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 bias the motion of a particle (like an ion or polypeptide); the result would essentially be diffusion of a particle whose net motion is strongly biased in one direction. The translocation of the particle would only be loosely coupled to hydrolysis of ATP. The dynamics and activity of Brownian motors are current topics of study in theoretical and experimental biophysics. Brownian motors are sometimes modeled using the Fokker-Planck equation or with Monte Carlo methods. Many researchers are presently engaged in understanding how molecular-scale motors operate in environments with non-negligible thermal noise. The thermodynamics of such motors is constrained by the