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
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).