Brownian motion but rather a singular behavior in which the particle doesn't move at all. The Brownian motion can be modeled by a random walk. Random walks in porous media or fractals are anomalous. In the general case, Brownian motion is a non-Markov random process and described by stochastic integral equations. The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous R-valued stochastic process "X" to actually be "n"-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion. Let "X" = ("X", ..., "X") be a
Brownian motor at the nanometer scale. The term "Brownian motor" was originally coined by Peter Hänggi in 1995: A distinct feature of a Brownian motor is—in contrast to a molecular motor—that the output response is typically coupled only loosely to the input perturbation and action of fluctuations. In biology, many protein-based molecular motors in the cell may in fact be Brownian motors. These molecular motors convert the chemical energy present in ATP into mechanical energy. One example of a Brownian motor would be an ATPase motor that hydrolyzes ATP to generate fluctuating anisotropic energetic potentials. The anisotropic potentials along the path would