Paraphrase the following text using more academic and scientific language Use a neutral tone and avoid repetitions of words and phrases：The survival amplitude is used in the derivation of the kinetic

The determination of the kinetic energy of the kicking rotor involves the utilization of the survival amplitude. The average survival amplitude is characterized as the mean of the inner product of the noiseless kick-driven system's eigenstate $|r\rangle$ and the product of $N$ noise operators, denoted by $\hat{F}^{k_N}\hat{F}^{k_{N-1}}...\hat{F}^{k_{2}}\hat{F}^{k_{1}}$, as shown in Equation \ref{asr}. The average is calculated over noise, and $|r\rangle$ conforms to the condition that $\hat{F}$ acting on it yields the exponential of $-i\epsilon_r$. The survival amplitude serves as a measure of the extent of coupling between the initial and evolved states after the Nth kicks.