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 = deS + diS entropy flow € deS diS ≥ 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 F1-ATPase NANOMOTOR H. Noji, R. Yasuda, M. Yoshida, & K. Kinosita Jr., Nature 386 (1997) 299 protein: F1 = (αβ)3γ (Courtesy Professor K. Kinosita Jr.) R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr. & H. Itoh, Nature 410 (2001) 898 chemical fuel of the F1 nanomotor: ATP adenosine triphosphate power = 10−18 Watt OUT-OF-EQUILIBRIUM NANOSYSTEMS Nanosystems sustaining fluxes of matter or energy, dissipating energy supply Examples: - electronic nanocircuits - heterogeneous catalysis at the nanoscale - molecular motors - ribosome - RNA polymerase: information processing Structure in 3D space: Dynamics in 4D space-time: • no flux Jγ = 0 • flux Jγ ≠ 0 d S d S • no entropy production i = 0 • entropy production i > 0 dt dt €• no energy supply needed € • energy supply required • equilibrium • nonequilibrium € € • 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. d 2r Phase space: velocity v m 2 = F(r) dt trajectory 1 = Θ (trajectory 2) € position r 0 time reversal Θ trajectory 2 = Θ (trajectory 1) BREAKING OF TIME-REVERSAL SYMMETRY Selecting the initial condition typically breaks the time-reversal symmetry. d 2r Phase space: velocity v m 2 = F(r) dt This trajectory is selected by the initial condition. € initia*l condition position r 0 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. d 2r Phase space: velocity v m 2 = −kr dt € position r 0 time reversal Θ self-reversed trajectories FREE PARTICLE Almost all of the trajectories are distinct from their time reversal. d 2r Phase space: velocity v m 2 = 0 dt Antares € position r self-reversed trajectories at zero velocity 0 time reversal Θ Aldebaran BREAKING OF TIME-REVERSAL SYMMETRY IN NONEQUILIBRIUM STEADY STATES weighting each trajectory with a probability Phase space: velocity v r 1 e r s i e o position r r v 0 v r o e i s time reversal Θ r e 2 r 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 ∂p = {H, p} = Lˆ p is also time-reversal symmetric. ∂t The solution of an equation may have a lower symmetry than the equation itself (spontaneous symmetry breaking). Typical Newtonian trajectories T are differe€nt 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: Peq(Θ A) = Peq(A) (detailed balancing) nonequilibrium: Pneq(Θ A) ≠ Pneq(A) STEADY-STATE FLUCTUATION THEOREM FOR CURRENTS t fluctuating currents: 1 Jγ = ∫ jγ (t') dt' ex: • electric currents in a nanoscopic conductor t 0 • rates of chemical reactions • velocity of a molecular motor ΔG G − Geq De Donder affinities or thermodynamic forces: A = γ = γ γ ( free energy sources) € γ T T Stationary probability distribution P : t P J ∑ Aγ Jγ {+ γ } kB time interval: • No directionality at equilibrium A = 0 ≈ e γ γ € P J t → +∞ • Directionality out of equilibrium Aγ ≠ 0 {− γ } valid far from equilibrium as well as close to equilibrium € d S c thermodynamic entropy production: i € = ∑ Aγ Jγ ≥ 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, Fluctuat€i on theorem for currents and Schnakenberg network theory, J. Stat. Phys. 127 (2007) 107. GENERATING FUNCTION OF THE CURRENTS: FULL COUNTING STATISTICS t fluctuation theorem for the currents: A J P +J k ∑ γ γ { γ } B γ with the probability distribution P ≈ e P{−Jγ } 1 t generating function: Q( λγ ,Aγ ) = lim− ln exp− λγ jγ (t')dt' { € } t →∞ ∑ ∫ t γ 0 noneq. fluctuation theorem for the currents bis: Q({λγ ,Aγ }) = Q({Aγ − λγ ,Aγ }) € ∂Q 1 1 average currents: J = = L A + M A A + N A A A + α ∑ €α,β β ∑ α,βγ β γ ∑ α,βγδ β γ δ L ∂λα β 2 β ,γ 6 β ,γ ,δ λα = 0 € LINEAR & NONLINEAR RESPONSE THEORY linear response coefficients Green-Kubo formulas: 2nd cumulants 1 ∂ 2Q 1 +∞ L 0,0 j (t) j j (0) j dt α,β = − ({ }) = ∫ [ α − α ][ β − β ] 2 ∂λα∂λβ 2 −∞ Onsager reciprocity relations: Lα,β = Lβ ,α is totally symmetric nonlinear response coefficients at 2nd order € ∂ 3Q 2nd responses of currents: Mα,βγ ≡ ({0,0}) ∂λ ∂A ∂A € α β γ ∂ 3Q 1st responses of 2nd cumulants: Rαβ ,γ ≡ − ({0,0}) ∂λα∂λβ∂Aγ € +∞ = 1st responses of diffusivities: ∂ Rαβ ,γ = ∫ [ jα (t) − jα ] jβ (0) − jβ dt ∂A [ ] noneq. γ −∞ A= 0 € 2nd responses of currents = M = 1 R + R 1st responses of 2nd cumulants: α,βγ 2 ( αβ ,γ αγ ,β ) € 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({λγ ,Aγ }) = Q({Aγ − λγ ,Aγ }) average current: ∂Q 1 1 Jα = = ∑Lα,β Aβ + ∑ Mα,βγ Aβ Aγ + ∑Nα,βγδ Aβ Aγ Aδ +L ∂λα β 2 β ,γ 6 β ,γ ,δ λα = 0 € nonlinear response coefficients at 3rd order ∂ 4Q 3rd responses of currents: N ≡ 0,0 € α,βγδ ({ }) ∂λα∂Aβ∂Aγ∂Aδ ∂ 4Q 2nd responses of 2nd cumulants: Tαβ ,γδ ≡ − ({0,0}) ∂λα∂λβ∂Aγ∂Aδ 1st responses of 3rd cumulants =€ 4t h cumulants: ∂ 4Q 1 ∂ 4Q Sαβγ ,δ ≡ ({0,0}) = − ({0,0}) € ∂λα∂λβ∂λγ∂Aδ 2 ∂λα∂λβ∂λγ∂λδ relations at 3nd order: 1 Nα,βγδ = 2 (Tαβ ,γδ + Tαγ ,βδ + Tαδ ,βγ − Sαβγ ,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): VQD =1.2 mV IQD =127 aA real-time detection with a quantum point contact (QPC) VQPC = 0.5 mV IQPC = 4.5 nA limit of a large bias voltage: ±eV /2 −ε >> kBT bidirectionality not observed € € € € T. Fujisawa et al., Bidi€re ctional 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). µL ε µR = µL− eV charging rate from side : ρ α ρ = Wρ (01) = Γρ f ρ discharging rate from side ρ: βρ = Wρ (10) = Γρ (1− f ρ ) 1 ρ = L,R Fermi-Dirac distributions: fρ = 1+ e(ε −µρ )/(kBT ) € € € FLUCTUATION THEOREM FOR FERMION TRANSPORT f f α L = ΓL L α R = ΓR R graph: β = Γ 1− f β = Γ 1− f L L ( L ) R R ( R ) ρ = L µL = eV 0 1 ε µR µL− • • € € € ρ = R α β affinity or thermodynamic force defined over the cycle: L R = eβ (µL −µR ) = eβeV ≡ eA α RβL € dP − α L −α R βL + βR Pauli master equation: = ⋅ P dt α L + α R −βL − βR € eλ eigenvalue equation: −α L −α R βL + βR −λ ⋅ F = −QF α Le + α R −βL − βR generating function€: 1 2 −λ λ Q(λ) = α L + α R + βL + βR − (α L + α R − βL − βR ) + 4 α Le + α R βLe + βR 2 ( )( ) € P(k,t) fluctuation theorem: Q(λ) = Q(A − λ) ≈ exp(βeVk) t → ∞ P(−k,t) € € € FULL COUNTING STATISTICS OF FERMIONS: LARGE BIAS VOLTAGE α L ≈ ΓL α R ≈ 0 βL ≈ 0 βR ≈ ΓR µL = eV ε µR µL− € € ρ = L,R 1 limit of a large bias voltage: ±eV /2 −ε >> kBT fρ = 1+ e(ε −µρ )/(kBT ) Fermi-Dirac distributions: fL ≈1 fR ≈ 0 € α β A fully irreversible limit:€ inf inite nonequilibrium driving force: L R = e → ∞ € α RβL P(k,t) fluctuation theorem: ≈ ∞ € P(−k,€t) generating function: € 1 2 −λ € Q(λ) ≈ ΓL + ΓR − (ΓL − ΓR ) + 4ΓLΓRe 2 € QUANTUM FLUCTUATION THEOREM WITH A MAGNETIC FIELD B D.