Fluctuation theorem in systems in contact with different heath - PowerPoint PPT Presentation

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

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 2008. 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, 2014 2 / 34

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, 2014 3 / 34

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 ) − Γ p 1 + ξ 1 , dp N = − ∂ q N U ( q 1 , . . . q N ) − Γ p 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, 2014 4 / 34

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 − Γ p 1 + ξ 1 , dQ 1 = p 1 ( − Γ p 1 + ξ 1 ) dt − Γ p 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, 2014 5 / 34

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 � 1 P ss ( Q ) � − 1 �� P ss ( − Q ) = exp − Q 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, 2014 6 / 34

Generating function The math for P ( Q, t ) is far too complicated, so one introduces the cumulant generating function µ ( λ ) � ∞ d Q e λQ P ( Q, t → ∞ ) ≡ e tµ ( λ ) −∞ Example: � Q 1 � = t ∂ λ µ ( λ ) | λ =0 Requiring the FT 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, 2014 7 / 34

General Interaction potential U p 2 H = � N 2 + U ( q 1 , q 2 , . . . , q N ) i i =1 ∂P = L 0 P ∂t = { P, H } + Γ [ ∂ p 1 ( p 1 + T 1 ∂ p 1 ) + ∂ p N ( p N + T N ∂ p N )] P � ∂P N ∂H − ∂P ∂H � � { P, H } = ∂p n ∂q n ∂q n ∂p n n =1 A. Imparato (IFA) Heat fluctuations May 28, 2014 8 / 34

Joint probability distribution function φ ( q , p , Q, t ) ∂φ ∂ Q ( p 2 1 + T 1 ) + T 1 p 2 1 ∂ 2 � � = L 0 φ + Γ Q + 2 T 1 p 1 ∂ Q ∂ p 1 φ ∂t d λ e λQ φ ( q , p , Q, t ) � ψ ( q , p , λ, t ) ≡ ∂ψ − λT 1 + λ ( λT 1 − 1) p 2 � � = L 0 ψ + Γ 1 − 2 λT 1 p 1 ∂ p 1 ψ ∂t = L λ ψ � Ψ( λ, t ) = d q d p 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, 2014 9 / 34

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 � 1 P ss ( Q ) � − 1 �� P ss ( − Q ) = exp − Q 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. 2012 AI, H. Fogedby, in prepar. A. Imparato (IFA) Heat fluctuations May 28, 2014 10 / 34

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 � 1 P ss ( Q ) � − 1 �� P ss ( − Q ) = exp − Q 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. 2012 AI, H. Fogedby, in prepar. A. Imparato (IFA) Heat fluctuations May 28, 2014 10 / 34

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 ˜ ψ n ( 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, 2014 11 / 34

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   �  AI, H. Fogedby, in prepar. P ( Q ) /P ( − Q ) = exp  − Q i /τ ik i ( i � = k ) A. Imparato (IFA) Heat fluctuations May 28, 2014 12 / 34

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, 2014 13 / 34

Generating function In the limit N → ∞ � π � 1 + 8Γ κ − 1 / 2 sin( p/ 2) sin( p ) f ( λ ) � √ κ cos( p/ 2) ln dp µ ( λ ) = − 1 + 4(Γ 2 /κ ) sin 2 ( p/ 2) 2 π 0 0.3 0.3 0.25 0.25 0.2 0.2 µ(λ) µ(λ) 0.15 0.15 0.1 0.1 0.05 0.05 0 0 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 λ λ N = 2 N = 10 A. Imparato (IFA) Heat fluctuations May 28, 2014 14 / 34

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 -2 1 -4 0 ln[ P ss ( Q )] -6 -1 -8 -2 -150 0 150 -10 simulations -12 exact -14 ~ exp(Q/T N ) ~ exp(-Q/T 1 ) -16 -2000 -1500 -1000 -500 0 500 1000 1500 2000 Q Γ = 10 , κ = 60 , T 1 = 100 , T N = 120 , N = 10 , t = 100 A. Imparato (IFA) Heat fluctuations May 28, 2014 15 / 34

An electric circuit with viscous coupling S. Ciliberto, AI, A. Naert e M. Tanase, 2013 − V 1 ( C 1 + C ) ˙ + C ˙ V 1 = V 2 + η 1 R 1 − V 2 ( C 1 + C ) ˙ + C ˙ V 2 = V 1 + η 2 R 2 � � η i η ′ = 2 δ ij T i R i δ ( t − t ′ ). where η i is the usual white noise: j A. Imparato (IFA) Heat fluctuations May 28, 2014 16 / 34

Nyquist effect The potential difference across a dipole fluctuates because of the thermal noise − V C ˙ V = R + η = 2 T η ( t ) η ( t ′ ) Rδ ( t − t ′ ) � � with C R C R ��� ��� ��� ��� η ��� ��� ��� ��� ��� ��� A. Imparato (IFA) Heat fluctuations May 28, 2014 17 / 34

Thermodynamic quantities Work done by circuit 2 on circuit 1 � t +∆ t � t +∆ t d t ′ C d V 2 d t ′ V 1 ( f 2 + ξ 2 ( t ′ )) d t ′ V 1 ( t ′ ) = W 1 ( t, ∆ t ) = t t Heat dissipated in resistor 1 � t +∆ t d t ′ CV 1 ( t ′ )d V 2 d t ′ − ( C 1 + C ) V 1 ( t ′ )d V 1 Q 1 ( t, ∆ t ) = d t ′ t � t +∆ t � V 1 ( t ′ ) � d t ′ V 1 ( t ′ ) − η 1 ( t ′ ) = R 1 t Analogous definition for W 2 and Q 2 A. Imparato (IFA) Heat fluctuations May 28, 2014 18 / 34

FT for Q 1 : slow convergence 1 1 0.01 0.01 0.0001 0.0001 1e-06 1e-06 1e-08 1e-08 1e-10 P ( Q 1 ) P ( Q 1 ) 1e-10 1e-12 P ( − Q 1 ) exp( − Q 1 /τ ) P ( − Q 1 ) exp( − Q 1 /τ ) 1e-12 1e-14 -4 -2 0 2 4 -6 -4 -2 0 2 4 6 Q 1 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 = 100 pF, C 1 = 680 pF, C 2 = 420 pF and R 1 = R 2 = 10 M Ω A. Imparato (IFA) Heat fluctuations May 28, 2014 19 / 34

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, 2014 20 / 34

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, 2014 21 / 34