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Extra info for Applied stochastic processes in science and engineering
In an article immediately following, Nyquist provided a theoretical framework for Johnson’s observations. His main tools were, like Langevin, the equipartition of energy theorem from equilibrium statistical mechanics and the ﬂuctuation-dissipation theorem. The consequence of Nyquist’s analysis Random processes 45 was that the linear dependence of the ﬂuctuation power on the ambient temperature yields Boltzmann’s constant (which is proportional to the slope of the line). 5). , the noise is white).
Curves (and contours) of this type are obtained by experiment, that is, they are the result of observations. They represent realizations of the random functions; they are also called sample functions or sample paths. Their general appearance shows that a deterministic description of the phenomenon would be so complicated that the resulting mathematical model (if possible at all) would be practically impossible to solve. 1 Random processes - Basic concepts Suppose we are given an experiment speciﬁed by its outcomes ???? ∈ Ω (Ω = the set of all possible outcomes), and by the probability of occurrence of certain subsets of Ω.
It follows that the autocorrelation ???? depends upon time diﬀerence only, ⟨????(????)????(????)⟩ = ????(???? − ????) = ????(???? ). It turns out that there exists a whole class of phenomena where the underlying stochastic process is completely characterized by the mean ⟨????(????)⟩ = ???? = constant, and by the correlation function ????(???? ). Such stochastic processes are said to be stationary in the wide sense (or in Khinchin’s sense). It should be clear that, in general, Eq. 13 above may not hold for stochastic processes which are only stationary in the wide sense.