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PHY 770 -- Statistical Mechanics 12:00 * - 1:45 P M TR Olin 107

PHY 770 -- Statistical Mechanics 12:00 * - 1:45 P M TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: http://www.wfu.edu/~natalie/s14phy770. Lecture 16 Chap. 7 – Brownian motion and other non-equilibrium phenomena Overview Langevin equation

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PHY 770 -- Statistical Mechanics 12:00 * - 1:45 P M TR Olin 107

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  1. PHY 770 -- Statistical Mechanics 12:00*- 1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: http://www.wfu.edu/~natalie/s14phy770 • Lecture 16 • Chap. 7 – Brownian motion and other non-equilibrium phenomena • Overview • Langevin equation • Correlation function and spectral density • Fokker-Planck equation *Partial make-up lecture -- early start time PHY 770 Spring 2014 -- Lecture 16

  2. PHY 770 Spring 2014 -- Lecture 16

  3. PHY 770 Spring 2014 -- Lecture 16

  4. PHY 770 Spring 2014 -- Lecture 16

  5. http://famousbiologists.org/robert-brown/ PHY 770 Spring 2014 -- Lecture 16

  6. Brownian motion: • Phonomenon: Under a microscope a large particle (~1 m in diameter) immersed in a fluid with the same density as the particle, appears to be in a state of agitation, undergoing rapid and random motions. http://upload.wikimedia.org/wikipedia/commons/c/c2/Brownian_motion_large.gif PHY 770 Spring 2014 -- Lecture 16

  7. Brownian motion: • Description based on the Langevin equation of motion friction coefficient PHY 770 Spring 2014 -- Lecture 16

  8. Brownian motion and Langevin equation of motion -- continued PHY 770 Spring 2014 -- Lecture 16

  9. Brownian motion and Langevin equation of motion – continued Note that since x(t) is a stochastic variable, so is v(t) and x(t) PHY 770 Spring 2014 -- Lecture 16

  10. Brownian motion and Langevin equation of motion – continued Note that since x(t) is a stochastic variable, so is v(t) and x(t) PHY 770 Spring 2014 -- Lecture 16

  11. Brownian motion and Langevin equation of motion – continued • It is interesting to take the Fourier transform of the correlation function PHY 770 Spring 2014 -- Lecture 16

  12. Example of Brownian system • Consider a particle of mass m attached to a harmonic spring with spring constant mw02 constrained to move in one dimension: PHY 770 Spring 2014 -- Lecture 16

  13. Example of Brownian system -- continued PHY 770 Spring 2014 -- Lecture 16

  14. Example of Brownian system -- continued PHY 770 Spring 2014 -- Lecture 16

  15. Example of Brownian system -- continued PHY 770 Spring 2014 -- Lecture 16

  16. Example of Brownian system -- continued w PHY 770 Spring 2014 -- Lecture 16

  17. Probability analysis of Brownian motion Fokker-Planck equation PHY 770 Spring 2014 -- Lecture 16

  18. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation v S x PHY 770 Spring 2014 -- Lecture 16

  19. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

  20. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

  21. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

  22. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

  23. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

  24. Probability analysis of Brownian motion Fokker-Plank equation • Justification of Fokker-Plank equation PHY 770 Spring 2014 -- Lecture 16

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