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. 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: A0 ≡ β1 − β2

Aα ≡ −β1µ1α + β2µ2α for α =1,2,...,c quantum steady-state fluctuation theorem for currents:

Q ,A ;B Q A , ; B ({λ€γ γ } ) = ({ γ − λγ λγ } − )

linear response coefficients Casimir-Onsager reciprocity relations: L (B) L ( B) € α,β = β,α − nonlinear response coefficients: magnetic-field asymmetry ∂ 3Q ∂ 3Q (0,0;B) = − (0,0;−B) A A A A ∂λα∂ β∂ γ €∂ λα∂ β∂ γ ∂ 3Q Rαβ ,γ (B) = Rαβ ,γ (−B) + (0,0;B) ∂λα∂Aβ∂Aγ ∂ 3Q Mα,βγ (B) + Mα,βγ (−B) = Rαβ ,γ (−B) + Rαγ ,β (−B) + (0,0;B) ∂λα∂Aβ∂Aγ

€ 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 α = Γ f α = Γ f Bose-Einstein distributions: L L L R R R βL = ΓL (1+ f L ) βR = ΓR (1+ f R ) 1 f = β β ρ β ρ hω 0 L R e −1 ω0 ρ = L,R € € α β affinity or thermodynamic force: L R = e−hω 0 (β L −β R ) ≡ eA € α β R L master equation: dP(N,t) € = (βL + βR )(N +1)P(N +1,t) − (βL + βR )NP(N,t) dt

+ (α L + α R )NP(N −1,t) − (α L + α R )(N +1)P(N,t) eigenvalue equation: € λ −Q(λ)F(N,t) = (βLe + βR )(N +1)F(N +1,t) − (βL + βR )NF(N,t) −λ + (α L + α R )NF(N −1,t) − α Le + α R (N +1)F(N,t) € ( ) fluctuation theorem: Q( ) Q(A ) P(k,t) λ = − λ ≈ exp[−hω0 (βL − βR )k] t → ∞ P(−k,t) €

€ € STATISTICS OF PATHS / HISTORIES

Coarse-graining: cell ω in the phase space

stroboscopic observation of the trajectory with sampling time Δt : Γ(nΔt;r0,p0) in cell ωn path or history: ω = ω0ω1ω2…ωn−1

R If ω = ω0ω1ω2…ωn−1 is a possible path, then ω = ωn−1…ω2ω1ω0 is also a possible path. But, again, ω ≠ ωR.

Stationary probability distribution as solution of Liouville equation:

P(ω0ω1ω2 ...ωn−1) = lim dΓ p(Γ,t) I − Δt − ( n−1)Δt (Γ) t →∞ ∫ ω 0 ∩Φ (ω1 )...∩Φ (ω n−1 ) € Statistical description: probability of a path or history:

equilibrium steady state: Peq(ω0ω1ω2…ωn−1) = Peq(ωn−1…ω2ω1ω0) (detailed balancing) nonequilibrium steady state: P (ω ω ω …ω ) ≠ P (ω …ω ω ω ) € neq 0 1 2 n−1 neq n−1 2 1 0

In a nonequilibrium steady state, ω and ωR have different probability weights: breaking of the time-reversal symmetry by the nonequilibrium probability distribution. TEMPORAL DISORDER

nonequilibrium steady state: P (ω0 ω1ω2 … ωn−1) ≠ P (ωn−1 … ω2 ω1 ω0)

If the probability of a typical path decays as (coin tossing Bernoulli process: h = log 2)

P(ω) = P(ω0 ω1 ω2 … ωn−1) ~ exp( −h Δt n ) the probability of the time-reversed path decays as

R R R P(ω ) = P(ωn−1 … ω2 ω1 ω0) ~ exp( −h Δt n ) with h ≠ 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

R R h = lim n→∞ (−1/nΔt) ∑ω P(ω) ln P(ω )

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: deS entropy flow Second law of thermodynamics: entropy S d S ≥ 0 entropy production dS d S d S d S i = e + i with i ≥ 0 dt dt dt dt € Out-of-equilibrium process €

€ Entropy production: 1 d S i = h R − h ≥ 0 kB dt

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. €

nΔt d i S nΔt h R −h P(ω) P(ω0ω1ω2 ... ωn−1) ( ) kB dt R = ≈ e = e P(ω ) P(ωn−1... ω2ω1ω0 )

thermodynamic entropy production = time asymmetry of temporal disorder

€ OUT-OF-EQUILIBRIUM DIRECTIONALITY thermodynamic entropy production = temporal disorder hR 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 < hR.

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 Erasing information in the memory of a computer 0 is irreversible and dissipates energy. 0 1

€ €

thermodynamic entropy production per erased bit: R Δ iS = kB(h − h) = kBD 1 Shannon disorder per bit: D ≡ lim − ∑µ(σ1σ 2 ...σ l ) ln µ(σ1σ 2 ...σ l ) l →∞ l σ 1σ 2 ...σ l € Landauer’s case of a random binary sequence: R Δ iS = kB(h − h) = kB ln2 D. Andrieux & P. Ga€spa rd, 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 S α u2 i = dt 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 Joule law: trajectories d S R I 2 i = dt 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

at equilibrium: …212132131223132… Random trajectories random observed in experiments: R. Yasuda, H. Noji, M. Yoshida, out of equilibrium: K. Kinosita Jr. & H. Itoh, …123123123123123… Nature 410 (2001) 898 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 S thermodynamic arrow of time i = h R − h ≥ 0 = time asymmetry in temporal disorder kB dt

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