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Extra resources for Applied Stochastic Processes in Science and Engineering
23 is −βt y(t) = y0 e t ′ e−β(t−t ) η(t′ )dt′ . 27) since the average of η(t) vanishes by Eq. 24. 28) where y 2 eq is given by the variance of the equilibrium distribution, as calculated by equilibrium statistical mechanics (equipartition of energy). The spectral density is simply, Syy (ω) = 1 2 y π eq β . 29) Notice that up to this point, Eq. 25 has not been used. From the fundamental relation above (Eq. 19), (β 2 + ω 2 )Syy (ω) = Sηη (ω). 30) Now using Eq. 25 and Eq. 31) relating magnitude of the fluctuations (Γ) to the magnitude of the phenomenological dissipation (β).
1: Examples of Random Functions A) Height of a protein molecule above a catalytic surface. B) Fluctuations in the magnetic field around the North pole. C) Example of a random field – The cosmic background radiation. homogeneities in the refractive index of the atmosphere (fading) and the influence of random electrical discharges (meteorological and industrial noise). Electrical noise, albeit very important, is far from a unique case. As other examples, we may cite the pressure, temperature and velocity vector of a fluid particle in a turbulent flow – in particular, at a point in the Earth’s atmosphere.
Xn ; t1 , . . , tn ) gives the joint probability distribution of ξt1 (ω), . . , ξtn (ω). 7) f (x1 , t1 ) f (x1 , . . , xn ; t1 , . . , tn ) . f (xk+1 , . . , xn ; tk+1 , . . , tn |x1 , . . , xk ; t1 , . . , tk ) = f (x1 , . . , xk ; t1 , . . , tk ) f (x2 , t2 |x1 , t1 ) = In particular, we have, f (x1 , . . , xn ; t1 , . . , tn ) = f (xn , tn |xn−1 , . . , x1 ; tn−1 , . . , t1 )× . . × f (x2 , t2 |x1 , t1 ) × f (x1 , t1 ). Intuitively, the relationship between the conditional and joint probabilities is very straightforward.