particles

This repository contains Python implementations and simulations related to the Fokker-Planck equation and Langevin dynamics, two fundamental mathematical frameworks for modeling stochastic processes in physics, chemistry, finance, and machine learning.

Fokker-Planck Equation 1d Simulation

See notebooks/fokker_planck_equation_1d_simulation.ipynb for a simulation of the Fokker-Planck equation in 1d.

The Fokker-Planck equation is given by:

$$ \frac{\partial P(x, t)}{\partial t} = -\frac{\partial}{\partial x} [A(x) P(x, t)] + \frac{\partial^2}{\partial x^2} [B(x) P(x, t)] $$

• where $( P(x, t) )$ is the probability density function
• $( A(x) )$ is the drift coefficient.
• $( B(x) )$ is the diffusion coefficient.

    # define negative log likelihood function
    def negative_log_likelihood(params, data, dt):
        mu, sigma = params
        returns = data['Return'].values
        N = len(returns)
        nll = 0.5 * N * np.log(2 * np.pi * sigma**2 * dt) + \
            np.sum((returns - mu * dt)**2) / (2 * sigma**2 * dt)
        return nll

    # set initial parameters and bounds
    initial_params = [0.0, 0.01]
    bounds = [(-np.inf, np.inf), (1e-6, np.inf)]

    # optimize negative log likelihood function
    result = minimize(
        negative_log_likelihood, 
        initial_params, 
        args=(data, dt), 
        bounds=bounds, 
        method='L-BFGS-B'
    )
    estimated_mu, estimated_sigma = result.x

Langevin Dynamics 1d Simulation

See notebooks/langevin_dynamics_1d_simulation.ipynb for a simulation of Langevin dynamics in 1d.

The Langevin equation is given by:

$$ m \frac{d^2 x}{dt^2} = -\gamma \frac{dx}{dt} + F(x) + \eta(t), $$

where:

• $m$: mass of the particle
• $\gamma$: friction coefficient
• $F(x)$: force derived from the external potential $(F(x) = -\nabla U(x))$
• $\eta(t)$: Gaussian white noise with mean 0 and $\langle \eta(t) \eta(t{\prime}) \rangle = 2\gamma k_B T \delta(t - t{\prime})$.

Normalizing Flow models Simulation

See notebooks/normalizing_flow_models_simulation.ipynb for a simulation of Normalizing Flow models.

$$ \mathbf{x} = f_K \circ f_{K-1} \circ \cdots \circ f_1(\mathbf{z}) $$

• where $(\mathbf{z})$ is the latent variable

$$ p_X(\mathbf{x}) = p_Z(\mathbf{z}) \left| \det \left( \frac{\partial \mathbf{z}}{\partial \mathbf{x}} \right) \right| $$