Introduction to Stochastic Processes: Comparison between Non-interacting and Interacting Systems

In this introductory seminar, the one-dimensional systems in which each particle can hop to the right nearest neighbour site only are presented. If the particles do not interact with each other, each particle can be treated independently, reducing the problem to one body. In this non-interacting system, the particles obey the Poisson process [1], which is exactly solvable. However, once one restricts the particles so that each site can have at most one particle, that is, a particle can not hop to the site which is occupied by another particle, then the story is drastically different. Due to the hard-core exclusion, the particles can no longer be treated independently, but they obey the totally asymmetric simple exclusion process (TASEP) [2], which is a current research topic. The density profile and current of the above two systems are discussed. Undergraduates who are interested in studying Statistical Mechanics further are also welcome.
[1] N.G. van Kampen, Stochastic Processes in Physics and Chemistry North-Holland, (1992)
[2] B. Derrida, M.R. Evans, V. Hakim and V. Pasquier, J. Phys. A 26, 1493 (1993)