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PHY Statistical Mechanics 12:00* - 1:45 PM TR Olin 107

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Presentation on theme: "PHY Statistical Mechanics 12:00* - 1:45 PM TR Olin 107"— Presentation transcript:

1 PHY 770 -- Statistical Mechanics 12:00* - 1:45 PM TR Olin 107
Instructor: Natalie Holzwarth (Olin 300) Course Webpage: 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 3/25/2014 PHY 770 Spring Lecture 16

2 3/25/2014 PHY 770 Spring Lecture 16

3 3/25/2014 PHY 770 Spring Lecture 16

4 3/25/2014 PHY 770 Spring Lecture 16

5 http://famousbiologists.org/robert-brown/ 3/25/2014
PHY 770 Spring 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. 3/25/2014 PHY 770 Spring Lecture 16

7 Description based on the Langevin equation of motion
Brownian motion: Description based on the Langevin equation of motion friction coefficient 3/25/2014 PHY 770 Spring Lecture 16

8 Brownian motion and Langevin equation of motion -- continued
3/25/2014 PHY 770 Spring 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) 3/25/2014 PHY 770 Spring 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) 3/25/2014 PHY 770 Spring Lecture 16

11 Brownian motion and Langevin equation of motion – continued
It is interesting to take the Fourier transform of the correlation function 3/25/2014 PHY 770 Spring 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: 3/25/2014 PHY 770 Spring Lecture 16

13 Example of Brownian system -- continued
3/25/2014 PHY 770 Spring Lecture 16

14 Example of Brownian system -- continued
3/25/2014 PHY 770 Spring Lecture 16

15 Example of Brownian system -- continued
3/25/2014 PHY 770 Spring Lecture 16

16 Example of Brownian system -- continued
3/25/2014 PHY 770 Spring Lecture 16

17 Probability analysis of Brownian motion Fokker-Planck equation
3/25/2014 PHY 770 Spring Lecture 16

18 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation x v S 3/25/2014 PHY 770 Spring Lecture 16

19 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

20 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

21 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

22 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

23 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

24 Probability analysis of Brownian motion Fokker-Plank equation
Justification of Fokker-Plank equation 3/25/2014 PHY 770 Spring Lecture 16

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