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Experimental results on the fluctuations in two out of equilibrium systems

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. Outline. Motivation for experiments. Fluctuations of injected power in an harmonic oscillator.

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Experimental results on the fluctuations in two out of equilibrium systems

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  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

  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. brass wire gold mirror The system of interest Torsion pendulum

  5. Experimental set up and the interferometer

  6. Typical fluctuation spectrum

  7. External Forcing

  8. Equation of motion In Fourier space The thermal fluctuation powerspectral density is given by FDT

  9. 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

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

  11. Energy Balance(II)

  12. Linear forcing

  13. t ti Transient and Steady State Fluctuation Theorem TFT SSFT t

  14. TFT

  15. SSFT

  16. S(t) for the SSFT

  17. SSFT periodic forcing

  18. SSFT periodic forcing: W

  19. SSFT periodic forcing: S

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

  21. SSFT periodic forcing: S for Q

  22. Theoretical description What are the fluctuations of Qwhen the system is out of equilibrium ? Linear ramp torque

  23. Theoretical description Langevin Equation

  24. Theoretical description: Fluctuation Theorem for W

  25. Theoretical description: Fluctuation Theorem for Q

  26. Theoretical description: Fluctuation Theorem for Q

  27. 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

  28. Trajectory dependent entropy

  29. Trajectory dependent entropy

  30. Fluctuations of the total entropy

  31. FT for total entropy FT for total entropy

  32. 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.

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

  34. Camera rapide AOD 75 MHz Multiple optical traps

  35. Examples of traps

  36. The non linear potential

  37. The non linear potential f=0.1Hz

  38. Stochastic Resonance

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

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

  41. Fluctuation Theorem for Ws f=0.25Hz

  42. PDF of DU

  43. Fluctuation Theorem for Q

  44. Slope at the origin

  45. Conclusions • We have experimentally investigated the powerinjected in a bistable colloidal system by an external oscillatingforce. • We find that the injected power in the stochastic resonanceregime presents a maximum when the frequency of the driving forcecorresponds to half of the Kramers' rate. • 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 is notsatisfied exactly for small n is certainly due to statisticaland numerical inaccuracy.

  46. References F. Douarche, S. Ciliberto, A. Petrosyan, I. Rabbiosi,Europhys. Lett. 70, 5, 593 (2005) F. Douarche, S. Ciliberto, A. Petrosyan, JSTAT,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, Submitted to Europhysics Letters.

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