analyze-diffusion-dynamics

# Analyze Diffusion Dynamics Characterize the behavior of diffusion processes by specifying their stochastic differential equations, deriving the corresponding Fokker-Planck equation, computing first-passage time distributions analytically or numerically, performing parameter sensitivity analysis, and validating analytical results against Monte Carlo simulation. ## When to Use - Deriving the probability density evolution of a continuous-time diffusion process - Computing mean first-passage times or full first-passage time distributions for bounded diffusion - Analyzing how drift, diffusion coefficient, and boundary parameters affect process behavior - Validating closed-form solutions against stochastic simulation - Building intuition for the dynamics underlying drift-diffusion models or generative diffusion processes ## Inputs - **Required**: SDE specification (drift function, diffusion coefficient, domain/boundaries) - **Required**: Parameter values or ranges for the drift and diffusion functions - **Required**: Boundary conditions (absorbing, reflecting, or mixed) - **Optional**: Time horizon for transient analysis (default: auto-detect from dynamics) - **Optional**: Spatial discretization resolution for numerical PDE solvers (default: dx=0.001) - **Optional**: Number of Monte Carlo trajectories for simulation validation (default: 10000) ## Procedure ### Step 1: Specify the SDE Model Define the drift function, diffusion coefficient, and boundary conditions for the p...