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Given a particle inside a fluid, it's known that its movement will be unpredictable due to the random collisions with the particles of the fluid. However, the distance from the origin of motion will follow a normal probability distribution (in 3 dimensions) given by

$$ p(\mathbf{r},t)\,dV = \dfrac{1}{(12\pi Dt)^{3/2}}\exp\left(-\dfrac{r^2}{12Dt}\right)\,dx_1dx_2dx_3 $$

where $D$ represents the diffusion coefficient of the fluid.

Therefore, if we simulate $N$ particles with random collisions, where the probabilities of the collisions are defined as a function of some fluid properties, could we recover the distance from the origin over time of the $N$ experiments to derive the $p(\mathbf{r},t)$ and get an approximation of $D$? Which properties of the fluid would take part in the definition of the collision probabilities (viscosity, molecular mass, ...)?

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You would need to use molecular dynamics rather than Monte Carlo (for which there's no 'time' variable). Also, you don't need to compute $\rho(\textbf{r},t)$ but simply the mean square displacement, which obeys $\langle \Delta r^2\rangle = 2nD\Delta t$ where $n$ is the spatial dimension.

This is the standard method for computing diffusion coefficients from molecular simulation.

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