MICROREVERSIBILITY AND TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS AND THERMODYNAMICS

MICROREVERSIBILITY AND TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS AND THERMODYNAMICS Pierre GASPARD Université Libre de Bruxelles, Brussels, Belgium David ANDRIEUX, Brussels & Yale Massimiliano ESPOSITO, Brussels Takaaki MONNAI, Tokyo Shuichi TASAKI, Tokyo INTRODUCTION: TIME-REVERSAL SYMMETRY BREAKING FLUCTUATION THEOREM FOR CURRENTS & NONLINEAR RESPONSE THEOREM OF NONEQUILIBRIUM TEMPORAL ORDERING CONCLUSIONS ICTS Program on Non-Equilibrium Statistical Physics, Indian Institute of Technology, Kanpur NESP2010 Ilya Prigogine Lecture, 30 January 2010

ILYA PRIGOGINE (1917-2003) Second law of thermodynamics: spiral waves in the BZ reaction Open system: ds = d e S + d i S entropy flow d e S d i S 0 entropy production Nobel Prize in Chemistry (1977) for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures.

THE SCALE OF THINGS: NANOMETERS AND MORE Publication of the US Government

F 1 -ATPase NANOMOTOR H. Noji, R. Yasuda, M. Yoshida, & K. Kinosita Jr., Nature 386 (1997) 299 protein: F 1 = (αβ) 3 γ (Courtesy Professor K. Kinosita Jr.) R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 chemical fuel of the F 1 nanomotor: ATP adenosine triphosphate power = 10 18 Watt

OUT-OF-EQUILIBRIUM NANOSYSTEMS Nanosystems sustaining fluxes of matter or energy, dissipating energy supply Examples: Structure in 3D space: - electronic nanocircuits - heterogeneous catalysis at the nanoscale - molecular motors - ribosome - RNA polymerase: information processing Dynamics in 4D space-time: no flux no entropy production no energy supply needed equilibrium J γ = 0 d i S dt = 0 flux J γ 0 entropy production energy supply required nonequilibrium d i S dt > 0 in contact with one reservoir in contact with several reservoirs

MICRODYNAMICS: TIME-REVERSAL SYMMETRY Θ(r,v) = (r, v) Newton s equation of mechanics is time-reversal symmetric. Phase space: velocity v m d 2 r dt 2 = F(r) trajectory 1 = Θ (trajectory 2) 0 position r time reversal Θ trajectory 2 = Θ (trajectory 1)

BREAKING OF TIME-REVERSAL SYMMETRY Selecting the initial condition typically breaks the time-reversal symmetry. Phase space: velocity v m d 2 r dt 2 = F(r) This trajectory is selected by the initial condition. * initial condition 0 position r time reversal Θ The time-reversed trajectory is not selected by the initial condition if it is distinct from the selected trajectory.

HARMONIC OSCILLATOR All the trajectories are time-reversal symmetric in the harmonic oscillator. Phase space: velocity v m d 2 r dt 2 = kr 0 time reversal Θ position r self-reversed trajectories

FREE PARTICLE Almost all of the trajectories are distinct from their time reversal. Phase space: velocity v m d 2 r dt 2 = 0 Antares self-reversed trajectories at zero velocity 0 time reversal Θ position r Aldebaran

reservoir 1 reservoir 2 BREAKING OF TIME-REVERSAL SYMMETRY IN NONEQUILIBRIUM STEADY STATES weighting each trajectory with a probability Phase space: velocity v 0 position r time reversal Θ Nonequilibrium stationary probability distribution: directionality

DETAILED BALANCING AT EQUILIBRIUM The time-reversal symmetry, e.g. detailed balancing, is restored at equilibrium. Phase space: velocity v 0 position r time reversal Θ Equilibrium stationary probability distribution: no directionality

BREAKING OF TIME-REVERSAL SYMMETRY Θ(r,v) = (r, v) Newton s equation of mechanics is time-reversal symmetric if the Hamiltonian H is even in the momenta. Liouville equation of statistical mechanics, ruling the time evolution of the probability density p is also time-reversal symmetric. The solution of an equation may have a lower symmetry than the equation itself (spontaneous symmetry breaking). Typical Newtonian trajectories T are different from their time-reversal image Θ T : Θ T T Irreversible behavior is obtained by weighting differently the trajectories T and their time-reversal image Θ T with a probability measure. Stationary probability distribution: (random event A) equilibrium: P eq (Θ A) = P eq (A) (detailed balancing) nonequilibrium: P neq (Θ A) P neq (A) p t = { H, p } = L ˆ p

STEADY-STATE FLUCTUATION THEOREM FOR CURRENTS fluctuating currents: J γ = 1 t t j γ (t') dt' 0 ex: electric currents in a nanoscopic conductor rates of chemical reactions velocity of a molecular motor De Donder affinities or thermodynamic forces: Stationary probability distribution P : No directionality at equilibrium A γ = 0 Directionality out of equilibrium A γ 0 A γ = ΔG γ T { } { } e P +J γ P J γ = G G eq γ γ T t k B γ A γ J γ ( free energy sources) time interval: t + valid far from equilibrium as well as close to equilibrium thermodynamic entropy production: c d i S = A γ J γ 0 dt st γ =1 D. Andrieux & P. Gaspard, Fluctuation theorem and Onsager reciprocity relations, J. Chem. Phys. 121 (2004) 6167. D. Andrieux & P. Gaspard, Fluctuation theorem for currents and Schnakenberg network theory, J. Stat. Phys. 127 (2007) 107.

GENERATING FUNCTION OF THE CURRENTS: FULL COUNTING STATISTICS fluctuation theorem for the currents: with the probability distribution P { } { } e P +J γ P J γ t k B γ A γ J γ generating function: Q( { λ γ,a γ }) = lim 1 t t t ln exp λ γ j γ ( t' )dt' γ 0 noneq. fluctuation theorem for the currents bis: Q( { λ γ, A γ }) = Q( { A γ λ γ, A γ }) average currents: J α = Q λ α λ α = 0 = L α,β A β + 1 M α,βγ A β A γ + 1 2 6 β β,γ N α,βγδ A β A γ A δ +L β,γ,δ

LINEAR & NONLINEAR RESPONSE THEORY linear response coefficients Green-Kubo formulas: 2nd cumulants L α,β = 1 2 Q 0,0 2 λ α λ β Onsager reciprocity relations: ({ }) = 1 2 + [ j α (t) j α ] j β (0) j β dt L α,β = L β,α nonlinear response coefficients at 2nd order 3 Q 2nd responses of currents: M α,βγ 0,0 λ α A β A γ ({ }) [ ] is totally symmetric 1st responses of 2nd cumulants: = 1st responses of diffusivities: 2nd responses of currents = 1st responses of 2nd cumulants: 3 Q R αβ,γ 0,0 λ α λ β A γ R αβ,γ = A γ + ({ }) [ j α (t) j α ] j β (0) j β dt M α,βγ = 1 ( R + R ) 2 αβ,γ αγ,β [ ] noneq. A= 0 D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Mech. (2006) P01011.

NONLINEAR RESPONSE THEORY (cont d) fluctuation theorem for the currents: Q( { λ γ,a γ }) = Q( { A γ λ γ, A γ }) average current: J α = Q λ α λ α = 0 = L α,β A β + 1 M α,βγ A β A γ + 1 2 6 β β,γ N α,βγδ A β A γ A δ +L β,γ,δ nonlinear response coefficients at 3rd order 4 Q 3rd responses of currents: N α,βγδ 0,0 λ α A β A γ A δ ({ }) 4 Q 2nd responses of 2nd cumulants: T αβ,γδ 0,0 λ α λ β A γ A δ 1st responses of 3rd cumulants = 4th cumulants: S αβγ,δ 4 Q 0,0 λ α λ β λ γ A δ ({ }) = 1 2 ({ }) 4 Q 0,0 λ α λ β λ γ λ δ ({ }) relations at 3nd order: N α,βγδ = 1 ( T + T + T S ) 2 αβ,γδ αγ,βδ αδ,βγ αβγ,δ D. Andrieux & P. Gaspard, J. Stat. Mech. (2007) P02006.

QUANTUM NANOSYSTEMS de Broglie quantum wavelength: λ = h/(mv) electrons are much lighter than nuclei -> quantum effects are important in electronics S. Gustavsson et al., Counting Statistics of Single Electron Transport in a Quantum Dot, Phys. Rev. Lett. 96, 076605 (2006). T = 350 mk current fluctuations in a GaAs-GaAlAs quantum dot (QD): real-time detection with a quantum point contact (QPC) V QD =1.2 mv V QPC = 0.5 mv I QD =127 aa I QPC = 4.5 na limit of a large bias voltage: ±ev /2 ε >> k B T bidirectionality not observed T. Fujisawa et al., Bidirectional Counting of Single Electrons, Science 312, 1634 (2006). current fluctuations in a AlGaAs/GaAs double quantum dot: real-time detection with a quantum point contact bidirectionality observed

FULL COUNTING STATISTICS OF FERMIONS L. S. Levitov & G. B. Lesovik, Charge distribution in quantum shot noise, JETP Lett. 58, 230 (1993) D. A. Bagrets and Yu. V. Nazarov, Full counting statisticsof charge transfer in Coulomb blockade systems, Phys. Rev. B 67, 085316 (2003). J. Tobiska & Yu. V. Nazarov, Inelastic interaction corrections and universal relations for full counting statistics in a quantum contact, Phys. Rev. B 72, 235328 (2005). D. Andrieux & P. Gaspard, Fluctuation theorem for transport in mesoscopic systems, J. Stat. Mech. P01011 (2006). U. Harbola, M. Esposito, and S. Mukamel, Quantum master equation for electron transport through quantum dots and single molecules, Phys. Rev. B 74, 235309 (2006). M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009). µ L ε µ R = µ L ev charging rate from side ρ: discharging rate from side ρ: α ρ = W ρ 01 β ρ = W ρ 10 ( ) = Γ ρ f ρ ( ) = Γ ρ ( 1 f ρ ) 1 Fermi-Dirac distributions: f ρ = 1+ e (ε µ ρ )/(k B T ) ρ = L,R

α L = Γ L f L β L = Γ L 1 f L FLUCTUATION THEOREM FOR FERMION TRANSPORT ( ) α R = Γ R f R ( ) β R = Γ R 1 f R graph: ρ = L µ L µ R = µ L ev ε 0 1 ρ = R affinity or thermodynamic force defined over the cycle: Pauli master equation: eigenvalue equation: generating function: Q(λ) = 1 2 α L + α R + β L + β R dp dt = α α L R α L + α R α L α R α L e λ + α R β L + β R P β L β R α L β R α R β L = e β (µ L µ R ) = e βev e A β L e λ + β R F = QF β L β R ( α L + α R β L β R ) 2 + 4( α L e λ + α R )( β L e λ + β R ) fluctuation theorem: Q(λ) = Q(A λ) P(k,t) P( k,t) exp( βevk) t

FULL COUNTING STATISTICS OF FERMIONS: LARGE BIAS VOLTAGE α L Γ L β L 0 α R 0 β R Γ R limit of a large bias voltage: µ L ε µ = µ R L ev ±ev /2 ε >> k B T Fermi-Dirac distributions: f L 1 f R 0 f ρ = ρ = L,R 1 1+ e (ε µ ρ )/(k B T ) fully irreversible limit: infinite nonequilibrium driving force: fluctuation theorem: P(k,t) P( k,t) α L β R α R β L = e A generating function: Q(λ) 1 2 Γ + Γ L R ( ) 2 + 4Γ L Γ R e λ Γ L Γ R

QUANTUM FLUCTUATION THEOREM WITH A MAGNETIC FIELD B D. Andrieux, P. Gaspard, T. Monnai & S. Tasaki, The fluctuation theorem for currents in open quantum systems, New J. Phys. 11 (2009) 043014; Erratum 109802 affinities or thermodynamic forces: A 0 β 1 β 2 A α β 1 µ 1α + β 2 µ 2α for α =1,2,...,c quantum steady-state fluctuation theorem for currents: Q( { λ γ, A γ };B) = Q { A γ λ γ,λ γ }; B Casimir-Onsager reciprocity relations: 3 Q λ α A β A γ ( 0,0;B) = R αβ,γ (B) = R αβ,γ ( B) + 3 Q λ α A β A γ ( ) linear response coefficients 3 Q λ α A β A γ L α,β (B) = L β,α ( B) nonlinear response coefficients: magnetic-field asymmetry ( 0,0; B) ( 0,0;B) M α,βγ (B) + M α,βγ ( B) = R αβ,γ ( B) + R αγ,β ( B) + 3 Q λ α A β A γ ( 0,0;B)

FLUCTUATION THEOREM FOR BOSON TRANSPORT U. Harbola, M. Esposito, and S. Mukamel., Statistics and fluctuation theorem for boson and fermion transport through mesoscopic junctions, Phys. Rev. B 76, 085408 (2007). M. Esposito, U. Harbola, and S. Mukamel., Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems, Rev. Mod. Phys. 81, 1665 (2009). A single oscillator mode between two heat baths at different temperatures Bose-Einstein distributions: f ρ = 1 e β ρ hω 0 ρ = L,R master equation: 1 eigenvalue equation: β L affinity or thermodynamic force: dp(n,t) dt fluctuation theorem: Q(λ) = Q(A λ) α L = Γ L f L ( ) β L = Γ L 1+ f L α R = Γ R f R β R = Γ R 1+ f R = ( β L + β R )( N +1)P(N +1,t) ( β L + β R )NP(N,t) + ( α L + α R )NP(N 1,t) ( α L + α R )( N +1)P(N,t) Q(λ)F(N,t) = β L e λ + β R ( )F(N +1,t) ( β L + β R )NF(N,t) ( ) N +1 ω 0 α L β R α R β L = e hω 0 (β L β R ) e A + ( α L + α R )NF(N 1,t) α L e λ + α R P(k,t) P( k,t) exp hω 0( β L β R )k ( ) ( ) N +1 β R ( )F(N,t) [ ] t

STATISTICS OF PATHS / HISTORIES Coarse-graining: cell ω in the phase space stroboscopic observation of the trajectory with sampling time Δt : Γ(nΔt;r 0,p 0 ) in cell ω n path or history: ω = ω 0 ω 1 ω 2 ω n 1 If ω = ω 0 ω 1 ω 2 ω n 1 is a possible path, then ω R = ω n 1 ω 2 ω 1 ω 0 is also a possible path. But, again, ω ω R. Stationary probability distribution as solution of Liouville equation: P(ω 0 ω 1 ω 2...ω n 1 ) = lim t dγ p(γ,t) I ω 0 Φ Δt ( ω 1 )... Φ ( n 1)Δt ( ω n 1 ) (Γ) Statistical description: probability of a path or history: equilibrium steady state: P eq (ω 0 ω 1 ω 2 ω n 1 ) = P eq (ω n 1 ω 2 ω 1 ω 0 ) (detailed balancing) nonequilibrium steady state: P neq (ω 0 ω 1 ω 2 ω n 1 ) P neq (ω n 1 ω 2 ω 1 ω 0 ) In a nonequilibrium steady state, ω and ω R have different probability weights: breaking of the time-reversal symmetry by the nonequilibrium probability distribution.

TEMPORAL DISORDER nonequilibrium steady state: P (ω 0 ω 1 ω 2 ω n 1 ) P (ω n 1 ω 2 ω 1 ω 0 ) If the probability of a typical path decays as (coin tossing Bernoulli process: h = log 2) the probability of the time-reversed path decays as P(ω) = P(ω 0 ω 1 ω 2 ω n 1 ) ~ exp( h Δt n ) P(ω R ) = P(ω n 1 ω 2 ω 1 ω 0 ) ~ exp( hr Δt n ) with h R h temporal disorder per unit time: (dynamical randomness) h = lim n ( 1/nΔt) ω P(ω) ln P(ω) time-reversed temporal disorder per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 h R = lim n ( 1/nΔt) ω P(ω) ln P(ω R ) The time-reversed temporal disorder per unit time characterizes the dynamical randomness of the time-reversed paths.

THERMODYNAMIC ENTROPY PRODUCTION Open system interacting with its environment: d e S entropy flow Second law of thermodynamics: entropy S ds dt = d es dt + d is dt with d is dt 0 d i S 0 entropy production Out-of-equilibrium process Entropy production: 1 d i S k B dt = h R h 0 P. Gaspard, J. Stat. Phys. 117 (2004) 599 P. Gaspard and G. Nicolis, Phys. Rev. Lett. 65 (1990) 1693 Time-reversed paths are less probable. P(ω) P(ω R ) = P(ω nδt 0ω 1 ω 2... ω n 1 ) P(ω n 1... ω 2 ω 1 ω 0 ) enδt ( h R h) = e k B d i S dt thermodynamic entropy production = time asymmetry of temporal disorder

OUT-OF-EQUILIBRIUM DIRECTIONALITY thermodynamic entropy production = temporal disorder h R of time-reversed paths temporal disorder h of typical paths = time asymmetry in temporal disorder = measure of the breaking of the time-reversal symmetry in the probability distribution Theorem of nonequilibrium temporal ordering as a corollary of the second law: In nonequilibrium steady states, the typical paths are more ordered in time than the corresponding time-reversed paths, in the sense that h < h R. Temporal ordering is possible out of equilibrium at the expense of the increase of phase-space disorder. There is thus no contradiction with Boltzmann s interpretation of the second law. Nonequilibrium processes can generate dynamical order and information. Remark: This dynamical order is a key feature of biological phenomena.

DEDUCTION OF LANDAUER S PRINCIPLE 0 Erasing information in the memory of a computer is irreversible and dissipates energy. 1 0 thermodynamic entropy production per erased bit: Shannon disorder per bit: D lim l 1 l σ 1 σ 2...σ l Δ i S = k B ( h R h) = k B D µ(σ 1 σ 2...σ l ) ln µ(σ 1 σ 2...σ l ) Landauer s case of a random binary sequence: Δ i S = k B ( h R h) = k B ln2 D. Andrieux & P. Gaspard, Dynamical randomness, information, and Landauer s principle, EuroPhysics Letters 81 (2008) 28004

OUT-OF-EQUILIBRIUM FLUCTUATING SYSTEMS Energy supply Brownian particle in an optical trap and a flow (Ciliberto et al., 2008) laser D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, J. Stat. Mech. (2008) P01002 RC electric circuit (Nyquist thermal noise) (Ciliberto et al., 2008) D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, J. Stat. Mech. (2008) P01002 Molecular motor F1-ATPase (Kinosita et al., 2001) R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898

BROWNIAN PARTICLE OUT OF EQUILIBRIUM D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, J. Stat. Mech. (2008) P01002 particle of 2 µm diameter in an optical trap and a flow of speed u laser trajectories for u and u probability distributions of position Temporal disorders of typical and reversed trajectories d i S dt = α u2 T Their difference is the thermodynamic production entropy Irreversibility is observed down to the nanoscale.

RC ELECTRIC CIRCUIT OUT OF EQUILIBRIUM D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, J. Stat. Mech. (2008) P01002 R = 9.22 MΩ C = 278 pf τ R = RC = 2.56 ms paths for I and I probability distributions of charges Temporal disorders of typical and reversed trajectories Joule law: d i S dt = R I 2 T Their difference is the thermodynamic entropy production. Irreversibility is observed down to fluctuations of several thousands of electrons.

OUT-OF-EQUILIBRIUM TRAJECTORIES OF THE MOLECULAR MOTOR Power of the motor: 10 18 Watt Random trajectories simulated by a model: P. Gaspard & E. Gerritsma, J. Theor. Biol. 247 (2007) 672 Random trajectories observed in experiments: R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 at equilibrium: 212132131223132 random out of equilibrium: 123123123123123 regular: directionality

CONCLUSIONS Breaking of time-reversal symmetry in the statistical description of nonequilibrium systems Statistical thermodynamics for nonequilibrium nanosystems: Fluctuation theorem for currents in steady states Fluctuation theorem for quantum systems Extensions of Onsager-Casimir reciprocity relations to nonlinear response Thermodynamic entropy production: 1 d i S k B dt = h R h 0 thermodynamic arrow of time = time asymmetry in temporal disorder Theorem of nonequilibrium temporal ordering as a corollary of the second law: Directionality and lowering of randomness in the motion. The farther from equilibrium, the lower the temporal disorder. Thermodynamic arrow of time down to the nanoscale Explanation of directionality in nonequilibrium systems