Fluctuation theorem in systems in contact with different heath baths: theory and experiments.

Transcription

1 Fluctuation theorem in systems in contact with different heath baths: theory and experiments. Alberto Imparato Institut for Fysik og Astronomi Aarhus Universitet Denmark Workshop Advances in Nonequilibrium Statistical Mechanics GGI-Firenze May 28, 2014

2 Motivations and Overview Fluctuation theorems set constraints on energy and matter fluctuations 1D system of particles in contact with heath (or particle) reservoirs as typical example of out-of-equilibrium systems see, e.g., S. Lepri, R. Livi, and A. Politi, Phys. Rep. (2003); A. Dhar, Adv. in Phys Review the asymptotic FT in a system with a general potential a few remarks on the harmonic chain Correction to the asymptotic limit: exact result Experimental verification Another exact result A. Imparato (IFA) Heat fluctuations May 28, / 34

3 A prototypical example: linear chain of harmonic oscillators Q 1 Q N T 1 T N Two stochastic heat baths harmonic springs Exact solution for the position and momentum PDF Z. Rieder, J.L. Lebowitz, F. Lieb (1967) Q 1 t = Q N t One expects Q 1 t t(t 1 T N ) Q 1 t does not depend on the system size N while the Fourier s law predicts Q 1 t L 1 A. Imparato (IFA) Heat fluctuations May 28, / 34

4 The equations dq n dt = p n, dp n dt = q n U(q 1,... q N ), n = 2, N 1, dp 1 dt = q 1 U(q 1,... q N ) Γp 1 + ξ 1, dp N = qn U(q 1,... q N ) Γp N + ξ N, dt ξ l (t)ξ m (t ) = 2Γ T l δ lm δ(t t ), l, m = 1, N A. Imparato (IFA) Heat fluctuations May 28, / 34

5 Definition of Q 1 (Q N ) Heat flow Q 1 because of the coupling to the reservoirs The heat Q 1 is the work done by the left reservoir on the first particle dp 1 dt = q 1 U Γp 1 + ξ 1, dq 1 dt = p 1 ( Γp 1 + ξ 1 ) Γp 1 is the friction force, and ξ 1 is the stochastic force Analogous definition for Q N In the following Q Q 1 A. Imparato (IFA) Heat fluctuations May 28, / 34

6 Heat probability distribution function We are interested in the steady state probability distribution function (PDF) P ss (Q) = P (Q, t ) We expect the fluctuation theorem (FT) to hold [ ( P ss (Q) 1 P ss ( Q) = exp Q 1 )] T 1 T N see, e.g., G. Gallavotti and E. G. D. Cohen (1995); J. L. Lebowitz and H. Spohn, (1999) A. Imparato (IFA) Heat fluctuations May 28, / 34

7 Generating function The math for P (Q, t) is far too complicated, so one introduces the cumulant generating function µ(λ) Example: Q 1 = t λ µ(λ) λ=0 Requiring the FT dq e λq P (Q, t ) e tµ(λ) P ss (Q)/P ss ( Q) = e Q/τ with τ = (1/T 1 1/T N ) 1 is equivalent to require the symmetry µ(λ) = µ(1/τ λ) A. Imparato (IFA) Heat fluctuations May 28, / 34

8 General Interaction potential U H = N p 2 i i=1 2 + U(q 1, q 2,..., q N ) P t = L 0 P = {P, H} + Γ [ p1 (p 1 + T 1 p1 ) + pn (p N + T N pn )] P {P, H} = N [ P H P ] H p n q n q n p n n=1 A. Imparato (IFA) Heat fluctuations May 28, / 34

9 Joint probability distribution function φ(q, p, Q, t) φ t = L 0 φ + Γ [ Q (p T 1 ) + T 1 p Q + 2T 1 p 1 Q p1 ] φ ψ(q, p, λ, t) dλ e λq φ(q, p, Q, t) ψ t = L 0 ψ + Γ [ λt 1 + λ(λt 1 1)p 2 1 2λT 1 p 1 p1 ] ψ = L λ ψ Ψ(λ, t) = dq dp ψ(q, p, λ, t) If µ 0 (λ) is the largest eigenvalue of L λ : Ψ(λ, t ) e tµ 0(λ) A. Imparato (IFA) Heat fluctuations May 28, / 34

10 Proof of the asymptotic FT in the general case µ 0 (λ) maximal eigenvalue of L λ L λ adjoint of L λ L λ and L λ have the same max. eigenvalue One can prove that L λ and L 1/τ λ have identical spectra µ n (λ) = µ n (1/τ λ) holds for each eigenvalue in particular µ 0 (λ) = µ 0 (1/τ λ) which proves the FT [ ( P ss (Q) 1 P ss ( Q) = exp Q 1 )] T 1 T N A similar symmetry can be proved for the eigenfunctions ψ n (q, p, λ) = exp [ H(q, p)/t N ] ψ n(q, p, 1/τ λ) AI, H. Fogedby, J. Stat. Mec AI, H. Fogedby, in prepar. A. Imparato (IFA) Heat fluctuations May 28, / 34

11 Proof of the asymptotic FT in the general case µ 0 (λ) maximal eigenvalue of L λ L λ adjoint of L λ L λ and L λ have the same max. eigenvalue One can prove that L λ and L 1/τ λ have identical spectra µ n (λ) = µ n (1/τ λ) holds for each eigenvalue in particular µ 0 (λ) = µ 0 (1/τ λ) which proves the FT [ ( P ss (Q) 1 P ss ( Q) = exp Q 1 )] T 1 T N A similar symmetry can be proved for the eigenfunctions ψ n (q, p, λ) = exp [ H(q, p)/t N ] ψ n(q, p, 1/τ λ) AI, H. Fogedby, J. Stat. Mec AI, H. Fogedby, in prepar. A. Imparato (IFA) Heat fluctuations May 28, / 34

12 Properties of the operator L λ time-reversal operator T P (q, p) = P (q, p) new operator L λ = T 1 e βh L λ e βh T, if ψ n (q, p, λ) is an eigenfunction of L λ : L λ ψ n = µ n (λ)ψ n, then ψ n (q, p, λ) = T 1 e βh ψ n, is an eigenfunction for L λ with the same eigenvalue µ n (λ) L λ ψn (q, p, λ) = µ n (λ) ψ n, One can prove that, if β = 1/T N L λ = L 1/τ λ where L λ is the adjoint operator of L λ A. Imparato (IFA) Heat fluctuations May 28, / 34

13 Can be generalized Q 1 Q 2 Q N T 1 T 2 T 3 T N define the vector Q = (Q 1, Q 2,... Q N ) Define τ ij = (1/T i 1/T j ) 1 Fix any reservoir number k P (Q)/P ( Q) = exp Q i /τ ik AI, H. Fogedby, in prepar. i(i k) A. Imparato (IFA) Heat fluctuations May 28, / 34

14 Generating function for the harmonic chain The problem of finding µ 0 (λ) for the harmonic chain can be solved exactly K. Saito and A. Dhar, (2007), (2011). A. Imparato (IFA) Heat fluctuations May 28, / 34

15 Generating function In the limit N π µ(λ) = 0 dp 2π κ cos(p/2) ln [1 + 8Γκ 1/2 sin(p/2) sin(p)f(λ) 1 + 4(Γ 2 /κ) sin 2 (p/2) ] µ(λ) µ(λ) λ λ N = 2 N = 10 A. Imparato (IFA) Heat fluctuations May 28, / 34

16 Exponential tails? µ 0 (λ) exhibits two branch points at λ = 1/T N, λ + = 1/T 1, where µ 0 (λ) diverges AI and H. Fogedby (2012). 0 ln[pss(q)] simulations exact ~ exp(q/t N ) ~ exp(-q/t 1 ) Γ = 10, κ = 60, T 1 = 100, T N = 120, N = 10, t = 100 Q A. Imparato (IFA) Heat fluctuations May 28, / 34

17 An electric circuit with viscous coupling S. Ciliberto, AI, A. Naert e M. Tanase, 2013 where η i is the usual white noise: (C 1 + C) V 1 = V 1 R 1 + C V 2 + η 1 (C 1 + C) V 2 = V 2 R 2 + C V 1 + η 2 η i η j = 2δ ij T i R i δ(t t ). A. Imparato (IFA) Heat fluctuations May 28, / 34

18 Nyquist effect The potential difference across a dipole fluctuates because of the thermal noise C V = V R + η with η(t)η(t ) = 2 T R δ(t t ) C R C R η A. Imparato (IFA) Heat fluctuations May 28, / 34

19 Thermodynamic quantities Work done by circuit 2 on circuit 1 W 1 (t, t) = t+ t Heat dissipated in resistor 1 t dt C dv 2 dt V 1(t ) = t+ t t dt V 1 (f 2 + ξ 2 (t )) Q 1 (t, t) = = t+ t t t+ t t dt CV 1 (t ) dv 2 dt (C 1 + C)V 1 (t ) dv 1 dt ( dt V 1 (t V1 (t ) ) ) η 1 (t ) R 1 Analogous definition for W 2 and Q 2 A. Imparato (IFA) Heat fluctuations May 28, / 34

20 FT for Q 1 : slow convergence e-06 1e-06 1e-08 1e-08 1e-10 1e-12 P (Q 1 ) P ( Q 1 ) exp( Q 1 /τ) Q 1 1e-10 1e-12 1e-14 P (Q 1 ) P ( Q 1 ) exp( Q 1 /τ) Q 1 t = 0.2 s, t = 0.5 s log P ss(q 1 ) P ss ( Q 1 ) = Q 1 τ T 1 = 88 K, T 2 = 296 K, C = 100pF, C 1 = 680pF, C 2 = 420pF and R 1 = R 2 = 10MΩ A. Imparato (IFA) Heat fluctuations May 28, / 34

21 A FT that holds at any time? So far we considered the limit t Is there a FT for any t > 0? Consider the total entropy variation for the system A. Imparato (IFA) Heat fluctuations May 28, / 34

22 A few definitions S r, t : the entropy due to the heat exchanged with the reservoirs up to the time t S r, t = Q 1, t /T 1 + Q 2, t /T 2 the reservoir entropy S r, t is not the only component of the total entropy production: entropy variation of the system? A. Imparato (IFA) Heat fluctuations May 28, / 34

23 A trajectory entropy The system follows a stochastic trajectory through its phase space, the dynamical variables are the voltages V i (t). V i Following Seifert, PRL 2005, for such a system we can define a time dependent trajectory entropy S s (t) = k B log P (V 1 (t), V 2 (t)) Thus, the system entropy variation reads [ ] P (V1 (t + t), V 2 (t + t)) S s, t = k B log. P (V 1 (t), V 2 (t)) t A. Imparato (IFA) Heat fluctuations May 28, / 34

24 These are measurable quantities Q i can be measured as discussed earlier P (V 1, V 2 ) can be easily sampled Left: T 1 = 296 K (eq.) right: T 1 = 88 K The system is in a steady state: P (V 1, V 2 ) does not change with t A. Imparato (IFA) Heat fluctuations May 28, / 34

25 Total entropy Measure the voltages V i at time t = 0 and t = t, and thus obtain [ ] P (V1 ( t), V 2 ( t)) S s, t = k B log. P (V 1 (0), V 2 (0)) Measure the heats Q 1 and Q 2 flowing from/towards the reservoirs in the time interval [0, t] and thus obtain Define the total entropy as S r, t = Q 1, t /T 1 + Q 2, t /T 2 S tot, t = S r, t + S s, t A. Imparato (IFA) Heat fluctuations May 28, / 34

26 FT for the total entropy one can show that the following equality holds exp( S tot /k B ) = 1, which implies that P ( S tot ) should satisfy a fluctuation theorem of the form log[p ( S tot )/P ( S tot )] = S tot /k B, t, T, A. Imparato (IFA) Heat fluctuations May 28, / 34

27 FT for the total entropy: experimental verification [ ] e Stot/k B P ( Stot ) = 1, Sym( S tot ) = log P ( S tot ) = S tot, t, T, k B Sym( S tot ) T 1 1 T 1 =164K T 1 =88K Theory T 1 =140K < exp ( S tot /k B ) > S [k ] tot B A. Imparato (IFA) Heat fluctuations May 28, / 34

28 Second law? Jensen s inequality: e X e X S tot 0 A. Imparato (IFA) Heat fluctuations May 28, / 34

29 Recap and next question An asymptotic FT for the heat current alone P ss (Q) = P (Q, t ) The system is already in a steady state at t < 0, and at t = 0 one starts sampling the heat currents P ss (Q) P ss ( Q) = exp An exact FT that holds t > 0 S s, t = k B log [ P (x( t)) P (x(0)) [ Q S tot, t = S r, t + S s, t ; ( 1 1 )] T 1 T N ] ; S r, t = i e S r, t/k B Can we find a FT for the heat currents Q i, t alone and that holds t > 0? Q i, t /T i = 1 A. Imparato (IFA) Heat fluctuations May 28, / 34

30 Recap and next question An asymptotic FT for the heat current alone P ss (Q) = P (Q, t ) The system is already in a steady state at t < 0, and at t = 0 one starts sampling the heat currents P ss (Q) P ss ( Q) = exp An exact FT that holds t > 0 S s, t = k B log [ P (x( t)) P (x(0)) [ Q S tot, t = S r, t + S s, t ; ( 1 1 )] T 1 T N ] ; S r, t = i e S r, t/k B Can we find a FT for the heat currents Q i, t alone and that holds t > 0? Q i, t /T i = 1 A. Imparato (IFA) Heat fluctuations May 28, / 34

31 Recap and next question An asymptotic FT for the heat current alone P ss (Q) = P (Q, t ) The system is already in a steady state at t < 0, and at t = 0 one starts sampling the heat currents P ss (Q) P ss ( Q) = exp [ Q An exact FT that holds t > 0 [ ] P (x( t)) S s, t = k B log ; P (x(0)) S tot, t = S r, t + S s, t ; ( 1 1 )] T 1 T N Can we find a FT for the heat currents Q i, t alone and that holds t > 0? S r, t = Q i, t /T i i e S r, t/k B = 1 A. Imparato (IFA) Heat fluctuations May 28, / 34

32 A different approach At t < 0 the system is at equilibrium with the bath at T 1 Q N T N At t = 0 connect the other bath at T N and start sampling Q 1 (or Q N ) One finds [ ( P (Q 1, t) 1 P ( Q 1, t) = exp Q 1 1 )] T 1 T N t > 0 G. B. Cuetara, M. Esposito, A. I A. Imparato (IFA) Heat fluctuations May 28, / 34

33 A different approach At t < 0 the system is at equilibrium with the bath at T 1 Q N T N At t = 0 connect the other bath at T N and start sampling Q 1 (or Q N ) T 1 Q 1 Q N T N One finds [ ( P (Q 1, t) 1 P ( Q 1, t) = exp Q 1 1 )] T 1 T N G. B. Cuetara, M. Esposito, A. I t > 0 A. Imparato (IFA) Heat fluctuations May 28, / 34

34 A different approach At t < 0 the system is at equilibrium with the bath at T 1 Q N T N At t = 0 connect the other bath at T N and start sampling Q 1 (or Q N ) T 1 Q 1 Q N T N One finds [ ( P (Q 1, t) 1 P ( Q 1, t) = exp Q 1 1 )] T 1 T N G. B. Cuetara, M. Esposito, A. I t > 0 A. Imparato (IFA) Heat fluctuations May 28, / 34

35 More in general A system S in contact with N energy and particle reservoirs with β ν = Tν 1 and µ ν at t < 0, S is at equilibrium with reservoir ν = 1 and disconnected from reservoirs ν = 1,..., N. At t > 0 connect all the reservoirs and start sampling the energy and particle currents Even more general, for t > 0 perform some work w λ on the system by changing some parameter λ(t) (e.g. pressure, magnetic field...) A. Imparato (IFA) Heat fluctuations May 28, / 34

36 More in general A system S in contact with N energy and particle reservoirs with β ν = Tν 1 and µ ν at t < 0, S is at equilibrium with reservoir ν = 1 and disconnected from reservoirs ν = 1,..., N. At t > 0 connect all the reservoirs and start sampling the energy and particle currents Even more general, for t > 0 perform some work w λ on the system by changing some parameter λ(t) (e.g. pressure, magnetic field...) A. Imparato (IFA) Heat fluctuations May 28, / 34

37 More in general A system S in contact with N energy and particle reservoirs with β ν = Tν 1 and µ ν at t < 0, S is at equilibrium with reservoir ν = 1 and disconnected from reservoirs ν = 1,..., N. At t > 0 connect all the reservoirs and start sampling the energy and particle currents Even more general, for t > 0 perform some work w λ on the system by changing some parameter λ(t) (e.g. pressure, magnetic field...) A. Imparato (IFA) Heat fluctuations May 28, / 34

38 More in general A system S in contact with N energy and particle reservoirs with β ν = Tν 1 and µ ν at t < 0, S is at equilibrium with reservoir ν = 1 and disconnected from reservoirs ν = 1,..., N. At t > 0 connect all the reservoirs and start sampling the energy and particle currents Even more general, for t > 0 perform some work w λ on the system by changing some parameter λ(t) (e.g. pressure, magnetic field...) A. Imparato (IFA) Heat fluctuations May 28, / 34

39 More in general II The following FT holds t > 0 ln P (w λ, {j ɛ ν}, {j n ν }) P ( w λ, { j ɛ ν}, { j n ν }) = β 1 (w λ Φ 1 ) + t where the thermodynamic forces read N (A ɛ νjν ɛ + A n ν jν n ), ν=2 A ɛ ν = β 1 β ν, A n ν = β ν µ ν β 1 µ 1, and the energy and particle currents read j ɛ ν = ɛ ν /t j n ν = n ν /t Involves only measurable currents The knowledge of the PDF P (x(t), λ(t), t), for t 0 is not required. A. Imparato (IFA) Heat fluctuations May 28, / 34

40 Prefactors matter!! See, e.g. van Zon, Cohen 2004 ψ = L λ ψ t Ψ(λ, t) = dq dp ψ(q, p, λ, t) = = b n (λ)e tµn(λ) n b n (λ) = c n (λ) dq dp ψ n (q, p, λ) dq dp n c n (λ)ψ n (q, p, λ)e tµn(λ) we know that µ n (λ) = µ n (1/τ λ) with our choice for the initial condition, also the prefactors satisfy b n (λ) = b n (1/τ λ) A. Imparato (IFA) Heat fluctuations May 28, / 34

41 Summary and perspectives FT for Q in the limit t in 1D systems coupled with stochastic heat baths Not restricted in 1D: can be easily generalized to the 3D case and for any potential, based on the FP operator symmetries, see AI, H. Fogedby 2012 FT for the total entropy that holds for any t > 0 FT for the currents alone that holds for any t > 0 Experimental check in single electron boxes? See, e.g., J. Koski et al, Nat. Phys. (2013) A. Imparato (IFA) Heat fluctuations May 28, / 34

42 Acknowledgments Hans Fogedby, AU H. Fogedby, AI, J. Stat. Mec. 2011, 2012 S. Ciliberto, A. Naert e M. Tanase, ENS Lyon S. Ciliberto, AI, A. Naert e M. Tanase, PRL + J. Stat. Mec Ambassade de France au Danemark/den Franske Ambassade i Danmark G. B. Cuetara, M. Esposito, University of Luxembourg G. B. Cuetara, M. Esposito, AI PRE 2014 A. Imparato (IFA) Heat fluctuations May 28, / 34