Fluctuation response inequality out of equilibrium

Transcription

1 Fluctuation response inequality out of equilibrium Shin-ichi Sasa (Kyoto), Andreas Dechant (Tohoku) Non-equilibrium statistical physics of complex 2018/07/10 1

2 "The energy of the universe is constant, the entropy of the universe tends toward a maximum." (Clausius, 1865) Fundamental and Universal inequality! (The second law of thermodynamics) Beyond the second law of thermodynamics -- equalities -- (fluctuation theorem 1993-; Jarzynski equality ; and many..) Beyond the second law of thermodynamics -- inequalities -- Thermodynamic uncertain relations, thermodynamic bound, trade-off relations Frenecy bound, large deviation bound.. Barato, Seifert; Chetrite, Touchette; Gingrich, Horowitz; Maes; Shraishi,Saito, Tasaki Pietzonka, Ritort, Seifert; Pigolotti, Neri, Roldan,Julicher; Dechant, Sasa.. ( ) 2

3 A simple example : setting up A Brownian particle under a tilted potential constant external force Periodic potential Gaussian white noise V.Blickle, T. Speck, C. Lutz, U. Seifert, C. Bechinger, PRL,

4 The second law of thermodynamics Equilibrium condition External operation Heat from the bath (Sekimoto, 1997) Entropy Thermodynamic processes (from an equilibrium state to another equilibrium state) 4

5 Fluctuation theorem Total entropy production (integral) fluctuation theorem Local detailed balance The second law of thermodynamics (non-)linear response relations Reciprocity Zubarev measure External operation Jarzynksi equality,

6 Fluctuation response relation (FRR) drift velocity differential mobility Diffusion constant Einstein relation 6

7 FRR out of equilibrium (after 2000) Several versions Harada,Sasa (2005);Speck,Seifert (2006); Prost,Joanny,Parrondo (2009); Baisel,Maes,Wynants (2009).. One idea: effective temperature Cf. Cugliandolo, Kurchan, Peliti, 1997 Thermometer Hayashi,Sasa,

8 On conjectures (2003 private communication) 8

9 On conjectures (2003 private communication) Counter examples!! Sasaki, Amari, 2005 See also Nakamura,Ooguri, 2011 using holography 9

10 On conjectures (2003 private communication) Counter examples!! Sasaki, Amari, 2005 See also Nakamura,Ooguri, 2011 using holography Recent results are helpful? Entropy production ratio Its lower bound Thermodynamic uncertain relations Barato, Seifert (2015) Gingrich, Horowitz (2016) for Markov jump processes Dechant, Sasa (2018) 10 for a wide class of systems

11 In this talk, I show Dechant, Sasa Arixiv:

12 In this talk, I will show Dechant, Sasa Arixiv: (***, ***, 2005) using a special equality associated with coarse-graining 12

13 In this talk, I will show Dechant, Sasa Arixiv: (Hayashi, Sasa, 2005) using a special equality associated with coarse-graining 13

14 Highlight I show a very elegant proof so as to see (i) This inequality is closely related to thermodynamic uncertain relations and recent many inequalities (ii) This type of inequality holds for a wide class of systems such as under-damped systems, higher-dimension systems, Markov chain, flashing ratchet, Buttiker ratchet, Feynman ratchet, time-dependent driving systems,.. 14

15 Outline of my talk I. Introduction 2. Results 3. Proof of the main result 4. Summary 15

16 Main result : fluctuation response inequality Probability density of trajectory for a system a trajectory Slight modification (perturbation) of a in the sense that For any observable, 16

17 Application I: mobility bound (the simplest example) Set Consider a perturbation 17

18 Application I: mobility bound (the simplest example) Set Consider a perturbation 18

19 Set Application 2: thermodynamic uncertain relations Consider a perturbation 19

20 Rule of the game Set Consider a perturbation 20

21 Rule of the game Set Consider a perturbation You can write a paper! 21

22 Outline of my talk I. Introduction 2. Results 3. Proof of the main result 4. Summary 22

23 Main result : fluctuation response inequality Probability density of trajectory for a system a trajectory Slight modification (perturbation) of a in the sense that For any observable, 23

24 Step 1. A bound on the relative entropy 24

25 Who gave this inequality?? 25

26 Kullback! The Annals of Mathematical Statistics, 1954 ; citation 33 Kullback inequality: On information and sufficiency S Kullback, RA Leibler, The annals of mathematical statistics, 1951; citation

27 Step 2. linear response Kullback inequality: Linear response: 27

28 Outline of my talk I. Introduction 2. Results 3. Proof of the main result 4. Summary 28

29 Summary A very useful and universal inequality Arixiv: A universal inequality on mobility and effective temperature The simplest derivation of (finite time) thermodynamic uncertain relation 29