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# 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.