Rotational Brownian motion the absorption tends toward a maximum and then declines with increasing frequency. The breakdown in Debye's theory in these regimes can be corrected by including inertial effects; allowing the molecules to be non-spherical; including dipole-dipole interactions between molecules; etc. These are computationally very difficult problems and rotational Brownian motion is a topic of much current research interest. Rotational Brownian motion Rotational Brownian motion is the random change in the orientation of a polar molecule due to collisions with other molecules. It is an important element of theories of dielectric materials. The polarization of a dielectric material is a competition between
Brownian excursion Brownian excursion process, formula_1, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. Another representation of a Brownian excursion formula_1 in terms of a Brownian motion process "W" (due to Paul Lévy and noted by Kiyoshi Itō and Henry P. McKean, Jr.) is in terms of the last time formula_3 that "W" hits zero before time 1 and the first time formula_4 that Brownian motion formula_5 hits zero after time 1: Let formula_7 be the time that a