By Don S Lemons; Paul Langevin

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Additional resources for An introduction to stochastic processes in physics : containing "On the theory of Brownian motion" by Paul Langevin, translated by Anthony Gythiel

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Autocorrelated Process. Let X (t) and X (t ) be the instantaneous random position of a Brownian particle at times for which t ≤ t. a. Find cov{X (t), X (t )}. b. Find cor{X (t), X (t )}. c. Evaluate cor{X (t), X (t )} in the limits t /t → 0 and t /t → 1. 7]. 4. Local particle density N0 p(x, t) versus time at x = x1 > 0, given that all the particles are initialized at x = 0. Here δ 2 = 1, x1 = 10, and N0 = 100. 2. Concentration Pulse. Suppose that N0 particles of dye are released at time t = 0 in the center (at x = 0) of a fluid contained within an essentially one-dimensional pipe, and the dye is allowed to diffuse in both directions along the pipe.

B. Determine mean{N }, var{N }, and the coefficient of variation var{N }/ mean{N } in terms of No , Vo , and V . 1 Brownian Motion Described We are ready to use our knowledge of how random variables add and multiply to model the simplest of all physical processes—a single particle at rest. If at one instant a particle occupies a definite position and has zero velocity, it will, according to Newton’s first law of motion, continue to occupy the same position as long as no forces act on it. Consider, though, whether this deterministic (and boring) picture can ever be a precise description of any real object.

Alternatively, the √ realization x(t + dt) is the sum of the sure variable x(t) and the product of δ 2 dt and a realization of √ the unit normal N (0, 1). 3) seems odd. Are dt and dt allowed in the same differential equation? If one’s standard is the √ ordinary calculus of sure processes, certainly not. Terms proportional to dt are in- 44 EINSTEIN’S BROWNIAN MOTION √ definitely larger than terms proportional to dt as dt → 0. However, here dt is multiplied by the unit normal Ntt+dt (0, 1), which in different subintervals assumes different positive and negative√ values.

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