aberut / BrownianSimulation1D

A Python script that generates trajectories of Brownian particles, by integrating the overdamped Langevin equation, following Heun's method.

BrownianSimulation1D

A Python code that generates trajectories of Brownian particles, by integrating the overdamped Langevin equation, following Heun's method.

The main function, called Brownian_simu, is available in the script Numerical_Simulations_Langevin_Equation.
It can be used to simulate 1D trajectories (position $x$ as a function of time $t$), of a Brownian particle that is submitted to an external force $F(x,t)$, in addition to the Gaussian white noise corresponding to the equilibrium thermal agitation $\xi$.
The simulations is done by integrating the following overdamped Langevin equation:

$\gamma \frac{\mathrm{d}x}{\mathrm{d}t} = F_{ext}(x,t) + \xi(t)$

where, $\gamma$ is the Stokes friciton term (given by $\gamma= 6 \pi R \mu$ with $R$ the particle's radius, and $\mu$ the fluid's viscosity), and $\xi$ is a Gaussian white noise which verifies $\langle \xi(t) \rangle$ = 0 and $\langle \xi(t) \xi(t') \rangle = 2\gamma k_\mathrm{B}T$ $\delta(t-t')$ (with $k_\mathrm{B}$ the Botlzmann constant, $T$ the temperature, and $\delta$ the Dirac function).

The script also contains a function, called colored_noise_simu that also integrates an overdamped Langevin equation to generate exponentially correlated Gaussian noises, with a given variance $\alpha$ and correlation time $\tau_c$.

To use a functions, simply download the script in your working directory, then import in your Python scripts with:

from Numerical_Simulations_Langevin_Equation import Brownian_simu

Both functions have a detailed help, available by typing:

help(Brownian_simu)

Tutorial

A step-by-step example is provided as a Jupyter Notebook.

This example shows how to:

1. Compute the trajectories of free particles

2. Compute the trajectories of particles held in an optical trap

3. Compute the trajectories of particles submitted to a sinusoidal forcing

4. Compute the trajectories of particles held in an optical trap with a time-dependent trap stiffness

5. Compute the trajectories of particles held in an optical trap and submitted to a colored noise

Note : you can open this example with Binder to directly run it and interact with it in your internet browser:

Example of use in scientific articles

These functions were used to compute the numerical results presented in the article Comment on "Harvesting information to control non-equilibrium states of active matter", arXiv:2212.06825

The complete code is provided as a Jupyter Notebook.

Citation

If you would like to use this code in a scientific work, please cite as:

Antoine Bérut. aberut/BrownianSimulation1D: v1.9. (2023) doi:10.5281/zenodo.7526269

License

This work is licensed under a Creative Commons Attribution 4.0 International License.