Glossary

Discretizating continuous information

All quantities defined over a space, velocity, and time are understood to be described by their discretized equivalents (to the order of the truncation).

\[x = x_i, v = v_\alpha, t = t_n,\]

where \(i, \alpha, n\) represent an integer index of arrays corresponding to space, velocity, and time, respectively.

Fourier Transforming in Space and Velocity Space

VlaPy relies on representing phase space in it’s Fourier domain equivalent.

Given that \(f=f^n(x,v)\) is the discretized distribution function, VlaPy uses the following definitions throughout this documentation

\[\begin{split}\mathcal{F}_x(k_x, v) = \sum_{k_x=0}^{k_x=N_x/2} \exp{(-i k_x x) f(x,v)} \\ \mathcal{F}_v(x, k_v) = \sum_{k_v=0}^{k_v=N_v/2} \exp{(-i k_v v) f(x,v)}\end{split}\]

where \(\mathcal{F}_x, \mathcal{F}_v\) are the discrete-Fourier-transform equivalents in configuration-space, and velocity-space, respectively. These may also be performed simultaneously such that

\[\mathcal{F}_{x,v}(k_x, k_v) = \sum_{k_v=0}^{N_v/2} \exp{(-i k_v v)} \sum_{k_x=0}^{N_x/2} \exp{(-i k_x x)} f(x,v).\]