1. Experimental results on the fluctuations in two out of equilibrium systems S. Joubaud, P.Jop, A. Petrossyan and S.C. Laboratoire de Physique, ENS de Lyon, UMR5672 CNRS Firenze, 16 February 2009

2. Outline  Motivation for experiments.  Fluctuations of injected power in an harmonic oscillator.  Fluctuation Theorems for work, heat and total entropy.  Injected power in the stochastic resonance  Conclusions

3. Experimental study of Fluctuations in Mechanical Systems Motivation Study of thermodynamics laws in weakly out of equlibrium systems.For example the application of Fluctuation Dissipation Theorem (FDT) in aging materials Applications of fluctuation theorems to estimate thermal losses in these systems. Estimate the reversible and irreversible part of the work done by an external force on the system.  Check the hypothesis done to prove FT  Test in experiments the analytical and numerical studies on FT What is the good variable to look at ?

4. The system of interest Torsion pendulum brass wire gold mirror

7. Experimental set up and the interferometer

8. External Forcing

9. In Fourier space The thermal fluctuation power spectral density is given by FDT Equation of motion

10. Typical Fluctuation spectra Spectrum of the torsion pendulum made by a brass wire and a mirror Spectrum of the torsion pendulum inside a viscous oil

11. Work during periodic forcing

12. PDF of the work

13. Energy Balance (I) Sekimoto K, Progress of Theoretical Phys. supplement (130), 17 (1998).

14. Energy Balance (II) Sekimoto K, Progress of Theoretical Phys. supplement (130), 17 (1998).

15. PDF of heat

16. PDF of the work and of the heat

17. Stationary State Fluctuation Theorem (SSFT)  Fluctuation Theorem fixes the symmetry of P(X) around zero

18. SSFT periodic forcing: W

19. SSFT periodic forcing: 

20. SSFT periodic forcing: Q Energy disipated in a time 

21. SSFT periodic forcing:  for Q

22. Linear forcing

23. Transient and Steady State Fluctuation Theorem  TFT  titi SSFT

24. TFT

25. SSFT

26.  for the SSFT

27. Theoretical description What are the fluctuations of  when the system is out of equilibrium ? Linear ramp torque

28. Theoretical description Langevin Equation

29. Theoretical description: Fluctuation Theorem for W

30. Theoretical description: Fluctuation Theorem for Q

32. Trajectory dependent entropy  U. Seifert, Phys. Rev. Lett., 95, 040602, (2005), for Langevin dynamics  A. Puglisi, L. Rondoni, A. Vulpiani, J. Stat. Mech.: Theory and Experiment, P08010,(2006) for Markov process

33. Trajectory dependent entropy

35. Fluctuations of the total entropy

36. FT for total entropy

37. Conclusions  We have experimentally perturbed an harmonic oscillator very close to equilibrium.  We have computed and measured the finite time corrections for SSFT and TFT.  The ‘’ trajectory dependent entropy ‘’ has been computed  We have checked that for the total entropy SSFT is verified for all times.  We have shown that in this specific example the ‘’total entropy’’ takes into account only the entropy produced by the external driving and does not take into account the entropy fluctuations at equilibrium.

38. The stochastic resonance and Fluctuatio Theorem  P.Jop. A. Petrosian, S. Ciliberto, EPL March 2008

39. Camera rapide AOD 75 MHz Multiple optical traps

40. Examples of traps

41. The non linear potential

42. f=0.1Hz

43. Stochastic Resonance

44. Fluctuations of W and Q at Stochastic Resonance  Stochastic work  Classical work  Heat

45. PDF of the Work at various frequencies f=0.05Hz f=2Hz f=0.375Hz f=0.1Hz

46. Fluctuation Theorem for Ws f=0.25Hz

47. PDF of  U

48. Fluctuation Theorem for Q

49. Slope at the origin

50. Theoretical comparison I A. Imparato, P. Jop, A. Petrosyan and S. Ciliberto, J. Stat. Mech. (2008) P10017 PDF of the work computed on a single period : (initial phase=0)(averaged over different initial phases) Experimental data Theoretical prediction based on Fokker-Planck equation

51. Theoretical comparison II Symmetry function for the work PDF of the work

52. Theoretical comparison III A. Imparato, P. Jop, A. Petrosyan and S. Ciliberto, J. Stat. Mech. (2008) P10017 PDF of the heat computed on a single period : (initial phase=0)(averaged over different initial phases) Experimental data Theoretical prediction based on Fokker-Planck equation

53. Conclusions We have experimentally investigated the power injected in a bistable colloidal system by an external oscillating force. We find that the injected power presents a maximum when the frequency of the driving force corresponds to half of the Kramers' rate. (stochastic resonance) SSFT is valid for a non-linear potential. We have shown that FT rapidly converge to the asymptotic value for rather small n. The fact that for the total entropy FT is not satisfied exactly for small n is due to statistical and numerical inaccuracy. A theoretical description based on Fokker-Planck equation is in perfect agreeement with the experimental results

54. F. Douarche, S. Ciliberto, A. Petrosyan, I. Rabbiosi, Europhys. Lett. 70, 5, 593 (2005) F. Douarche, S. Ciliberto, A. Petrosyan, J. Stat. Mech., P09011 (2005). Experimental study of work fluctuations in a harmonic oscillator, S. Joubaud, N. B. Garnier, F. Douarche, A. Petrosyan, S. Ciliberto, C. R. Physique 8 (2007) 518–527 Fluctuation theorems for harmonic oscillators, S. Joubaud, N. B. Garnier, S. Ciliberto, J. Stat. Mech., P09018 (2007) Work Fluctuation Theorems for Harmonic Oscillators F. Douarche, S. Joubaud, N. B. Garnier, A. Petrosyan, and S. Ciliberto Phys. Rev. Lett. 97, 140603 (2006). Fluctuations of the total entropy production in stochastic systems, S. Joubaud, N. B. Garnier, S. Ciliberto, Eur. Phys. Lett. 82, 3 (2008) 30007 Probability density functions of work and heat near the stochastic resonance of a colloidal particle A. Imparato, P. Jop, A. Petrosyan and S. Ciliberto, J. Stat. Mech. (2008) P10017 References