# 1 D Brownian Motion

From the beginning of the last century, starting from the Einstein, Smoluchowski and Perrin analyses of Brownian motion had great relevance in theoretical physics. The history of the scientific debate about the Brownian motion and the role it played in the success of the molecular kinetic theory of matter is a fascinating subject worth learning in high school. Probability is typically used to account for some practical limitations, in particular the impossibility to describe at any level of accuracy the mechanical state of the Brownian particle environment, which is assumed to be made up of a huge number of particles. Notice that are nowadays available high quality physics simulation softwares, based on object oriented programming, that students can easily use to simulate and visualize the motion of a massive large particle triggered by elastic collisions with a large number of light small particles.

We consider a particle moving on a one-dimensional lattice with uniform motion but we ignore the initial position. We only know that it has equal probability of being to the right or to the left of the origin, taken as the center of the lattice. We then assign to our particle a probability profile or rather two probability distributions identical and symmetrical with respect to the origin. By assigning them equal and opposite momentum, the packets travel in uniform motion towards the origin. When the two packets cross, the probability that the particle is at the origin at a certain instant will be given by the sum of the probabilities that the particle arrives from the right or from the left (step probability), multiplied respectively for the probability that this was initially on one side or the other.