New PDF release: An introduction to stochastic processes in physics,

By Don S. Lemons

ISBN-10: 0801868661

ISBN-13: 9780801868665

ISBN-10: 080186867X

ISBN-13: 9780801868672

ISBN-10: 0801876389

ISBN-13: 9780801876387

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

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

Example text

3, Multistep Walk. 3 Critique and Prospect In spite of its attractions, the random step process is deficient as a physical model of Brownian motion. 7), seems to depend separately upon the arbitrary magnitudes x and t through the ratio ( x 2 / t). Unless ( x 2 / t) is itself a physically meaningful constant, the properties of the total displacement X will depend on the fineness with which it is analyzed into subincrements. That ( x 2 / t) is, indeed, a characteristic constant—equal to twice the diffusion 20 RANDOM STEPS constant—will, in chapter 6, be shown to follow from the requirement of continuity, but in the present oversimplified account this claim remains unmotivated.

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. The diffusion constant is positive definite, that is, D ≥ 0, because a gradient always drives an oppositely directed flux in an effort to diminish the gradient. 5) with D replacing δ 2 /2. In his famous 1905 paper on Brownian motion, Albert Einstein (1879–1955) constructed the diffusion equation in yet another way—directly from the continuity and Markov properties of Brownian motion.

PROBLEMS 21 b. A steady wind blows the Brownian particle, causing its steps to the right to be larger than those to the left. That is, the two possible outcomes of each step are X 1 = xr and X 2 = − xl where xr > xl > 0. Assume the probability of a step to the right is the same as the probability of a step to the left. Find mean{X }, var{X }, and X 2 after n steps. 4. Autocorrelation. According to the random step model of Brownian motion, the particle position is, after n random steps, given by n X (n) = Xi i=1 where the X i are independent displacements with X i = 0 and X i2 = x 2 for all i.

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An introduction to stochastic processes in physics, containing On the theory of Brownian notion by Don S. Lemons


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