Heat conduction induced by non-Gaussian athermal fluctuations

1. 2013/07/03 Physics of Granular Flows @YITP Heat conduction induced by non-Gaussian athermal fluctuations Difference between thermal & athermal fluctuation Kiyoshi Kanazawa (YITP) Takahiro Sagawa (Tokyo Univ.) Hisao Hayakawa (YITP)

2. Introduction:Thermodynamics for small systems

3. To understand small systems Thermal noise • small systems (μm～nm)(Ex. Colloidal particles)→manipulation of small systems (Ex. Optical tweezers) • Role of fluctuations→Feynman Ratchet (heat engine/ heat conduction) • Framework for small system(Theoretical limit of manipulation) laser

4. Analogy between Macro & Micro Macroscopic thermodynamics Microscopic thermodynamics • macro bath & macro system • Work& Heat (The 1st law) • Irreversibility(The 2nd law) • Efficiency(Carnotefficiency) • macro bath& micro system(Ex) water& colloidal particle • Manipulation (Optical tweezers) • The 1st & 2nd laws? • Efficiency of small systems Heat Bath (Macroscopic) Heat Bath (Macroscopic) Microscopic Macroscopic System cmorder order

5. Brownian particle trapped by optical tweezers(“Single particle gas”) Langevin Eq. with a potential( width of the potential) white Gaussian noise Can we define thermodynamic quantities (work) & (heat)? (for this manipulation process) laser Colloidal particle water small⇔ compression big ⇔ expansion

6. The 1st law for small systems(Stochastic energetics) • (work) = macro degrees of freedom • (heat) = micro degrees of freedom Environmental effect (micro degrees of freedom) K. Sekimoto, Prog. Theor. Phys. Supp. 130, 17 (1998).K. Sekimoto, Stochastic Energetics (Springer). K. Kanazawa et. al., Phys. Rev. Lett.108, 210601 (2012).

7. The 2nd law for small systems • Average of workobeys the 2nd law! • Maximum efficiency is achievedfor quasi-staticprocesses! laser Equality holds for quasi-static processes Colloidal particle water

8. Heat conduction for small systems • Small systems ()Ex.) Biological motors Nanotubes • Characterized byNon-equilibrium equalities Non-equilibrium equalities Fourier law Average Heat Fluctuation relation Fluctuation

9. Electrical circuits (LRC) Experimental realization (S. Cilibertoet al. (2013)) Thermal Thermal Brownian motor (spring) • Vanes (Angles ) driven by noises () & viscosity • Spring synchronizes the angles Heat Gaussian noise Detail

10. Summary of introduction • The 1st law→ • The 2nd law→ Thermodynamics for small systems • Fourier law • Fluctuation relation Non-equilibrium equalities

11. Main: Heat conduction induced by non-Gaussian fluctuations

12. Thermal & athermal fluctuations Thermal noise (Gauss noise) Fluctuation • Thermal fluctuation→from eq. environmentEx.)Nyquistnoise Brownian noise Athermal noise (Non-Gaussian) • Athermal fluctuation→from noneq. environmentEx.)Electrical shotnoiseBiological fluctuation Granular noise

13. Poisson noise(shot noise) • Weak electrical current→particle property→”come” or “do not come”(spike noise） • A typical of non-Gaussian noise • Noises happen times per unit time.Intensity (fight distanse) = E

14. Athermal env. & non-Gaussian noise Fluctuations in athermal env. ⇒ non-Gaussian noise (i) Shot & burst noise in electrical circuits Ex. (ii) Membrane of Red Blood Cell with ATP receptions Apply shot noise (zero-mean) Non-Gaussian Gaussian Thermal Env. Athermal Env. Abstraction water ATP

15. What corrections appear in the Fourier law? Between thermal systems Between athermal systems • Fourier law (FL) • Fluctuation theorem (FT) • Extension of FL & FT? • Correction terms? + Correction? Correction? Thermal() Thermal() Athermal Athermal conductingwire conductingwire

16. Non-equilibrium Brownian motor conducting wire Non-Gaussian noise（） Non-Gaussiannoise（） Non-Gaussian athermal fluctuations A conducting wire synchronizing the angles Heat() Athermal Athermal Langevin Eq. Non-Gaussian K. Kanazawa et. al., PRL, 108, 210601 (2012) Heat Characterization of non-Gaussianity

17. (i) Generalized Fourier Law • Perturbation in terms of • Not only but also contribute to . coupling

18. Harmonic potential Quartic potential Corresponding correction The ordinary Fourier law Correspondence coupling

19. Numerical check of GFL(Setup) Gaussian vs. Two-sided Poisson Gaussian noise (thermal) Two-sided Poisson noise (athermal) • Variance = 2T • High order cumulants = 0 • Flight distance = • Transition rate =

20. Numerical check of GFL(Results) • Quartic potential • Changing the parameter • The direction of heat current depends on We can change the direction of heat current by choosing an appropriate conducting wire

21. (ii)Absence of the 0th law B • Does the 0th law exists? （Equilibrium between A and B, B and C → A and C） • The direction of heat depends on the device.←Violation of the 0th law • But, the 0th law recovers if we fix the device.(+we can define effective temperature.) eq. eq. Ａ Ｃ eq. Absent for athermal systems

22. (iii) Generalized Fluctuation theorem • Perturbation in terms of • Harmonic Potential → Ordinary Fourier law • But, the fluctuation theorem is modified.

23. In a case of the Gaussian & two-sided Poisson noise We can further sum up the expansion! A special case of the Gaussian and two-sided Poisson noise

24. Numerical check of the linear part ofthe generalized fluctuation relation • The Gaussian vs. two-sided Poisson case • Consistent with our generalized FTnot with the conventional FT Conventional Modified

25. Conclusion • Generalized Fourier law • Generalized fluctuation theorem • Violation of the 0st lawThe direction of heat depends on the contact device(If we fix the contact device, the 0th law recovers) Non-Gaussian Brownian motor K. Kanazawa, T. Sagawa, and H. Hayakawa, Phys. Rev. E 87, 052124 (2013)