# An introduction to stochastic processes in physics, by Don S. Lemons

By Don S. Lemons

A textbook for physics and engineering scholars that recasts foundational difficulties in classical physics into the language of random variables. It develops the ideas of statistical independence, anticipated values, the algebra of ordinary variables, the primary restrict theorem, and Wiener and Ornstein-Uhlenbeck methods. solutions are supplied for a few difficulties.

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Extra resources for An introduction to stochastic processes in physics, containing On the theory of Brownian notion

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5. Single-slit diffraction. PROBLEMS 31 The random variable so defined is called the exponential random variable E(λ). a. Show that mean{E(λ)} = 1/λ. b. Find the moment-generating function M E (t) of E(λ) for t < λ. c. Use the moment-generating function to find var{E(λ)}. d. Also, find E(λ)n for arbitrary integer n. 4. Poisson Random Variable. The probability that n identical outcomes are realized in a very large set of statistically independent and identically distributed random variables when a each outcome is extremely improbable is described by the Poisson probability distribution Pn = e−µ µn , n!

5) PROBLEMS 29 and that of a normal variable N (m, a 2 ) is ∞ 1 M N (t) = √ 2πa 2 −∞ d x exp t x − (x − m)2 . 6) By completing the square in the argument of the exponential, the latter reduces to 2 2 exp mt + t 2a M N (t) = √ 2πa 2 ∞ d x exp −∞ −(x − m − ta 2 )2 . 2, Moments of a Normal). Since only random variables with finite moments have a moment-generating function, the Cauchy variable C(m, a) does not have one except in the special case when a = 0, in which case it collapses to the sure variable m.

B. The light intensity produced by diffraction through a single, narrow slit, as found in almost any introductory physics text, is proportional to 1 sin2 [(πa/λ) sin θ )] r2 sin2 θ where r is the distance from the center of the slit to an arbitrary place on the screen, a is the slit width, and λ the light wavelength. Show that for slits so narrow that πa/λ 1, the above light intensity is proportional to the photon probability density derived in part a. 2. Moments of a Normal. 8), show that N (0, σ 2 )n = 1 · 3 · 5 .