# Derive properties of fluids using Monte Carlo method on brownian motion

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, ...)?

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.