Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise

1. Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise SCI-B1-0910 Blok 1, 2009 Aud M (NBI Blegdamsvej), kl 10-12 Mondays and Fridays

2. Non-equilibrium Statistical Mechanics: The Physics of Fluctuations and Noise SCI-B1-0910 Blok 1, 2009 Aud M (NBI Blegdamsvej), kl 10-12 Mondays and Fridays John Hertz office: Kc-10 (NBI Blegdamsvej) email:hertz@nbi.dk tel.3532 5236 (office Kbh),+46 8 5537 8808 (office Sth), 2055 1874 (mobil) http://www.nbi.dk/~hertz/noisecourse/coursepage.html

3. Source material “text”: N G van Kampen, Stochastic Processes in Physics and Chemistry (North –Holland) [very clear, good on general formal methods, but little recent stuff. I will not follow it slavishly in the lectures, but I recommend you buy and read it.] These two are good for anomalous diffusion: L Vlahos et al, Normal and Anomalous Diffusion: a Tutorial arXiv.org/abs/0805.0419 R Metzler and J Klafter, The Random Walker’s Guide to Anomalous Diffusion, Physics Reports 339, 1-77 (2000) On first-passage-time problems: S Redner, A Guide to First-Passage Problems (Cambridge U Press) [library reserve] On finance-theoretical applications: J-P Bouchaud and M Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge U Press) [library reserve] (and more to be mentioned as we go along)

4. Lecture 1: A random walk through the course

5. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states.

6. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy:

7. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy: V1(x) x

8. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy: V1(x) x

9. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy:  V1(x) V2(x) x

10. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy:  V1(x) V2(x) x

11. Lecture 1: A random walk through the course The ubiquity of noise (especially in biology): Changing conditions does not change states; rather, it changes the relative probabilities of states. Example: protein conformational change Potential energy: The real story: P1(x)  V1(x) P2(x) V2(x) x

12. From (equilibrium) stat mech: This course: how P(x) changes from P1(x) to P2(x):

13. From (equilibrium) stat mech: This course: how P(x) changes from P1(x) to P2(x): Dynamics of P(x,t) www.nbi.dk/hertz/noisecourse/demos/Pseq.mat www.nbi.dk/hertz/noisecourse/demos/runseq.m

14. From (equilibrium) stat mech: This course: how P(x) changes from P1(x) to P2(x): Dynamics of P(x,t) www.nbi.dk/hertz/noisecourse/demos/Pseq.mat www.nbi.dk/hertz/noisecourse/demos/runseq.m or etc.

15. Random walks www.nbi.dk/~hertz/noisecourse/demos/brown.m

16. Random walks www.nbi.dk/~hertz/noisecourse/demos/brown.m

17. Random walks

18. Random walks independent steps

19. Random walks independent steps

20. Random walks independent steps i.e, rms distance

21. Random walks and diffusion Step length distribution χ(y)

22. Random walks and diffusion Step length distribution χ(y) Change in P from one step:

23. Random walks and diffusion Step length distribution χ(y) Change in P from one step: P P χ

24. Random walks and diffusion Step length distribution χ(y) Change in P from one step: P P χ

25. Random walks and diffusion Step length distribution χ(y) Change in P from one step: P P χ

26. Random walks and diffusion Step length distribution χ(y) Change in P from one step: P Diffusion equation: P χ

27. Random walks and diffusion Step length distribution χ(y) Change in P from one step: P Diffusion equation: diffusion constant P χ

28. Solution of the diffusion equation:

29. Solution of the diffusion equation:

30. Solution of the diffusion equation: Gaussian, spreading with time, variance 2Dt http;//www.nbi.dk/~hertz/noisecourse/gaussspread.m

31. Solution of the diffusion equation: Gaussian, spreading with time, variance 2Dt http;//www.nbi.dk/~hertz/noisecourse/gaussspread.m

32. Distribution obtained by simulating 20000 random walks:

33. Anomalous diffusion Normal diffusion:

34. Anomalous diffusion Normal diffusion: An experimental counterexample:

35. Anomalous diffusion Normal diffusion: An experimental counterexample: Motion of lipid granules in yeast cells Tolic-Nørrelykke et al, Phys Rev Lett 93, 078102 (2004)

36. Anomalous diffusion Normal diffusion: An experimental counterexample: Motion of lipid granules in yeast cells Tolic-Nørrelykke et al, Phys Rev Lett 93, 078102 (2004)

37. Sub- and superdiffusion

38. Sub- and superdiffusion H: Hurst exponent H < 1: subdiffusion H > 1: superdiffusion

39. Sub- and superdiffusion H: Hurst exponent H < 1: subdiffusion H > 1: superdiffusion One way to get superdiffusion: long-time correlations between steps

40. Sub- and superdiffusion H: Hurst exponent H < 1: subdiffusion H > 1: superdiffusion One way to get superdiffusion: long-time correlations between steps One way to get subdiffusion: long-time anti-correlations between steps

41. Levy walks Step length distribution χ(y): _______________

42. Levy walks Step length distribution χ(y): _______________ power law tail in step length distribution:

43. Levy walks Step length distribution χ(y): _______________ power law tail in step length distribution: Example: Cauchy (Lorentz) distribution

44. Levy walks Step length distribution χ(y): _______________ power law tail in step length distribution: Example: Cauchy (Lorentz) distribution http://www.nbi.dk/~hertz/noisecourse/levy.m (a = 5/2)

45. Levy walks Step length distribution χ(y): _______________ power law tail in step length distribution: Example: Cauchy (Lorentz) distribution http://www.nbi.dk/~hertz/noisecourse/levy.m (a = 5/2)

46. Levy walks Step length distribution χ(y): _______________ power law tail in step length distribution: Example: Cauchy (Lorentz) distribution http://www.nbi.dk/~hertz/noisecourse/levy.m (a = 5/2) Note: <x2> = ∞ for all t

47. Brown vs Levy

48. Ising model (an example of a system with many degrees of freedom)

49. Ising model (an example of a system with many degrees of freedom) Binary “spins” Si(t) = ±1

50. Ising model (an example of a system with many degrees of freedom) Binary “spins” Si(t) = ±1 Dynamics: at every time step,