Probability

An introduction to stochastic processes in physics, by Don S. Lemons

By Don S. Lemons

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

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Content point » decrease undergraduate

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Argomenti correlati Scienze sociali e diritto - Statistica computazionale - Teoria e metodi statistici

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

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5) PROBLEMS 29 and that of a normal variable N (m, a 2 ) is ∞ 1 M N (t) = √ 2πa 2 −∞ d x exp t x − (x − m)2 . 6) By completing the square in the argument of the exponential, the latter reduces to 2 2 exp mt + t 2a M N (t) = √ 2πa 2 ∞ d x exp −∞ −(x − m − ta 2 )2 . 2, Moments of a Normal). Since only random variables with finite moments have a moment-generating function, the Cauchy variable C(m, a) does not have one except in the special case when a = 0, in which case it collapses to the sure variable m.

3, to the mathematically equivalent result X (t) − X (0) = N0t (0, 2Dt) has been via the algebra of random variables. We use the phrase Einstein’s Brownian motion to denote both these configuration-space descriptions (involving only position x or X ) of Brownian motion. In chapters 7 and 8, we will explore their relationship to Newton’s Second Law and possible velocity-space descriptions (involving velocity v or V as well as position). 1. Autocorrelated Process. Let X (t) and X (t ) be the instantaneous random position of a Brownian particle at times for which t ≤ t.

Imagine the Brownian particle starts at the origin x = 0 and is free to move in either direction along the x-axis. The net effect of many individual molecular impacts is to displace the particle a random amount X i in each interval of duration t. Assume each displacement X i realizes one of two possibilities, X i = + x or X i = − x, with equal probabilities ( 12 ) and that the various X i are statistically independent. After n such intervals the net displacement X is X = X1 + X2 + · · · + Xn. 1) This is the random step or random walk model of Brownian motion.

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