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MICROREVERSIBILITY and TIME ASYMMETRY in NONEQUILIBRIUM STATISTICAL MECHANICS and THERMODYNAMICS Pierre GASPARD Université Libre De Bruxelles, Brussels, Belgium

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MICROREVERSIBILITY and TIME ASYMMETRY in NONEQUILIBRIUM STATISTICAL MECHANICS and THERMODYNAMICS Pierre GASPARD Université Libre De Bruxelles, Brussels, Belgium 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.
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