From  the work of Jean Perrin, who won the Nobel Prize for Physics for his work,

http://www.nobelprize.org/nobel_prizes/physics/laureates/1926/perrin-lecture.html

Dear Nicolis,

Thank you very much for your reply and the reference. Except that Jean Perrin has carried the assumption of random motion further to base his theory on, no logical reason, cause or mechanism of action for this phenomenon is given by him. Every action should have a cause and a mechanism of action. If random motions of molecules in gaseous or liquid bodies are real, they also should have cause, medium of action and mechanism of action. That is what I am interested in. If you wish, kindly see http://vixra.org/abs/1103.0117 for Brownian movements.

Nainan

In a celebrated paper published in 1905  by A.Einstein, the motion of pollen grains in water (i.e. Brownian motion) was calculated on the basis of molecular nature of water and random collisions of these molecules with the grains that results in a mean squared displacement proportional to time and diffusion constant, this paper historically proved the molecular nature of water and relation of their mean velocity with  temperature. On the other hand the definition of "temperature"  based on the motion (kinetic) of molecules and famous works conducted by Boltzmann and Maxwell had been shown that the assumption of atomic or molecular nature plus the random motion of them could obtain a consistent theory for definite description of  macroscopic thermodynamical parameters such as temperature, pressure, heat conduction and internal energies of systems compatible with thermodynamic laws.Moreover this assumptive motion has been observed directly by electron microscopy and STM etc.