PySTFP: Python module for the short-time Fokker-Planck propagator

About

This module contains sympy-based symbolic tools for the short-time propagator for one-dimensional Fokker-Planck dynamics. This python module accompanies the paper Ref. <a href="#ref_1">[1]</a>.

Introduction

For a given drift $a(x)$ and diffusivity $D(x)$ we consider the Fokker-Planck equation

$$\partial_t P = - \partial_x \left( a P\right) + \partial_x^2 \left(D P\right),$$

where $P \equiv P(x,t \mid x_0,t_0)$ is the transition density, or propagator, to find a particle that started at time $t_0$ at location $x_0$ at a later time $t$ at a location $x$. This means we consider the initial conditions $P(x,t_0\mid x_0, t_0) = \delta (x-x_0)$, where $\delta$ denotes the Dirac-delta distribution. As spatial domain we consider $x \in \mathbb{R} = (-\infty,\infty)$, with boundary conditions $P(x,t\mid x_0, t_0) \rightarrow 0$ as $|x| \rightarrow \infty$.

In this module, we implement an approximate short-time propagator $P_K(x,t\mid x_0,t_0)$, in both the normalization-preserving representation

$$P_K(x,t \mid x_0 ,t_0) = \frac{1}{\sqrt{ 4 \pi D(x_0) \Delta t}} \exp\left[ - \frac{ \Delta x^2}{4 D(x_0) \Delta t} \right] \times \left[ 1 +\sum_{k=1}^K \sqrt{\Delta t}^k \mathcal{Q}_k\left(\tilde{x},x_0\right)\right],$$

and the positivity-preserving representation

$$P_K(x,t \mid x_0 ,t_0) = \frac{1}{\sqrt{ 4\pi D(x_0) \Delta t}} \exp\left[ - \frac{ \Delta x^2}{4 D(x_0) \Delta t} + \sum_{k=1}^K \sqrt{\Delta t}^k \hat{\mathcal{Q}}_k\left(\tilde{x},x_0\right)\right].$$

Here, $\Delta x \equiv x - x_0$, $\Delta t = t - t_0$, are the spatial and temporal increments, $\tilde{x} = \Delta x/ \sqrt{ 2 D(x_0) \Delta t}$ is the rescaled spatial increment, and $K \in \mathbb{N}_0$ is the truncation order for the approximate propagator. The explicit form of the coefficients $\mathcal{Q}_k$, $\hat{\mathcal{Q}}_k$, are a main result of Ref. <a href="#ref_1">[1]</a>. The module contains the precalculated coefficients up to $K = 8$ (corresponding to an accuracy $\Delta t^4$ for the propagator), but the module also includes code to calculate the coefficients to even higher order. Note that for a typical spatial increment $\Delta x$, for which the support of the propagator is non-negligible, we have $\tilde{x} \lesssim 1$; this is the reason why the power-series coefficients $\mathcal{Q}_k$, $\hat{\mathcal{Q}}_k$ have $\tilde{x}$ as an argument, see Ref. <a href="#ref_1">[1]</a> for more details.

In PySTFP we furthermore implement the explicit perturbation expansions for the spatial moments $\langle \Delta x^n \rangle$ ($n = 0, 1, 2, 3, 4$), as well as the medium entropy production rate, the total entropy production rate, and the Gibbs entropy. See Ref. <a href="#ref_1">[1]</a> and the examples below for more details.

Examples

In the subfolder examples/ we provide several notebooks that illustrate the use of PySTFP, as well as allow to re-derive and extend the symbolic perturbative expressions used.

Here is a list of the notebooks:

Subfolder examples/

These notebooks showcase basic functionality of PySTFP:

• plot propagator.ipynb: Plot the short-time propagator for a given drift and diffusivity profile.
• propagator in physical units.ipynb: Access the symbolic propagator stored in PySTFP, which are expressed in dimensionless variables, and rewrite them in physical dimensions.
• entropy production in physical units.ipynb: Access the symbolic entropy production (rates) stored in PySTFP, which are expressed in dimensionless variables, and rewrite them in physical dimensions.

Subfolder examples/derivations

These notebooks implement the symbolic derivations of all the perturbative results from Ref. <a href="#ref_1">[1]</a>:

• perturbative propagators and moments.ipynb: Symbolically evaluate the perturbation series for both the normalization-preserving and positivity-preserving propagator. Additionally, use the normalization-preserving propagator to calculate some moments $\langle \Delta x^k \rangle$.
• entropy production rates.ipynb: Symbolically calculate the medium entropy production rate, the total entropy production rate, and the Gibbs entropy.

Subfolder examples/numerical example

These notebooks allow to reproduce the plots of Ref. <a href="#ref_1">[1]</a>:

• analytical solution.ipynb: Evaluate the drift, diffusivity, and propagator exactly.
• Fig1 diffusivity and drift.ipynb: Plot the drift and diffusivity.
• Fig2 exact vs approximate propagator.ipynb: Compare the exact and perturbative propagators.
• Fig3 exact vs approximate Kramers-Moyal coefficients.ipynb: Compare the exact and perturbative finite-time Kramers-Moyal coefficients.
• Fig4 exact vs approximate medium entropy production.ipynb: Compare the exact and perturbative medium entropy production rate.
• Fig5 exact vs second order propagators.ipynb: Compare exact and various second-order perturbative propagators.

Installation

PySTFP requires sympy and numpy, the example notebooks furthermore use matplotlib and h5py. To install these requirements as well as PySTFP, you can run the following commands.

>> git clone https://github.com/juliankappler/PySTFP.git .
>> cd PySTFP
>> pip install -r requirements.txt
>> pip install .

References

<a id="ref_1">[1] Short-time Fokker-Planck propagator beyond the Gaussian approximation. J. Kappler. arXiv: 2405.18381.</a>

Acknowledgements

We acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101068745.