Transition Path Theory

struct Langevin{H, D, KE, T}

A struct representing a Langevin dynamical system with damping rate gammaand temperaturebeta``.

Fields

• Hamiltonian::Any: Function that computes the total energy (kinetic + potential) of the system.

• driftfree::Any: Function representing the divergence-free part of the drift.

• kinetic::Any: Function computing the kinetic energy of the system.

• gamma::Any: Damping coefficient that determines the strength of friction.

• beta::Any: Inverse temperature parameter (β = 1/kT) that sets the noise intensity.

Constructors

Langevin(Hamiltonian, driftfree, kinetic, gamma, beta)

defined at /home/runner/work/CriticalTransitions.jl/CriticalTransitions.jl/src/transition_path_theory/langevin.jl:14.

committor(
    sys::Langevin,
    mesh::CriticalTransitions.Mesh,
    Aind,
    Bind
) -> Vector{Float64}

Solve the committor equation for a two-dimensional Langevin system using finite elements.

Arguments

• sys::Langevin: The Langevin system containing kinetic energy, drift-free function, beta, and gamma.
• mesh::Mesh: The mesh containing points and triangles.
• Aind::Vector{Int}: Indices of mesh points corresponding to set A (Dirichlet boundary condition = 0).
• Bind::Vector{Int}: Indices of mesh points corresponding to set B (Dirichlet boundary condition = 1).

Returns

• A vector of committor values of length N, where each entry corresponds to a mesh node.

Implementation Details

This function assembles a global matrix A and right-hand-side vector b for the finite element discretization of the committor equation in Langevin dynamics. The code imposes Dirichlet boundary conditions on specified nodes (Aind set to 0, Bind set to 1). It computes elementwise contributions with stima_Langevin and stimavbdv, applying exponential factors involving beta and gamma to incorporate potential and damping effects. Finally, it solves the resulting linear system for the committor values on the free (non-boundary) nodes.

invariant_pdf(
    sys::Langevin,
    mesh::CriticalTransitions.Mesh,
    Amesh::CriticalTransitions.Mesh,
    Bmesh::CriticalTransitions.Mesh
) -> Any

Compute the invariant probability density function (PDF) for a Langevin system over a given mesh.

Arguments

• sys::Langevin: The Langevin system containing the Hamiltonian and beta parameters.
• mesh::Mesh: The main mesh containing points and triangles.
• Amesh::Mesh: The mesh corresponding to region A.
• Bmesh::Mesh: The mesh corresponding to region B.

Returns

• A scalar value representing the normalization constant Z of the invariant PDF.

Implementation Details

This function calculates the invariant PDF by integrating the exponential of the negative Hamiltonian over the areas of the triangles in the provided meshes. The normalization constant Z is computed by summing the contributions from the main mesh, region A mesh, and region B mesh.

probability_reactive(
    sys::Langevin,
    mesh::CriticalTransitions.Mesh,
    q,
    qminus,
    Z
) -> Any

Calculate the probability that a trajectory is reactive (transitions from A to B).

Arguments

• sys::Langevin: System with Hamiltonian and inverse temperature (beta)
• mesh::Mesh: Mesh structure containing points and triangulation
• q: Forward committor function values
• qminus: Backward committor function values
• Z: Partition function value

Returns

• Probability (float) of reactive trajectories normalized by partition function

probability_last_A(
    sys::Langevin,
    mesh::CriticalTransitions.Mesh,
    Ames::CriticalTransitions.Mesh,
    qminus,
    Z
) -> Any

Calculate the probability that the last visited metastable state was A.

Arguments

• sys::Langevin: System with Hamiltonian and inverse temperature (beta)
• mesh::Mesh: Main mesh structure containing points and triangulation
• Ames::Mesh: Mesh structure for region A
• qminus: Backward committor function values
• Z: Partition function value

Returns

• Probability (float) that the system was last in state A, normalized by partition function