Brownian motion on the discontinuous structure of matter". The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski are presented below. Another, pure probabilistic class of models is the class of the
Reflected Brownian motion "roughly speaking, "Z" is pushed in direction "R" whenever the boundary surface formula_5 is hit, where "R" is the "j"th column of the matrix "R"." Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where "R" is an M-matrix then necessary and sufficient conditions for stability are The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift "μ " and variance "σ" is for