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
Rotational Brownian motion imposed electric field. Rotational Brownian motion was first discussed by Peter Debye, who applied Einstein's theory of translational Brownian motion to the rotation of molecules having permanent electric dipoles. Debye ignored inertial effects and assumed that the molecules were spherical, with an intrinsic, fixed dipole moment. He derived expressions for the dielectric relaxation time and for the permittivity. These formulae have been successfully applied to many materials. However, Debye's expression for the permittivity predicts that the absorption tends toward a constant value when the frequency of the applied electric field becomes very large—the "Debye plateau". This is not observed; instead,