# Multi-success detrapping process

When presenting heterogeneous detrapping process at the DPG conference I got a question (or a suggestion to check) what would happen if instead of heterogeneous traps we would require multiple successes for the charge carrier to escape?

## Erlang distributed detrapping times

If we require the charge carrier to make \( N_{success} \) successful "escapes" before actually escaping the trap, and each individual escape times are sampled from exponential distribution (with rate given by \( \lambda_\tau \)), then the overall detrapping time will follow Erlang distribution with \( k = N_{success} \) and \( \lambda = \lambda_\tau \).

Probability density function of the Erlang distribution is given by

\begin{equation} p(\tau) = \frac{\lambda^{k}}{(k-1)!} \tau^{k-1} \exp(-\lambda \tau). \end{equation}

Observe that the asymptotic behavior of this probability density function is exponential. I.e., for extremely large \( \tau \), we have that \( p(\tau) \sim \exp(-\lambda \tau) \). Which leads us to predict that nothing interesting will happen - the power spectral density should remain almost the same as predicted in the earlier post.

When running the simulations on the interactive app below we see that larger \( N_{success} \) decreases the number of pulses in the observed signal. This leads to slightly smaller signal power spectral density, but does not induce other effects.

## Interactive app

This app is identical to the one presented in an earlier post, but now you are encouraged to verify insights of this posts by changing \( N_{success} \) parameter values.