# Distributions

GEMPIC.CosGaussianParamsType
CosGaussianParams( dims, k, α, σ, μ, δ )

Parameters of a distribution with is a product of a Cosine distribution along x and a Normal distribution along v.

• n_gaussians : Number of Gaussians
• n_cos : Number of cosines
• normal : Normalization constant of each Gaussian
GEMPIC.CosSumGaussianType
CosSumGaussian{D,V}( n_cos, n_gaussians, k, α, σ, μ, δ )

Data type for parameters of initial distribution

$$$(1+ \cos( \sum^{n_{cos}}_{i=1} k_i x)) \cdot \sum_{j=1}^{n_{gaussians}} \delta_j \exp \big( -\frac{1}{2} \frac{(v-\mu_j)^2}{\sigma_j^2} \big)$$$

Parameters

• k : values of the wave numbers (one array for each cosines)
• α : strength of perturbations
• σ : variance of the Gaussian (one velocity vector for each gaussian).
• μ : mean value of the Gaussian (one velocity vector for each gaussian).
• δ : portion of each Gaussian

Example

$$$f(x,v_1,v_2)=\frac{1}{2\pi\sigma_1\sigma_2} \exp \Big( - \frac{1}{2} \big( \frac{v_1^2}{\sigma_1^2} + \frac{v_2^2}{\sigma_2^2} \big) \Big) ( 1 + \alpha \cos(kx)),$$$
df = CosSumGaussian{1,2}([[k]],[α], [[σ₁,σ₂]], [[μ₁,μ₂]])
GEMPIC.SumCosGaussianType
SumCosGaussian( dims, n_cos, n_gaussians, k, α, σ, μ, δ )

Data type for parameters of initial distribution

$$$(1+ \sum_{i=1}^{n_{cos}} \alpha_i \cos( k_i \mathbf{x})) \cdot \sum_{j=1}^{n_{gaussians}} \delta_j \exp \big( -\frac{1}{2} \frac{(\mathbf{v}-\mu_j)^2}{\sigma_j^2} \big)$$$

Parameters

• k : values of the wave numbers (Array of vectors for multiple cosines)
• α : strength of perturbations
• σ : variance of the Gaussian ( Array of vectors for multiple Gaussians)
• μ : mean value of the Gaussian ( Array multiple Gaussians)
• normal : Normalization constant of each Gaussian
• n_gaussians : Number of Gaussians
• n_cos : Number of cosines
• δ : portion of each Gaussian

Example

$$$f(x,v_1,v_2) = \frac{1}{2\pi\sigma_1\sigma_2} \exp \Big( - \frac{1}{2} \big( \frac{v_1^2}{\sigma_1^2} + \frac{v_2^2}{\sigma_2^2} \big) \Big) ( 1 + \alpha_1 \cos(k_1 x) + \alpha_2 \cos(k_2 x) ),$$$
df = SumCosGaussian{1,2}([[k₁],[k₂]], [α₁, α₂], [[σ₁,σ₂]], [[0.0,0.0]])