillet 15
B. FLUCTUATIONS To be concrete, we will consider a linear Langevin equation, i.e. a stochastic harmonic oscillator submitted to a linear feedback p v t = − γ v t − kx t + k 0 x t � τ + m ˙ 2 γ T ξ ( t ) describes accurately the d y n a m i c s o f n a n o - + mechanical resonators (e.g. the cantilever of an AFM) used in feedback = cooling setups. − ( ) − Γ + Quality factor: In reduced units: 3 parameters p Q 0 = ω 0 τ 0 ( ω 0 = k/m, τ 0 = m/ γ ) v t + g v t = − x t − 1 x t − τ + ξ t ˙ Gain: g = ( k 0 /k ) Q 0 Q 0 Q 0 mercredi 29 juillet 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 corresponding moment generating functions Z + ∞ Z A ( λ , t ) = h e − λβ A [ X , Y ] i st = dA e − λ A P A ( A, t ) −∞ mercredi 29 juillet 15
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