'The Particle Theory':
The particle theory says that all matter consists of many, very small particles which are constantly moving or in a continual state of motion. The degree to which the particles move is determined by the amount of energy they have and their relationship to other particles. The particles might be atoms, molecules or ions.
'Brownian motion' :
The term Brownian motion (in honor of the botanist Robert Brown) refers to either
The physical phenomenon that minute particles immersed in a fluid move about randomly; or
The mathematical models used to describe those random movements.
The mathematical model can also be used to describe many phenomena not resembling (other than mathematically) the random movements of minute particles. An often quoted example is stock market fluctuations. Another example is the evolution of physical characteristics in the fossil record.