langevin¶

Tools for integrating the directed-percolation Langevin equation¶

The langevin package provides software tools to integrate a time-dependent density field described by Langevin equations of directed-percolation type. It can be extended to solve Langevin equations of absorbing phase transition (APT) type.

This is a work in progress

langevin is under active development as part of a research effort. If you are interested in using it, or even better, interested in collaborating in its development, please contact the maintainer cstarkjp@gmail.com.

Directed percolation (DP) is the type example of a non-equilibrium, absorbing phase transition. Its Langevin equation is: $$ \partial_t\rho = a \rho - b \rho^2 + D \nabla^2 \rho + \eta\sqrt{\rho}\,\xi $$ where $\rho(\mathbf{x},t)$ is a fluctuating meso-scale field evolving nonlinearly (with coefficients $a$ and $b$) subject to diffusion (with rate $D$) and multiplicative white noise $\sqrt{\rho}\,\xi(\mathbf{x},t)$ (with amplitude $\eta$).

Figure 1: Plot of grid-averaged density $\overline{\rho}(t)$ versus time, for an ensemble of simulations with $a$ taking values ranging symmetrically about criticality $a_c \approx 1.8857$ by up to $\Delta{a}=\pm 0.01$.

The langevin integrator employs the operator-splitting method originated largely by Dornic et al (2005). The software tools are implemented as a pip-installable Python package with a C++ core, a set of Jupyter notebooks, and related Python scripts.