1. TIME ASYMMETRY IN NONEQUILIBRIUM STATISTICAL MECHANICS Pierre GASPARD Brussels, Belgium J. R. Dorfman, College ParkS. Ciliberto, Lyon T. Gilbert, BrusselsN. Garnier, Lyon D. Andrieux, BrusselsS. Joubaud, Lyon A. Petrosyan, Lyon INTRODUCTION: THE BREAKING OF TIME-REVERSAL SYMMETRY FLUCTUATION THEOREMS FOR CURRENTS & NONLINEAR RESPONSE ENTROPY PRODUCTION & TIME ASYMMETRY OF NONEQUILIBRIUM FLUCTUATIONS CONCLUSIONS

2. BREAKING OF TIME-REVERSAL SYMMETRY  (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  T :  T ≠ T Irreversible behavior is obtained by weighting differently the trajectories T and their time-reversal image  T with a probability measure. Spontaneous symmetry breaking: relaxation modes of an autonomous system Explicit symmetry breaking: nonequilibrium steady state by the boundary conditions P. Gaspard, Physica A 369 (2006) 201-246.

3. STOCHASTIC DESCRIPTION IN TERMS OF A MASTER EQUATION Liouville’s equation of the Hamiltonian dynamics -> reduced description in terms of the coarse-grained states  -> master equation for the probability to visit the state  by the time t : P t (  ) rate of the transition due to the elementary process A trajectory is a solution of Hamilton’s equations of motion:  (t;r 0,p 0 ) Coarse-graining: cell  in the phase space stroboscopic observation of the trajectory with sampling time  t :  (n  t;r 0,p 0 ) in cell  n path or history:  0  1  2 …  n  1 -> statistical description of the equilibrium and nonequilibrium fluctuations = 0 steady state

4. FLUCTUATION THEOREM FOR THE CURRENTS steady state fluctuation theorem for the currents (2004): affinities or thermodynamic forces: fluctuating currents: thermodynamic entropy production: -> Onsager reciprocity relations and their generalizations to nonlinear response D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Phys. 127 (2007) 107. ex: electric currents in a nanoscopic conductor rates of chemical reactions velocity of a linear molecular motor rotation rate of a rotary molecular motor Schnakenberg network theory ( Rev. Mod. Phys. 1976 ): cycles in the graph of the process

5. BEYOND LINEAR RESPONSE & ONSAGER RECIPROCITY RELATIONS average current: fluctuation theorem for the currents: Onsager reciprocity relations: relations for nonlinear response: higher-order nonequilibrium coefficients: generating function of the currents: linear response coefficients: (Schnakenberg network theory) D. Andrieux & P. Gaspard, J. Chem. Phys. 121 (2004) 6167; J. Stat. Mech. (2007) P02006. linear response coefficients: (Green-Kubo formulas)

6. Microreversibility: Hamilton’s equations are time-reversal symmetric. If  (t;r 0,p 0 ) is a solution of Hamilton’s equation, then  (  t;r 0,  p 0 ) is also a solution. But, typically,  (t;r 0,p 0 ) ≠  (  t;r 0,  p 0 ). Coarse-graining: cell  in the phase space stroboscopic observation of the trajectory with sampling time  t :  (n  t;r 0,p 0 ) in cell  n path or history:  0  1  2 …  n  1 If  0  1  2 …  n  1 is a possible path, then  R  n  1 …  2  1  0 is also a possible path. But, again,  ≠  R. Statistical description: probability of a path or history: equilibrium steady state: P eq (  0  1  2 …  n  1 ) = P eq (  n  1 …  2  1  0 ) nonequilibrium steady state: P neq (  0  1  2 …  n  1 ) ≠ P neq (  n  1 …  2  1  0 ) In a nonequilibrium steady state,  and  R have different probability weights. Explicit breaking of the time-reversal symmetry by the nonequilibrium boundary conditions FLUCTUATIONS AND MICROREVERSIBILITY

7. DYNAMICAL RANDOMNESS OF TIME-REVERSED PATHS nonequilibrium steady state: P (  0  1  2 …  n  1 ) ≠ P (  n  1 …  2  1  0 ) If the probability of a typical path decays as P(  ) = P(  0  1  2 …  n  1 ) ~ exp(  h  t n ) the probability of the time-reversed path decays as P(  R ) = P(  n  1 …  2  1  0 ) ~ exp(  h R  t n ) with h R ≠ h entropy per unit time: dynamical randomness (temporal disorder) h = lim n  ∞ (  1/n  t) ∑  P(  ) ln P(  ) time-reversed entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 h R = lim n  ∞ (  1/n  t) ∑  P(  ) ln P(  R ) The time-reversed entropy per unit time characterizes the dynamical randomness (temporal disorder) of the time-reversed paths.

8. THERMODYNAMIC ENTROPY PRODUCTION Property: h R ≥ h (relative entropy) equality iff P(  ) = P(  R ) (detailed balance) which holds at equilibrium. Second law of thermodynamics: entropy S Entropy production: P. Gaspard, J. Stat. Phys. 117 (2004) 599 entropy flow entropy production

9. PROOF FOR CONTINUOUS-TIME JUMP PROCESSES Pauli-type master equation: nonequilibrium steady state:  -entropy per unit time: P. Gaspard & X.-J. Wang, Phys. Reports 235 (1993) 291 time-reversed  -entropy per unit time: P. Gaspard, J. Stat. Phys. 117 (2004) 599 thermodynamic entropy production: Luo Jiu-li, C. Van den Broeck, and G. Nicolis, Z. Phys. B- Cond. Mat. 56 (1984) 165 J. Schnakenberg, Rev. Mod. Phys. 48 (1976) 571

10. PROOF FOR THERMOSTATED DYNAMICAL SYSTEMS entropy per unit time: time-reversed entropy per unit time: thermodynamic entropy production:  is the diameter of the phase-space cells T. Gilbert, P. Gaspard, and J. R. Dorfman (2007)

11. INTERPRETATION nonequilibrium steady state: thermodynamic entropy production: If the probability of a typical path decays as the probability of the corresponding time-reversed path decays faster as The thermodynamic entropy production is due to a time asymmetry in dynamical randomness. entropy production dynamical randomness of time-reversed paths h R dynamical randomness of paths h P. Gaspard, J. Stat. Phys. 117 (2004) 599

12. relaxation time: DRIVEN BROWNIAN MOTION trap stiffness: trap velocity: Polystyrene particle of 2  m diameter in a 20% glycerol-water solution at temperature 298 K, driven by an optical tweezer. D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601 Langevin equation: dissipated heat: driving force: mean dissipated heat: u < 0 u > 0

13. comoving frame of reference: PATH PROBABILITIES OF NONEQUILIBRIUM FLUCTUATIONS thermodynamic entropy production: stationary probability density: path probability: ratio of probabilities for u>0 and u<0: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601 heat generated by dissipation:

14. path: RELATIONSHIP TO DYNAMICAL RANDOMNESS thermodynamic entropy production: (  )-entropy per unit time: path probability: time-reversed (  )-entropy per unit time: time-reversed (  )-entropy: (  )-entropy: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601 algorithm of time series analysis by Grassberger & Procaccia (1980’s)

15. DRIVEN BROWNIAN MOTION thermodynamic entropy production: sampling frequency: 8192 Hz resolution: (  )-entropy time-reversed (  )-entropy time series: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

16. DRIVEN BROWNIAN PARTICLE resolution: dissipated heat along the random path z t potentiel velocity: ratio of probabilities for u>0 and u<0: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

17. RC circuit: DRIVEN ELECTRIC CIRCUIT thermodynamic entropy production Joule law: D. Andrieux, P. Gaspard, S. Ciliberto, N. Garnier, S. Joubaud, and A. Petrosyan, Phys. Rev. Lett. 98 (2007) 150601

18. CONCLUSIONS Breaking of time-reversal symmetry in the statistical description Nonequilibrium work fluctuation theorem: systems driven by an external forcing Nonequilibrium modes of diffusion: relaxation rate  s k, Pollicott-Ruelle resonance Nonequilibrium transients: escape-rate formalism: fractal repeller diffusion D :  (1990) viscosity  :  (1995)

19. CONCLUSIONS (cont’d) thermodynamic entropy production = temporal disorder of time-reversed paths h R  temporal disorder of paths h = time asymmetry in dynamical randomness 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. Boltzmann’s interpretation of the second law: Out of equilibrium, the spatial disorder increases in time. Nonequilibrium steady states: Explicit breaking of time-reversal symmetry by the nonequilibrium conditions. Fluctuation theorem for the currents: Entropy production and temporal disorder: Toward a statistical thermodynamics for out-of-equilibrium nanosystems http://homepages.ulb.ac.be/ ~ gaspard