Gaussian distribution Brownian motion

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by... 

Abernathy and Sharp employed a similar idea, although in a more simplified form 130). They also worked in terms of a spin Hamiltonian varying with time in discrete steps and let the Hamiltonian contain the Zeeman and the ZFS interactions. They assumed, however, that the ZFS interaction was constant in the molecule-fixed (P) frame and that variation of the Hamiltonian originated only from fluctuation of the P frame with respect to the laboratory frame. These fluctuations were described in terms of Brownian reorientational motion, characterized by a time interval, x, (related to the rotational correlation time x ) and a Gaussian distribution of angular steps. 

It has been established that geometrical disorder has only a small effect on Brownian motion [S. Havlin, D. Ben Avraham (1987)]. Also, for thermally activated jumps, if the distribution of es and evv in a geometrically regular lattice is chosen to be Gaussian, as characterized by the variances as and crw, it has been ascertained [Y. Limoge, J. L. Bocquet (1990)] that there are two limiting diffusion coefficients ... 

Brownian motion is a characteristic of the movement of single colloidal particles, but this motion has important consequences when many particles are present. Suppose, for example, we consider a thin sheet in which there are initially c° particles in unit volume [Figure 6.3(a)] and examine the distribution of these particles after a time At. They will have spread out in both directions. The chance that a given particle will have reached a distance Ax is proportional to At1 2 the sharp initial concentration peak will spread out into a broad peak, which has the shape of a Gaussian probability curve [Figure 6.3(b)]. 

Let us return to the analysis of Brownian motion. For simplicity we begin by considering the continuous one-dimensional translational Brownian motion as represented by a one-dimensional random walk problem. The probability of a displacement between x and x + dx after n random steps of length I is given by the Gaussian distribution... 

This definition shows that Brownian motion is closely linked to the Gaussian/normal distribution. The formalism of Weiner processes opens stochastic processes to rigorous mathematical analysis and has enabled the use of Weiner processes in the field of stochastic differential equations. Stochastic differential equations are analogs of classical differential equations where... 

The connection between diffusion and random walks was given by Einstein in a study of Brownian motion [34]. Suppose a collection of points is initially sampled fromp(.Y, t = 0) and that after a short time, each point takes an independent random step sampled from a three dimensional Gaussian distribution ... 

The probability of finding the molecular fragment oscillating at the resonance frequency (o is determined by /(ffl). The oscillatory motion arises from random Brownian motion. The frequency distribution of Brownian motion is Gaussian and may cover... 

Although the Wiener integral formulation for the distribution functions of flexible polymer chains rests upon general considerations of random walks and Brownian motion, it is easily introduced, heuristically, through the concept of an equivalent chain. In this section, only those flexible polymer chains are considered which are composed of equivalent gaussian links. Here L is the maximum contour length of the real chain at full extension, and (R ) for the equivalent chain is taken to be that for the real chain. Thus we have... 

This network is composed of Gaussian chains which introduce forces between pairs of junctions so connected. The junctions fluctuate around their mean positions due to their Brownian motion. The instantaneous distribution of chain vectors r is not affine in the strain because it is the convolution of the distribution of the mean vector r (which is affine) with the distribution of the fluctuations Ar (which are independent of the strain). 

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