A Markov process allows in principle for a microscopic description of Brownian motion. As with the random walk studied in the previous section, we consider a particle which moves along the \( x \)-axis in the form of a series of jumps with step length \( \Delta x = l \). Time and space are discretized and the subsequent moves are statistically independent, i.e., the new move depends only on the previous step and not on the results from earlier trials. We start at a position \( x=jl=j\Delta x \) and move to a new position \( x =i\Delta x \) during a step \( \Delta t=\epsilon \), where \( i\ge 0 \) and \( j\ge 0 \) are integers. The original probability distribution function (PDF) of the particles is given by \( w_i(t=0) \) where \( i \) refers to a specific position on the grid in