# Notations

-   $B_0$: background magnetic field

-   $α$: pitch angle, $w = \cos(α)$ is the pitch angle cosine

-   $ψ$: gryo phase

-   θ : azimuthal angle, angle between $B_0$ and $B_N = B_0 \cos(θ)$

-   φ : polar angle

-   β : half of in plnae rotation angle $ω_{in}$

# Results

![](figures/example_tp.png)

Scan the parameter with angle difference by 10 degree and velocity by ...

![Transition matrix for different particle energies](figures/tm_stats_vPs.png)

## Examples of transition matrix

![](figures/tm/example.png)

# Updates

## 2024-10-30

![](figures/mixing_rate_sin2.png)

![](figures/mr/mixing_rate_sin2_ani.png)

## 2024-10-29

![](figures/pa_cdf.png)


![](figures/tm/tm_weighted.png)

![](figures/tm/%CE%94w_%CE%94t.png)

# Questions

- [ ] Order of magnitude of estimation of the scattering rate.

- The validality of small $κ$ in observations and its implications in quasi-adibatic approximation.
- The validality of approximate Hamiltonian form for off-center scattering.


- [ ] “How long does it take for an irreducible finite state Markov chain to converge to equilibrium?”
    - References: [The Spectral Gap](https://www.stat.berkeley.edu/~pitman/s205s03/lecture28.pdf), [Spectral gap of nonreversible Markov chains](https://arxiv.org/abs/2310.10876)

# TODOs

- [ ] Cite Juno observations
- [ ] Check the algorithm for energy conservation (Boris pusher)
- [x] GPU acceleration
- [ ] The independence of the parameters from discontinuities observations


DynamicalSystems.jl
- Currently, the limits can not be changed with time.


- Saving as a DataFrame is more space-efficient than saving as a vector of dictionaries