By Don S Lemons; Paul Langevin

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Extra 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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We now know that Brownian motion is a consequence of the atomic theory of matter. When a particle is suspended in any fluid media (air as well as water), the atoms or molecules composing the fluid hit the particle from different directions in unequal numbers during any given interval. While the human eye cannot distinguish the effect of individual molecular impacts, it can observe the net motion caused by many impacts over a period of time. 2 Brownian Motion Modeled Let’s model Brownian motion as a sum of independent random displacements.

Two variables are jointly normal when they are each linear combinations of a single set of independent normals. 1) X 2 = bN1 (0, 1) + cN2 (0, 1). 2) and Here a, b, and c are constants and N1 (0, 1) and N2 (0, 1) are, by specification, statistically independent unit normals. Here, as before, the different subscripts attached to N (0, 1) denote statistical independence; identical subscripts would denote complete correlation. Thus, the variables X 1 and X 2 are, by definition, jointly normal. The property of joint normality covers a number of possible relationships.

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. The diffusion constant D = δ 2 /2. At position X (t) and time t the density of dye particles is the product N0 p(x, t), where p(x, t) is the probability density of a single dye particle with initialization X (0) = 0.

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