The mathematics of Brownian motion and Johnson noise

American Journal of Physics

Volumn 64, Issue 3, 1996, Pages 225-240

  Gillespie, Daniel T ^a  

^a NAVAL AIR WARFARE CENTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0007324878     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18210     Document Type: Article

References (29)

1. A. Einstein, "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen," Ann. Phys. 17, 549-560 (1905); "Zur Theorie der Brownschen Bewegung," Ann. Phys. 19, 371-381 (1906).
2. A. Einstein, "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen," Ann. Phys. 17, 549-560 (1905); "Zur Theorie der Brownschen Bewegung," Ann. Phys. 19, 371-381 (1906).
3. M. von Smoluchowski, "Zarys teoryi kinetycznej ruchu Browna i roztworów metnych," Ann. Phys. 21, 756-780 (1906).
4. P. Langevin, "Sur la théorie du mouvement brownien," Comptes Rendues 146, 530-533 (1908).
5. J. B. Johnson, "Thermal agitation of electricity in conductors," Phys. Rev. 32, 97-109 (1928).
6. H. Nyquist, "Thermal agitation of electric charge in conductors," Phys. Rev. 32, 110-113 (1928).
7. Six especially important physics-oriented articles arc available in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). Three of those papers deserve particular mention in the present context: G. E. Uhlenbeck and L. S. Ornstein, "On the theory of Brownian motion,' Phys. Rev. 36, 823-841 (1930); S. Chandrasekhar, "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89 (1943); and Ming Chen Wang and G. E. Uhlenbeck, "On the theory of Brownian motion II," Rev. Mod. Phys. 17, 323-342 (1945).
8. Six especially important physics-oriented articles arc available in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). Three of those papers deserve particular mention in the present context: G. E. Uhlenbeck and L. S. Ornstein, "On the theory of Brownian motion,' Phys. Rev. 36, 823-841 (1930); S. Chandrasekhar, "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89 (1943); and Ming Chen Wang and G. E. Uhlenbeck, "On the theory of Brownian motion II," Rev. Mod. Phys. 17, 323-342 (1945).
9. Six especially important physics-oriented articles arc available in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). Three of those papers deserve particular mention in the present context: G. E. Uhlenbeck and L. S. Ornstein, "On the theory of Brownian motion,' Phys. Rev. 36, 823-841 (1930); S. Chandrasekhar, "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89 (1943); and Ming Chen Wang and G. E. Uhlenbeck, "On the theory of Brownian motion II," Rev. Mod. Phys. 17, 323-342 (1945).
10. Six especially important physics-oriented articles arc available in Selected Papers on Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). Three of those papers deserve particular mention in the present context: G. E. Uhlenbeck and L. S. Ornstein, "On the theory of Brownian motion,' Phys. Rev. 36, 823-841 (1930); S. Chandrasekhar, "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89 (1943); and Ming Chen Wang and G. E. Uhlenbeck, "On the theory of Brownian motion II," Rev. Mod. Phys. 17, 323-342 (1945).
11. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), Chap. 15.
12. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
13. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, New York, 1985).
14. D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists (Academic, San Diego, 1992).
15. R. L. Stratonovich, Nonlinear Nonequilibrium Thermodynamics I: Linear and Nonlinear Fluctuation-Dissipation Theorems (Springer-Verlag, New York, 1992).
16. D. T. Gillespie, "A theorem for physicists in the theory of random variables," Am. J. Phys. 51, 520-533 (1983). See also Ref. 10, Chap. 1.
17. 1/2 version will be seen to be more transparent and easier to work with.
18. n(t)〉 is what is commonly referred to as an "ensemble average."
19. See, e.g., Ref. 10, Chaps. 2-3.
20. 0) are more solid than those of Γ(t), we have chosen to proceed the other way around and define Γ(t) to be the derivative of W(t) [see Eq. (2.16)]. Electrical engineers tend to call the special Wiener process "Brownian motion," and denote it by B(t); however, we shall see in Sec. III that that appellation would be more appropriately applied to the integral of the Ornstein-Uhlenbeck process. (The Ornstein-Uhlenbeck process in turn is often called by electrical engineers "the Markov process," giving yet another example of the confusing intradisciplinary mismatch in terminology.)
21. 0 cos(2πvx)dv.
22. 2 in Eqs. (2.48a).
23. It is possible to construct a simulation algorithm for O-U processes that is exact for both Y and X, but to do that we need a little more random variable theory than is given in Sec. II B.
24. A nice account of Einstein's work in Ref. 1 is given in Ref. 9, pp. 2-6.
25. D. T. Gillespie, "Fluctuation and dissipation in Brownian motion," Am. J. Phys. 61, 1077-1083 (1993). A more sophisticated argument is given in Ref. 10, Secs. 4.5 and 4.6.
26. Nyquist's original analysis in Ref. 5 proceeds quite differently from the analysis that we give here; see Ref. 7, p. 592.
27. J. R. Reitz and F. J. Milford, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, MA, 1960), p. 176.
28. See, e.g., Ref. 7, Sec. 7.5.
29. A short proof of this "lemma" may be found in Ref. 10, p. 114.