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 options of statistical independence, anticipated values, the algebra of standard variables, the valuable restrict theorem, and Wiener and Ornstein-Uhlenbeck approaches. solutions are supplied for a few difficulties.
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Additional resources for An introduction to stochastic processes in physics, containing On the theory of Brownian notion
In proving these theorems, we will exploit the properties of momentgenerating functions. According to the normal linear transform theorem, a linear function of a normal variable is another normal variable with appropriately modified mean and variance. Thus α + β N (m, a 2 ) = N (α + βm, β 2 a 2 ). 1) we set m = 0 and a 2 = 1, we have α + β N (0, 1) = N (α, β 2 ). 2) Therefore, an arbitrary normal variable N (α, β 2 ) is a linear transform of a so-called unit normal N (0, 1) with a mean of zero and a variance of one.
Smoothness requires process-variable continuity, and processvariable continuity, in turn, requires time-domain continuity. 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.
For the former we have, by definition, Mα+β N (m,a 2 ) (t) = et (α+β N [m,a = etα etβ N (m,a 2 2 ]) ) = etα M N (m,a 2 ) (tβ). 3) yields Mα+β N (m,a 2 ) (t) = et (α+βm)+ β 2 a2 t 2 2 . 6) is, by definition, the moment-generating function of N (α + βm, β 2 a 2 ). 6) is equivalent to Mα+β N (m,a 2 ) (t) = M N (α+βm,β 2 a 2 ) (t). 1) is proved. 1, Uniform Linear Transform. 2 Normal Sum Theorem According to the normal sum theorem, two statistically independent normal variables sum to another normal variable.
An introduction to stochastic processes in physics, containing On the theory of Brownian notion by Don S. Lemons