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**

**Sample text**

The Russian mathematician A. A. Markov (1856–1922) even used memoryless processes to model the occurrence of short words in the prose of the great Russian poet Pushkin. 3) returns a unique value of q(t + dt) for each q(t). Many of the familiar processes of classical physics belong to the class of timedomain and process-variable continuous, smooth, and Markov sure processes. In the next section we investigate a particular random process that is continuous (in both senses) and Markov but neither smooth nor sure.

PROBLEMS 49 we find that p(x, t) solves the classical diffusion equation δ 2 ∂ 2 p(x, t) ∂ p(x, t) . 1). The latter equation governs the random variable X (t), while the former governs its probability density p(x, t). 1) reverses the usual order in modeling and problem solving. 6) J = −D . ∂x where the proportionality constant D is called the diffusion constant. Fick’s law, like F = ma and V = IR, both defines a quantity (diffusion constant, mass, or resistance) and states a relation between variables.

Suppose, as in chapter 3, the Brownian particle moves in one dimension along the x-axis. 1) in the interval (t, t + dt). These displacements are indifferently positive and negative, with size regulated by the parameter δ 2 . How does the net displacement of the Brownian particle evolve with time? 1) and answer this question. 2) 46 EINSTEIN’S BROWNIAN MOTION and when t = dt, X (2dt) = X (dt) + √ δ 2 dtNdt2dt (0, 1). 3) Dropping the former into the right-hand side of the latter produces X (2dt) = X (0) + √ δ 2 dtN0dt (0, 1) + √ δ 2 dtNdt2dt (0, 1).