By Don S Lemons; Paul Langevin

ISBN-10: 0801876389

ISBN-13: 9780801876387

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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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However, a continuous process need not be smooth. 3) with t1 replacing t. Alternatively stated, q(t1 ) alone predicts q(t1 + dt); no previous values q(t0 ) where t0 < t1 are needed. Most well-known processes in physics are Markov processes. Magnetic systems and others having longterm memory or hysteresis are exceptions. 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).

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.

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.

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