Gamma(fm::FrictionModel, at::AbstractSystem; filter=(_,_)->true, T=Float64)

Evaluates the friction tensor according to the friction model fm at the atomic configuration at::AbstractSystem. The friction tensor is the sum of the friction tensors of all matrix models in fm.matrixmodels.

Arguments:

• fm – the friction model of which the friction tensor is evaluated.
• at – the atomic configuration at which the basis is evaluated
• filter – (optional, default: (_,_)->true) a filter function of the generic form (i::Int,at::AbstractSystem) -> Bool. Only atoms at[i] for which filter(i,at) returns true are included in the evaluation of the friction tensor.

Output:

A friction tensor in the form of a sparse 3N x 3N matrix, where N is the number of atoms in the atomic configuration at.

Gamma(fm::FrictionModel{MODEL_IDS}, Σ_vec::NamedTuple{MODEL_IDS}) where {MODEL_IDS}

Computes the friction tensor from a pre-computed collection of diffusion coefficient matrices. The friction tensor is the sum of the squares of all diffusion coefficient matrices in the collection.

Arguments:

• fm – the friction model of which the friction tensor is evaluated. The friction tensor is the sum of the friction tensors of all matrix models in fm.matrixmodels.
• Σ – a collection of diffusion coefficient matrices. The friction tensor is the sum of the squares of all matrices in Σ.

Output:

A friction tensor in the form of a sparse 3N x 3N matrix, where N is the number of atoms in the atomic configuration at. The friction tensor is the sum of the symmetric squares $\Sigma\Sigma^T$ of all diffusion coefficient matrices $\Sigma$ in Σ_vec.

Sigma(fm::FrictionModel{MODEL_IDS}, at::AbstractSystem; filter=(_,_)->true, T=Float64) where {MODEL_IDS}

Computes the diffusion coefficient matrices for all matrix models in the friction model at a given configuration.

Arguments:

• fm – the friction model of which the diffusion coefficient matrices are evaluated
• at – the atomic configuration at which the diffusion coefficient matrices are evaluated
• filter – (optional, default: (_,_)->true) a filter function of the generic form (i::Int,at::AbstractSystem) -> Bool. Only atoms at[i] for which filter(i,at) returns true are included in the evaluation of the diffusion coefficient matrices.

Output:

A NamedTuple of diffusion coefficient matrices, where the keys are the IDs of the matrix models in the friction model.

randf(M::MatrixModel, Σ_vec::AbstractVector)

Random force of a single matrix model from its per-replica diffusion matrices Σ_vec = Sigma(M, at) (one sparse / Diagonal matrix per replica). Each replica draws an independent random force via the coupling-scheme-specific randf(M, Σ) method (defined in the model files), and the model's force is their sum. Its covariance equals the model's friction tensor Gamma(M, Σ_vec). Usually called through randf(fm, Σ) rather than directly.

randf(fm::FrictionModel{MODEL_IDS}, Σ::NamedTuple{MODEL_IDS}) where {MODEL_IDS}

Draws a ${\rm Normal}({\bm 0}, {\bm \Gamma})$-distributed Gaussian random force from a precomputed collection of diffusion coefficient matrices Σ = Sigma(fm, at). The sample covariance of the returned force equals the friction tensor ${\bm \Gamma} =$ Gamma(fm, Σ) (the fluctuation–dissipation relation); this holds for every coupling scheme, even where ${\bm \Gamma} \neq {\bm \Sigma}{\bm \Sigma}^{T}$ (e.g. the pair-wise coupling).

This is the cheap, per-step entry point for Langevin/GLE-type dynamics: assemble Σ once with Sigma, then call randf each timestep to obtain the stochastic force.

Arguments:

• fm – the friction model; the total force is the sum over its matrix models.
• Σ – a NamedTuple of per-model diffusion coefficient matrices (each a vector of per-replica matrices), as returned by Sigma(fm, at).

Output:

A Vector{SVector{3,Float64}} of length N (one 3-vector per atom), where N is the number of atoms in the configuration for which Σ was evaluated. Each matrix model draws its own independent noise (summed over replicas), and the model forces are summed.

basis(fm::FrictionModel{MODEL_IDS}, at::AbstractSystem; join_sites=false, filter=(_,_)->true, T=Float64) where {MODEL_IDS}

Evaluates the ACE-basis functions of the friction model fm at the atomic configuration at::AbstractSystem.

Arguments:

• fm – the friction model of which the basis is evaluated
• at – the atomic configuration at which the basis is evaluated
• join_sites – (optional, default: false) if true, the basis evaulations of all matrix models are concatenated into a single array. If false, the basis evaluations are returned as a named tuple of the type NamedTuple{MODEL_IDS}.
• filter – (optional, default: (_,_)->true) a filter function of the generic form (i::Int,at::AbstractSystem) -> Bool. The atom at[i] will be included in the basis iff filter(i,at) returns true.

params(fm::FrictionModel{MODEL_IDS}) where {MODEL_IDS}

Returns the parameters of all matrix models in the FrictionModel object as a NamedTuple.

params(m::FluxFrictionModel; transformed=true)

Return the parameters of the model m.

nparams(fm::FrictionModel{MODEL_IDS}) where {MODEL_IDS}

Returns the total number of scalar parameters of all matrix models in the FrictionModel object.

set_params!(fm::FrictionModel, θ::NamedTuple)

Sets the parameters of all matrix models in the FrictionModel object whose ID is contained in θ::NamedTuple to the values specified therein.

set_params!(m::FluxFrictionModel; sigma=1E-8, model_ids::Array{Symbol}=Symbol[])

Randomizes the parameters of the model m.

Arguments

• m::FluxFrictionModel: The model to set the parameters of.
• sigma::Float64=1E-8: The standard deviation of the random values.
• model_ids::Array{Symbol}=[]: The ids of the models to set the parameters of. If empty, all the parameters are set.

set_params!(m::FluxFrictionModel, c_new::NamedTuple)

Set the parameters of the model to the values in c_new.

Arguments

• m::FluxFrictionModel: The model to set the parameters of.
• c_new::NamedTuple: The new parameters. The keys of c_new should be a subset of m.model_ids.

set_zero!(fm::FrictionModel, model_ids)

Sets the parameters of all matrix models in the FrictionModel object to zero.