A module calculating the stochastic deviations in timepoints for atomic clocks.

This module is an application of the theory presented in Galleani et al. (2003), doi:10.1088/0026-1394/40/3/305.

The module's current functionality calculates stochastic deviations using the exact iterative solution to the stochastic differential equation in Galleani_exact()

$$\begin{equation*} \mathbf{X}(t_{n+1}) = \begin{pmatrix} 1 & \delta t \\\ 0 & 1 \end{pmatrix} \mathbf{X}(t_n) + \begin{pmatrix} \delta t \mu_1 + \frac{1}{2} \delta t^2 \mu_2 \\\ \delta t \mu_2 \end{pmatrix} + \mathbf{\Sigma}(t_n) \end{equation*}$$

$$\begin{equation*} \mathbf{\Sigma}(t_n) \sim \mathcal{N} \bigg( \mathbf{0}, \begin{bmatrix} \sigma_1^2 \delta t + \frac{1}{3} \sigma_2^2 \delta t^3 & \frac{1}{2}\sigma_2^2 \delta t^2 \\\ \frac{1}{2}\sigma_2^2 \delta t^2 & \sigma_2^2 \delta t \end{bmatrix} \bigg) \end{equation*}$$

Stochastic deviations can be visualised using clock_error(), and their distributions simulated with deviation_distribution().

Please consult the Jupyter notebook for a walkthrough of the package.