## 6. Probabilistic descriptionThe 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 by . 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 (), we now consider pulses with the following time dependence: ## 6.1. Energy after pulsesIn particular we are interested in the single contributions of the eigenfunctions to the total kinetic energy which is represented by Eq. (11) by: The time dependence appears in the second factor
. After some algebra we obtain the following
result for this time derivative of where and We can rewrite both summations in expression (15) as follows: where Substituting this result in Eqs. (14) and (15), we obtain the
following expression for Hence the final energy oscillates like the square of a sine, with a
specific amplitude dependent on the time intervals between the pulses,
## 6.2. Probabilistic approachTo work out our solution in an analytical way, we assume the
intervals between the pulses or, more specifically, in the mean value of this amplitude. We solve this problem in four steps: As a second step, we calculate the mean value of the first term of
Using the definition of mean value and the gamma distribution
function, the terms in the first summation can be calculated as The calculation of the double summation in Eq. (17) is less
obvious because We tackle this problem by rewriting the two successive summations
over is symmetric in If and if Following the derivation presented in the second step, we rewrite
the 'sine part' of where and if if Only the last but one term still has to be worked out. Therefore we
need the probability distribution function of where If we multiply this expression by the factor we obtain, according to Eq. (16), the mean value for the final
amplitude of as function of
.
We remark that although this figure shows a continuous graph, the results are only relevant for the discrete eigenfrequencies . Nevertheless the graph shows remarkable peaks which we try to explain in the next section. © European Southern Observatory (ESO) 1998 Online publication: June 12, 1998 |