diffusion

Solution of Diffusion Equation

Description

Let consider following one step transition probability :

$$ W(il-jl,\,\epsilon) = \begin{cases} \frac{1}{2} & (|i-j| = 1) \\ 0 & \text{else} \end{cases} $$

For small $\epsilon,\,l$, the probability distribution function satisfies diffusion equation :

$$ \frac{\partial}{\partial t}w(x,t) = D \frac{\partial^2 w(x,t)}{\partial x^2} ~~\text{ where } D = \frac{l^2}{2\epsilon} $$

And we can solve above equation with $w_i(0) = \delta_{i0}$ as initial state.

$$ w(x,t) = \frac{1}{\sqrt{4\pi D t}} \exp \left(-\frac{x^2}{4Dt}\right) $$

In this project, we want to compare distribution of end points of random walks via kernel density estimation (KDE) with this probability distribution function.

Setting

• Time step : $\epsilon = 10^{-2}$
• Unit Distance : $l = 10^{-1} \,\Rightarrow\, t=10,\,D=\frac{1}{2}$
• Length of Path : $n = 1000$
• Total number of trials : $N = 10000$
• Kernel : Epanechnikov Quadratic Kernel with window size $\lambda = 1$

Build Process

# Data Generation
cargo run --release

# Plot
python nc_plot.py

Result

References

• M. Chaichian, A. Demichev, Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics, CRC Press (2001)