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Background:

$$\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$$ is given by Fick's second law, in which $D$ is the diffusion coefficient. The solution to this equation (given the initial condition) $C(x, t)$ is the probability density of finding a particle at $(x, t)$.

This equation describes how a collection of independent Brownian particles diffuse.


In molecular dynamics (MD) simulation, one way to calculate the diffusion coefficient $D$ is to calculate the expectation $$\Bbb E(x^2)_t = \int_{-\infty}^{+\infty}x^2C(x, t)dx = \frac{1}{N}\sum_{i = 1}^N(x_i(t_0 + t) - x_i(t_0 ))^2$$

$N$ is the number of independent diffusive particles that is modeled. {$x_i(t_0)$} and {$x_i(t_0 + t)$} are positions of the $N$ particles at $t_0$ and $t_0 + t$ respectively. So, {$x_i(t_0 + t) - x_i(t_0)$} can be treated as $N$ samples drawn from $C(x, t)$.

However, in some MD simulations, the number of independent particles $N$ is limited to be less than $20$, so it will be insufficient to evaluate the expectation.

I wonder whether I could 'augment' the samples in such a way:

Since the diffusion of the particles are independent, and the particles are doing Brownian motion, the PDF $C(x, t)$ is invariant w.r.t $t_0$. So, I can start with say $M$ {$x_i^m(t_0)$}, then the expectation will be $$\Bbb E(x^2)_t = \frac{1}{N}\sum_{i = 1}^N \frac{1}{M}\sum_{m = 1}^M(x_i^m(t_0 + t) - x_i^m(t_0 ))^2$$

In MD simulation, the total simulation time is $t_{tot}$, following the spirit of multiple starting points, I want to randomly sample several time intervals $[t_0, t_0 + t]$ from $[0, t_{tot}]$ to calculate the expectation.

However, in a real MD simulation, the particles can only be approximated to be doing Brownie motion. And the motions of the particles are correlated to some extent. Can I still do such a 'resampling' for calculating the expectation? Above all, can the time intervals $[t_0, t_0 + t]$ sampled from $[0, t_{tot}]$ overlap?

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