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Astron. Astrophys. 335, 329-340 (1998)

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6. Probabilistic description

The aim of this section is to find a relationship between the mean value of the amplitude of the kinetic energy after a stochastic driving with identical pulses and the eigenfrequencies of the loop. To avoid complicated notations we denote the eigenfrequency [FORMULA] by [FORMULA].

We assume the time intervals between the identical pulses to satisfy a well-defined probability distribution. In order to show the effect of the stochastic driving most clearly, we modify the pulse. Instead of the model pulse of the previous section ([FORMULA]), we now consider pulses with the following time dependence:

[EQUATION]

6.1. Energy after [FORMULA] pulses

In particular we are interested in the single contributions of the eigenfunctions [FORMULA] to the total kinetic energy which is represented by Eq. (11) by:

[EQUATION]

The time dependence appears in the second factor [FORMULA]. After some algebra we obtain the following result for this time derivative of T after driving with [FORMULA] pulses:

[EQUATION]

where [FORMULA] and R.
R is the random time interval before the ith pulse.

We can rewrite both summations in expression (15) as follows:

[EQUATION]

where

[EQUATION]

Substituting this result in Eqs. (14) and (15), we obtain the following expression for E:

[EQUATION]

Hence the final energy oscillates like the square of a sine, with a specific amplitude dependent on the time intervals between the pulses, R. As this R dependence is included in the factor A, in the following subsection we study the influence of the frequency [FORMULA] on A.

6.2. Probabilistic approach

To work out our solution in an analytical way, we assume the intervals between the pulses R to satisfy a gamma distribution [FORMULA]. We are interested in the probability distribution of :

[EQUATION]

or, more specifically, in the mean value of this amplitude.

We solve this problem in four steps:
At first we need the probability distribution function of S. Since R for j going from 2 to k and since all R are independent, it is easy to check that S (Wonnacott & Wonnacott 1990).

As a second step, we calculate the mean value of the first term of A. It is convenient to rewrite this 'cosine part' of A as

[EQUATION]

Using the definition of mean value and the gamma distribution function, the terms in the first summation can be calculated as
[FORMULA]

[EQUATION]

The calculation of the double summation in Eq. (17) is less obvious because X and Y are two dependent stochastic variables and the mean of a product of two variables only equals the product of the two means if the two variables are independent.

We tackle this problem by rewriting the two successive summations over k and l as a sum of 3 summations where respectively k, k and k. Since the function

[EQUATION]

is symmetric in k and l, the first two terms in the sum of summations are equal and so we only have to consider the cases k and k.

If k we can calculate:

[FORMULA]

[EQUATION]

and if k we obtain:

[EQUATION]

Following the derivation presented in the second step, we rewrite the 'sine part' of A. The counterpart of Eq. (17) yields

[EQUATION]

where

[FORMULA]

[EQUATION]

and

[FORMULA]

[EQUATION]

if k and

[EQUATION]

if k.
Finally we put the two previous solutions together to find the mean value of A :

[EQUATION]

Only the last but one term still has to be worked out. Therefore we need the probability distribution function of X. Since in this term k is larger than l we know that X, such that X. So the double summation in Eq. (18) equals

[EQUATION]

where c.
Substituting this result in Eq. (18) we obtain the final result:

[EQUATION]

If we multiply this expression by the factor

[EQUATION]

we obtain, according to Eq. (16), the mean value for the final amplitude of [FORMULA] as function of [FORMULA].
In Fig. 9 we have plotted the mean value of this final amplitude as function of [FORMULA] for a, [FORMULA], [FORMULA] and [FORMULA]. A gamma distribution with the given parameters [FORMULA] and [FORMULA] is a strongly peaked distribution which is quasi symmetric around the value 0.5.

[FIGURE] Fig. 9. The mean value of the final amplitude of [FORMULA] after driving by a succession of five identical pulses with time intervals in between. These intervals satisfy a gamma distribution with parameters [FORMULA] and [FORMULA].

We remark that although this figure shows a continuous graph, the results are only relevant for the discrete eigenfrequencies [FORMULA]. Nevertheless the graph shows remarkable peaks which we try to explain in the next section.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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