1. Microscopic derivation of non- Gaussian Langevin equations for athermal systems ADVANCED STATISTICAL DYNAMICS GROUP KIYOSHI KANAZAWA Jan 28, 2015 Lunch seminar@YITP

2. Fluctuation in small systems  Experimental development (e.g., optical tweezers) → Manipulation & Observation of small systems  Single particle “ideal gas”  Thermodynamics for a single particle bead laser

3. Gaussian Langevin Equation (GL)  Motion of a fluctuating bead in water  The GL Eq. is universal and simple  A foundation for thermodynamics White Gaussian noise

4. My interest: Athermal fluctuation Non-Gaussian Langevin Eq. (NGL)  Athermal fluctuation  Originating from non-eq. environments  Characterized by non-Gaussianity  Electrical, biological, granular systems White non-Gaussian noise EX1) Avalanche noise EX2) Active noise Reverse voltage on diodes Chain-reaction Biological motor Fluctuation induced by ATP

5. Goal of this talk: Microscopic Derivation of NGL Eq.  Review of a derivation of GL Eq.  The central limit theorem (CLT) →Emergence of Gaussian noise  Why is the CLT violated for non- equilibrium systems? 1.Microscopic derivation of NGL Eq. 2.Application of a granular example Review on a derivation of GL Eq. Our study on a derivation of NGL Eq. KK, T.G. Sano, T. Sagawa, H. Hayakawa, to appear in PRL.

6. Derivation of GL Eq.: Example （ Rayleigh Piston ）

7. General derivation of GL Eq.: the system size expansion Weak coupling ・・・ Markov jump noise (ε-independent) Emergence of the NGL Eq. ＝ Simplification 1 & 2 are applicable, but simplification 3 is not applicable. ・・・ small parameter

8. Where is the CLT applied in the system size expansion? This condition can be violated for systems with multiple baths

9. An asymptotic derivation of the NGL Eq. The NGL Eq. is derived ε-independent

10. Violation of the CLT  (a) A single bath→Automatically weak friction Sufficiently frequent fluc. during relaxation ＝ the CLT is applicable  (b) Multiple baths→Origins of fluc. and diss. are separeted Not sufficiently frequent fluc. during relaxation ＝ the CLT is not applicable

11. Granular motor: Modeling, ε-independent

12. Granular motor: Results

13. Conclusion  A derivation of the GL. Eq.  A derivation of the NGL. Eq. 1.Weak coupling 2.Coexistence of both fluctuations 3.Strong thermal friction KK, T.G. Sano, T. Sagawa, H. Hayakawa, to appear in PRL (arXiv: 1407.5267).  Application to a granular 1.An exactly solvable model 2.Agreement with simulation Strong dissipation