Analyze Diffusion Dynamics

Characterize behavior of diffusion processes. Specify SDEs. Derive Fokker-Planck equation. Compute first-passage time distributions analytic or numeric. Perform parameter sensitivity analysis. Validate against Monte Carlo simulation.

When Use

• Deriving probability density evolution of continuous-time diffusion process
• Computing mean first-passage times or full first-passage time distributions for bounded diffusion
• Analyzing how drift, diffusion coefficient, boundary parameters affect process behavior
• Validating closed-form solutions against stochastic simulation
• Building intuition for 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 drift and diffusion functions
• Required: Boundary conditions (absorbing, reflecting, 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)

Steps

Step 1: Specify SDE Model

Define drift function, diffusion coefficient, boundary conditions for process.

1. Write SDE in standard Ito form:

dX(t) = mu(X, t) dt + sigma(X, t) dW(t)

where

mu

is drift function,

sigma

is diffusion coefficient,

W(t)

is standard Wiener process.

2. Implement SDE components in code:

import numpy as np

class DiffusionProcess:
    """A one-dimensional diffusion process specified by drift and diffusion functions."""

    def __init__(self, drift_fn, diffusion_fn, lower_bound=None, upper_bound=None,
                 boundary_type="absorbing"):
        self.drift = drift_fn
        self.diffusion = diffusion_fn
        self.lower_bound = lower_bound
        self.upper_bound = upper_bound
        self.boundary_type = boundary_type

# Example: Ornstein-Uhlenbeck process on [0, a]
ou_process = DiffusionProcess(
    drift_fn=lambda x, t: 2.0 * (0.5 - x),     # mean-reverting drift
    diffusion_fn=lambda x, t: 0.1,               # constant diffusion
    lower_bound=0.0,
    upper_bound=1.0,
    boundary_type="absorbing"
)

# Example: Standard DDM (constant drift and diffusion)
ddm_process = DiffusionProcess(
    drift_fn=lambda x, t: 0.5,        # drift rate v
    diffusion_fn=lambda x, t: 1.0,    # unit diffusion (s=1, convention)
    lower_bound=0.0,                   # lower absorbing boundary
    upper_bound=1.5,                   # upper absorbing boundary (a)
    boundary_type="absorbing"
)

3. Define initial condition:

# Point source at x0
x0 = 0.75  # starting point (e.g., midpoint between boundaries for DDM with z=a/2)

# Or a distribution
initial_distribution = lambda x: np.exp(-50 * (x - 0.75)**2)  # narrow Gaussian

4. Validate parameter consistency:

def validate_process(process, x0):
    """Check that the SDE specification is self-consistent."""
    assert process.lower_bound < process.upper_bound, "Lower bound must be less than upper bound"
    assert process.lower_bound <= x0 <= process.upper_bound, \
        f"Initial position {x0} outside bounds [{process.lower_bound}, {process.upper_bound}]"
    test_drift = process.drift(x0, 0)
    test_diff = process.diffusion(x0, 0)
    assert np.isfinite(test_drift), f"Drift is not finite at x0={x0}"
    assert test_diff > 0, f"Diffusion coefficient must be positive, got {test_diff}"
    print(f"Process validated: drift={test_drift:.4f}, diffusion={test_diff:.4f} at x0={x0}")

validate_process(ddm_process, x0=0.75)

Got: Fully specified SDE with finite drift values, strictly positive diffusion coefficient, initial condition within domain boundaries.

If fail: Diffusion coefficient zero or negative at any point in domain? Process degenerate -- check functional form. Drift infinite at boundary? Consider whether reflecting boundary more appropriate.

Step 2: Derive Fokker-Planck Equation

Convert SDE to its equivalent partial differential equation for probability density.

1. Write Fokker-Planck equation (FPE) for transition density p(x, t):

dp/dt = -d/dx [mu(x,t) * p(x,t)] + (1/2) * d^2/dx^2 [sigma(x,t)^2 * p(x,t)]

2. For constant coefficients (standard DDM case), this simplifies to:

dp/dt = -v * dp/dx + (s^2 / 2) * d^2p/dx^2

3. Implement numerical solution of FPE via finite differences:

from scipy.sparse import diags
from scipy.sparse.linalg import spsolve

def solve_fokker_planck(process, x0, t_max, dx=0.001, dt=None):
    """Solve the FPE numerically using Crank-Nicolson scheme."""
    x_grid = np.arange(process.lower_bound, process.upper_bound + dx, dx)
    N = len(x_grid)

    if dt is None:
        max_sigma = max(process.diffusion(x, 0) for x in x_grid)
        dt = 0.4 * dx**2 / max_sigma**2  # CFL-like stability condition

    # Initial condition: narrow Gaussian centered at x0
    p = np.exp(-((x_grid - x0)**2) / (2 * (2*dx)**2))
    p[0] = 0  # absorbing boundary
    p[-1] = 0  # absorbing boundary
    p = p / (np.sum(p) * dx)

    t_steps = int(t_max / dt)
    survival = np.zeros(t_steps)
    density_snapshots = []

    for step in range(t_steps):
        mu_vals = np.array([process.drift(x, step*dt) for x in x_grid])
        sigma_vals = np.array([process.diffusion(x, step*dt) for x in x_grid])
        D = 0.5 * sigma_vals**2

        # Finite difference operators (interior points)
        advection = -mu_vals[1:-1] / (2 * dx)
        diffusion_coeff = D[1:-1] / dx**2

        main_diag = 1 + dt * 2 * diffusion_coeff
        upper_diag = dt * (-diffusion_coeff[:-1] - advection[:-1])
        lower_diag = dt * (-diffusion_coeff[1:] + advection[1:])

        A = diags([lower_diag, main_diag, upper_diag], [-1, 0, 1], format="csc")
        p[1:-1] = spsolve(A, p[1:-1])
        p[0] = 0
        p[-1] = 0

        survival[step] = np.sum(p[1:-1]) * dx

        if step % (t_steps // 10) == 0:
            density_snapshots.append((step * dt, p.copy()))

    return x_grid, survival, density_snapshots

4. Run and plot evolving density:

import matplotlib.pyplot as plt

x_grid, survival, snapshots = solve_fokker_planck(ddm_process, x0=0.75, t_max=5.0)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
for t_val, density in snapshots:
    ax1.plot(x_grid, density, label=f"t={t_val:.2f}")
ax1.set_xlabel("x")
ax1.set_ylabel("p(x, t)")
ax1.set_title("Fokker-Planck Density Evolution")
ax1.legend()

t_vals = np.linspace(0, 5.0, len(survival))
ax2.plot(t_vals, survival)
ax2.set_xlabel("Time")
ax2.set_ylabel("Survival probability")
ax2.set_title("Survival Probability S(t)")
fig.tight_layout()
fig.savefig("fokker_planck_solution.png", dpi=150)

Got: Density starts as narrow peak at x0, spreads and drifts according to SDE coefficients, gradually decays as probability absorbed at boundaries. Survival probability decreases monotonic from 1 toward 0.

If fail: Density develops oscillations or negative values? Time step too large -- reduce dt. Density does not decay (survival stays near 1)? Boundaries may be too far from x0 or drift pushes away from both boundaries. Check boundary conditions in solver.

Step 3: Compute First-Passage Time Distributions

Derive distribution of times at which process first reaches boundary.

1. Compute first-passage time density from survival function:

def first_passage_time_density(survival, dt):
    """FPT density is the negative derivative of survival probability."""
    fpt_density = -np.gradient(survival, dt)
    fpt_density = np.maximum(fpt_density, 0)  # enforce non-negativity
    return fpt_density

2. For standard DDM with constant drift, use known analytic solution:

def ddm_fpt_upper(t, v, a, z, s=1.0, n_terms=50):
    """Analytic FPT density at the upper boundary for constant-drift DDM.

    Uses the infinite series representation (large-time expansion).
    """
    if t <= 0:
        return 0.0
    density = 0.0
    for k in range(1, n_terms + 1):
        density += (k * np.pi * s**2 / a**2) * \
            np.exp(-v * (a - z) / s**2 - 0.5 * v**2 * t / s**2) * \
            np.sin(k * np.pi * z / a) * \
            np.exp(-0.5 * (k * np.pi * s / a)**2 * t)
    return density

3. Compute summary statistics of FPT distribution:

def fpt_statistics(fpt_density, dt):
    """Compute mean, variance, and quantiles of the FPT distribution."""
    t_vals = np.arange(len(fpt_density)) * dt
    total_mass = np.sum(fpt_density) * dt

    # Normalize
    fpt_normed = fpt_density / total_mass if total_mass > 0 else fpt_density

    mean_fpt = np.sum(t_vals * fpt_normed) * dt
    var_fpt = np.sum((t_vals - mean_fpt)**2 * fpt_normed) * dt

    # Quantiles via CDF
    cdf = np.cumsum(fpt_normed) * dt
    quantile_10 = t_vals[np.searchsorted(cdf, 0.1)]
    quantile_50 = t_vals[np.searchsorted(cdf, 0.5)]
    quantile_90 = t_vals[np.searchsorted(cdf, 0.9)]

    return {
        "mean": mean_fpt,
        "std": np.sqrt(var_fpt),
        "q10": quantile_10,
        "q50": quantile_50,
        "q90": quantile_90,
        "total_probability": total_mass
    }

4. For two-boundary problems, separate FPT by boundary using probability flux at each absorbing wall (finite difference of density at boundary grid points).

Got: FPT density is right-skewed unimodal distribution. For DDM with positive drift, upper boundary FPT has more mass and shorter mode than lower boundary FPT. Mean FPT for typical DDM parameters (v=1, a=1.5, z=0.75) approximately 0.5-2.0 seconds.

If fail: FPT density has negative values? Numerical differentiation noisy -- apply small Gaussian smoothing kernel. Total probability at both boundaries does not sum to approximately 1.0? Either time horizon too short (increase t_max) or probability leakage in solver.

Step 4: Analyze Parameter Sensitivity

Quantify how changes in each parameter affect first-passage time distribution.

1. Define parameter grid for sensitivity analysis:

param_ranges = {
    "v": np.linspace(0.2, 3.0, 15),     # drift rate
    "a": np.linspace(0.5, 2.5, 15),      # boundary separation
    "z_ratio": np.linspace(0.3, 0.7, 9)  # starting point as fraction of a
}

base_params = {"v": 1.0, "a": 1.5, "z_ratio": 0.5}

2. Sweep each parameter while holding others at baseline:

sensitivity_results = {}

for param_name, param_values in param_ranges.items():
    means = []
    accuracies = []
    for val in param_values:
        params = base_params.copy()
        params[param_name] = val
        z = params["z_ratio"] * params["a"]

        process = DiffusionProcess(
            drift_fn=lambda x, t, v=params["v"]: v,
            diffusion_fn=lambda x, t: 1.0,
            lower_bound=0.0,
            upper_bound=params["a"],
            boundary_type="absorbing"
        )

        _, survival, _ = solve_fokker_planck(process, x0=z, t_max=10.0)
        fpt = first_passage_time_density(survival, dt=10.0/len(survival))
        stats = fpt_statistics(fpt, dt=10.0/len(survival))
        means.append(stats["mean"])
        accuracies.append(stats["total_probability"])  # proxy for upper boundary

    sensitivity_results[param_name] = {
        "values": param_values,
        "mean_fpt": np.array(means),
        "accuracy": np.array(accuracies)
    }

3. Plot sensitivity curves:

fig, axes = plt.subplots(1, 3, figsize=(18, 5))
for idx, (param_name, result) in enumerate(sensitivity_results.items()):
    ax = axes[idx]
    ax.plot(result["values"], result["mean_fpt"], "b-o", label="Mean FPT")
    ax.set_xlabel(param_name)
    ax.set_ylabel("Mean FPT")
    ax.set_title(f"Sensitivity to {param_name}")

    ax2 = ax.twinx()
    ax2.plot(result["values"], result["accuracy"], "r--s", label="P(upper)")
    ax2.set_ylabel("P(upper boundary)")
    ax.legend(loc="upper left")
    ax2.legend(loc="upper right")

fig.tight_layout()
fig.savefig("parameter_sensitivity.png", dpi=150)

4. Compute partial derivatives (local sensitivity at baseline):

for param_name, result in sensitivity_results.items():
    idx_base = np.argmin(np.abs(result["values"] - base_params[param_name]))
    if idx_base > 0 and idx_base < len(result["values"]) - 1:
        d_mean = (result["mean_fpt"][idx_base+1] - result["mean_fpt"][idx_base-1]) / \
                 (result["values"][idx_base+1] - result["values"][idx_base-1])
        print(f"d(mean_FPT)/d({param_name}) at baseline: {d_mean:.4f}")

Got: Drift rate (v) has strong negative effect on mean FPT and strong positive effect on accuracy. Boundary separation (a) has strong positive effect on mean FPT (speed-accuracy tradeoff). Starting point (z) shifts accuracy with smaller effect on mean FPT.

If fail: Sensitivity curves flat or non-monotonic? Check parameter range wide enough and solver time horizon captures full FPT distribution. Non-monotonic mean FPT with respect to drift rate would indicate solver bug.

Step 5: Validate Analytics Against Numerical Simulation

Run Monte Carlo simulations of SDE to confirm analytical and numerical PDE results.

1. Implement Euler-Maruyama simulation of SDE:

def simulate_sde(process, x0, dt_sim=0.0001, t_max=10.0, n_trajectories=10000):
    """Simulate SDE paths and record first-passage times."""
    n_steps = int(t_max / dt_sim)
    fpt_upper = np.full(n_trajectories, np.nan)
    fpt_lower = np.full(n_trajectories, np.nan)

    x = np.full(n_trajectories, x0)
    sqrt_dt = np.sqrt(dt_sim)

    for step in range(n_steps):
        t = step * dt_sim
        active = np.isnan(fpt_upper) & np.isnan(fpt_lower)
        if not active.any():
            break

        mu = np.array([process.drift(xi, t) for xi in x[active]])
        sigma = np.array([process.diffusion(xi, t) for xi in x[active]])
        dW = np.random.randn(active.sum()) * sqrt_dt

        x[active] += mu * dt_sim + sigma * dW

        # Check boundary crossings
        hit_upper = active & (x >= process.upper_bound)
        hit_lower = active & (x <= process.lower_bound)
        fpt_upper[hit_upper] = (step + 1) * dt_sim
        fpt_lower[hit_lower] = (step + 1) * dt_sim

    return fpt_upper, fpt_lower

2. Run simulation and compute empirical FPT distribution:

fpt_upper_sim, fpt_lower_sim = simulate_sde(ddm_process, x0=0.75, n_trajectories=50000)

# Empirical statistics
valid_upper = fpt_upper_sim[~np.isnan(fpt_upper_sim)]
valid_lower = fpt_lower_sim[~np.isnan(fpt_lower_sim)]
total_absorbed = len(valid_upper) + len(valid_lower)
accuracy_sim = len(valid_upper) / total_absorbed

print(f"Simulated accuracy: {accuracy_sim:.4f}")
print(f"Mean FPT (upper): {valid_upper.mean():.4f} +/- {valid_upper.std()/np.sqrt(len(valid_upper)):.4f}")
print(f"Mean FPT (lower): {valid_lower.mean():.4f} +/- {valid_lower.std()/np.sqrt(len(valid_lower)):.4f}")

3. Compare simulation against analytical or numerical PDE solution:

fig, ax = plt.subplots(figsize=(10, 6))

# Empirical histogram
ax.hist(valid_upper, bins=100, density=True, alpha=0.5, label="Simulation (upper)")
ax.hist(valid_lower, bins=100, density=True, alpha=0.5, label="Simulation (lower)")

# Analytical solution overlay
t_vals_analytic = np.linspace(0.01, 5.0, 500)
v, a, z = 0.5, 1.5, 0.75
fpt_analytic = [ddm_fpt_upper(t, v, a, z) for t in t_vals_analytic]
ax.plot(t_vals_analytic, fpt_analytic, "k-", linewidth=2, label="Analytic (upper)")

ax.set_xlabel("First-passage time")
ax.set_ylabel("Density")
ax.set_title("FPT Distribution: Simulation vs. Analytic")
ax.legend()
fig.savefig("fpt_validation.png", dpi=150)

4. Quantify agreement between methods:

from scipy.stats import ks_2samp

# Kolmogorov-Smirnov test between simulated and analytically-derived samples
analytic_cdf = np.cumsum(fpt_analytic) * (t_vals_analytic[1] - t_vals_analytic[0])
sim_sorted = np.sort(valid_upper)
sim_cdf = np.arange(1, len(sim_sorted)+1) / len(sim_sorted)

# Interpolate analytic CDF at simulation quantiles
from scipy.interpolate import interp1d
analytic_interp = interp1d(t_vals_analytic, analytic_cdf, bounds_error=False, fill_value=(0, 1))
max_diff = np.max(np.abs(sim_cdf - analytic_interp(sim_sorted)))
print(f"Max CDF difference (simulation vs. analytic): {max_diff:.4f}")
assert max_diff < 0.05, f"Simulation and analytic FPT differ by {max_diff:.4f} (threshold: 0.05)"

Got: Simulation histograms close match analytical FPT curves. KS-test maximum CDF difference below 0.05 for 50,000 trajectories. Mean FPT from simulation within 2 standard errors of analytical value.

If fail: Simulation disagrees with analytics? First check Euler-Maruyama step size -- dt_sim should be small enough that boundary crossings not missed (try dt_sim=0.00001). Analytical series does not converge? Increase n_terms. For non-constant coefficients where no analytic solution exists, compare two numerical methods (PDE solver vs. simulation) against each other.

Checks

• SDE specification passes consistency checks (finite drift, positive diffusion, x0 in domain)
• Fokker-Planck density integrates to value that decreases monotonic over time (survival function)
• Fokker-Planck solution shows no numerical artifacts (oscillations, negative values)
• FPT density non-negative and integrates to approximately 1.0 across both boundaries
• Sensitivity analysis shows expected monotonic relationships (v vs. accuracy, a vs. mean FPT)
• Monte Carlo simulation mean FPT within 2 standard errors of PDE/analytic solution
• KS-test maximum CDF difference between simulation and analytics below 0.05

Pitfalls

• Euler-Maruyama step size too large: Large dt_sim causes trajectories to overshoot boundaries. Leads to biased FPT estimates. Use dt_sim at most 1/10 of expected mean FPT, or use boundary-corrected scheme.
• Truncating FPT series too early: Analytic DDM FPT density uses infinite series. Too few terms (< 20) causes visible artifacts, especially at short times. Use at least 50 terms and check convergence.
• Ignoring numerical diffusion in PDE solver: First-order finite difference schemes introduce artificial diffusion that broadens FPT distribution. Use Crank-Nicolson or higher-order schemes for accuracy.
• Confusing Ito and Stratonovich forms: Fokker-Planck equation differs depending on SDE convention. Standard form above assumes Ito calculus. SDE written in Stratonovich form? Add noise-induced drift correction term.
• Not accounting for both boundaries: In two-boundary problems, total absorption probability must sum to 1.0. Reporting only upper boundary FPT without accounting for lower boundary gives incorrect statistics.

See Also

• fit-drift-diffusion-model

  - applies these dynamics to estimate parameters from behavioral data

• implement-diffusion-network

  - generative diffusion models discretize same SDE framework

• write-testthat-tests

  - testing numerical solvers and analytical implementations

• create-technical-report

  - documenting diffusion analysis results