Baumjohann, Basic spase plasma physics

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25) since the integrand does not depend on t2' Similarly, integration for the part tl < t2 yields T J(T o t) ¢(- t) e iwnl dt . 26) Fig. 4. 24) and T is made infinity. >(t)e-iWldt+J¢(-t)eiwldt 00 ]. 22). 29) -00 1 a(w) = - J z (t) e- 1wt dt 2n -00 00 . 30) as Fourier integrals. If z (t) is real, a(-w)=a*(w) is the complex conjugate of holds that a(w). 20). 31) is calculated as 1

46) again guarantees the equipartition law of energy. 22 1. 5) for all possible sets of tl, t2, ... , tn (n = 1,2, ... ). Probabilities of lower hierarchy are derived from those of higher hierarchy, but the latter generally contain new information not contained in the former. , tl I X2, t2)' This kind of stochastic process is considered below. Here we take up another class of simple processes, namely the Gaussian processes. This is an extension of the normal distribution discussed in Sect. 2 to stochastic processes.

8) becomes cf>(t) = exp {- A2rc [t - rc (1- e- tlrc )]} = exp [ - 0: 2 (L - + 1 e-I/r,)] . 1 Random Frequency Modulation 43 This function is shown in Fig. 1 and its Fourier transform I 00 I(w) = - J (t) 44 2. Physical Processes as Stochastic Processes When parameter (X decreases, the spectrum becomes narrower and narrower. 5]. 17) which means that it is Brownian motion on a unit circle in the complex plane.

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