Breaking and restoring the

Transcription

1 Breaking and restoring the fluctuation-response theorem. J. Parrondo, J.F. Joanny, J. Prost J. Barral, L. Dinis, P. Martin

2 Outline Example of fluctuation-dissipation theorem breaking: Hair cells The Hatano Sasa fluctuation theorem Restoring a general fluctuation-response relation Example revisited

3 BREAKING

4 The Ear

5 Organ of Corti (from Retznius) 20 µm (Source: R. Pujol,

6 The hair bundle: 2 µm (source: Hudspeth,

7 Experimental set-up P. Martin et al response stimulus

8 Hair-bundle oscillations

9 Linear response function ω / 2π < 7Hz χ '' (ω) < 0 The bundle gives work to the fiber! (here: A = 15 nm)

10 Fluctuation-dissipation effective temperature T eff T = ω C(ω) k B T χ '' (ω) (P. Martin, A.J. Hudspeth and F. Jülicher, PNAS (2001))

11 Hopf Bifurcation 1/3 1 SPL: 20 log P P 0 P 0 = 20µPa (source: Ruggero et al., J. Acoust. Soc. Am. 101, 2151 (1997))

12 Single Oscillator F. Jülicher,K. Dierkes, B. Lindner,J.P.,P. Martin Z(t) =X(t) +iy(t) t Z = (r +iω o )Z (u+iu a )Z 2 Z + f ext + η(t) η(t)η(t') = 0 η(t)η (t') = 4Dδ(t t')

13 P t = 1 ρ (rρ +uρ 3 f eff cos(ωt + φ))p +D P ρ + φ (ω +u o aρ 2 + f eff ρ sin(ωt + φ)p + D P ρ 2 φ r 2 >>ud;... ω o = ω o +u a ρ 2 o ;...r eff = r(2du/r 2 );...ρ 2 o = r u χ(ω) = X f ext 1 r eff + 2i(ω ω o ) C(ω) = X(ω)X (ω) C(ω) D r eff 2 + (ω ω o ) 2

14 Effective temperature T eff ( ω) ω C( ω) χ'' ( ω) = 2k BT ( eff ω) ω = D 2(ω ω o )

15 Hopf Bifurcators in higher dimensions tz= (r+iωo)z+(c+ica) Z (u+iua)z 2 Z+f +η d = 4 ε T.Risler, F. Jülicher,J.P., P.R.L Two loop order

16 Renormalization of parameters Flow diagram u u = 4 d ε l = lnb Effective frequency r ω ( ) 0 l eff ω 0 and phase l c a ω 0

17 4 d > 2 Effective temperature At threshold T eff ω ( ) 1 T (ω ω 0 ) µ µ =1 ε 5

18 In all cases for linearized response and two point correlation functions (renormalized quantities) For which: '' χ xx (ω) = ω (r 2 + ω 2 ω 0 2 ) (r 2 + ω 2 ω 2 ) 2 + 4ω 2 r 2 idem for : d = 0, d = 4 ε, d = 2? F. Jülicher,K. Dierkes, B. Lindner, J.P., P. Martin; T. Risler et al; D. Mukamel,G. Grinstein

19 FLUCTUATION THEOREMS

20 R random variable R, Two probability distributions P i (R), P j (R) P i (R) P J (R) = ϕ i,j (R) associated Fluctuation theorem: < ϕ i,j > j = 1 ϕ i,j (R) P J (R) dr = P i (R) dr = 1 Change variable to: R' = Log ( P j (R') P i (R') ) Then: < Exp( R') > = 1

21 Ω i Microcanonical ensemble Hamiltonian dynamics B. Cleuren et al, P. Pradhan et al. Ω f ω i Π(i f ;λ(t)) = ω i Ω i λ(t) ωf ω ' i λ(t f t) Π(f i;λ(t f t)) = ω ' f Ω f ω ' f ω i = ω f = ω ' f = ω ' i P(i f ) P(f i) = P( S) P( S) = Ω f Ω i = exp( S) with : S= S f S i and S= Log(Ω) C. Jarzynski, Crooks etc < exp( S) > = 1

22 Markovian Dynamics Steady State Continuum depending on external control parameters ρ(c; t + δt; λ ) = t =n δt d d c' G(c',c; λ(t)) ρ(c';t; λ) N 1 Π(i f ; λ t ) = G(c n,c n+1 ; λ n ) ρ(c 0 ;λ 0 ) n=0 Π Π t f ϕ(c,λ) = exp( dt λ ) λ t i G * (c',c; λ(t)) = G(c',c; λ(t)) ρss (c;λ(t)) ρ ss (c'; λ(t)) ϕ(c,λ) = Ln ρ ss (c,λ) 0 Π(f i; λ(t f t)) = G*(c n+1,c n ; λ n ) ρ(c N ;λ N ) i=n 1

23 Hatano Sasa Theorem Steady state stochastic entropy of U. Seifert

24 RESTORING A FLUCTUATION-RESPONSE RELATION Agarwal, P. Hänggi et al, U. Seifert et al, M. Baiesi et al, R. Chetrite et al G. Verley et al and us

25 FLUCTUATION RESPONSE THEOREM d dt 0

26 In general no Symmetry! Onsager relations only if detailed balance holds and φ(t) λ α = ε α φ( t) λ α Only symmetry (classical systems)

27 <A(t) >= A nn W n0...n N (λ. n 0..n N U. Seifert, T. Speck 2010 { [ ])P n0 (λ[.])} < δa(t) >= A nn ( W n0...n N (λ. n 0..n N { [ ] + δλ [.])P n0 (λ[.]+ δλ [.])} W n0...n N (λ. { [ ])P n0 (λ[.])}) Protocole λ(t): ] ] :λ(t) = λ 0 +δλ t,0 t > 0 :λ(t) = λ 0 +Markov: P <δa(t) >= n0 (λ 0 ) A nn W n0...n N (λ 0 ) λ n 0...n N χ(t t ') = d dt n 0...n N P n0 (λ 0 ) A nn W n0...n N (λ 0 ) = d λ dt <A(t) Φ(t = 0) λ G. Verley, R. Chétrite, D. Lacoste 2011 >

28 How to use it? Any dynamics around steady state: ( small f α (t) =m αβ δλ β (t)) In general: A αβ Σ βγ I αγ and: χ x x (t) d dt (C x x (t))

29 Correct choice: Why? or Irrespective of the original choice of variables there is one linear combination which restores the fluctuation response relation. More Important: this example shows that for linear response and two point fluctuations, linear Langevin equations suffice.

30 Back to example

31 Back to Hopf Bifurcation

32 C xx (ω) = 2 χ '' xx (ω) ω '' χ xx (ω) Check : T eff = 1! How about experiments?

33 Back to experiment. Back to Pascal Martin! In principle two variables. Bad news! x, dx dt? dx dt η x = rx ω 0 y =z dx dt = z+ f x + η x dz dt = (r 2 + ω 2 0 ) x 2r z+ f z + η z η z = (r η x + ω 0 η y )

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36 R R = ω C xx (ω ) / 2Im(χ xx (ω)

37 Suggestions and consequences Hair cells: do obey generalized FRT. Help identify number of slow variables Only one oscillator Hatano Sasa: severe constraints on renormalisation (Ward identities etc.) Non Markovian?