Brownian motion defined

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

It is our experience that to the first question, the most common student response is something akin to Because my teacher told me so . One is tempted to say that it is a pity that the scientific belief of so mat r students is sourced from an authority, rather than from empirical evidence - except that when chemists are asked question (ii), they find it not at all easy to answer. There is, after all, no single defining experiment that conclusively proves the claim, even though it was the phenomenon of Brownian motion that finally seems to have clinched the day for the atomists 150 or so years ago. Of course, from atomic forced microscopy (AFM), we see pictures of gold atoms being manipulated one by one - but the output from AFM is itself the result of application of interpretive models. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Here r and v are respectively the electron position and velocity, r = —(e2 /em)(r/r3) is the acceleration in the coulombic field of the positive ion and q = /3kBT/m. The mobility of the quasi-free electron is related to / and the relaxation time T by p = e/m/3 = et/m, so that fi = T l. In the spherically symmetrical situation, a density function n(vr, vt, t) may be defined such that n dr dvr dvt = W dr dv here, vr and vt and are respectively the radical and normal velocities. Expectation values of all dynamical variables are obtained from integration over n. Since the electron experiences only radical force (other than random interactions), it is reasonable to expect that its motion in the v space is basically a free Brownian motion only weakly coupled to r and vr by the centrifugal force. The correlations1, K(r, v,2) and fc(vr, v(2) are then neglected. Another condition, cr(r)2 (r)2, implying that the electron distribution is not too much delocalized on r, is verified a posteriori. Following Chandrasekhar (1943), the density function may now be written as an uncoupled product, n = gh, where... 

It is possible however to analyze mathematically well defined models which we hope will give a correct approximation to real physical systems. In this section, we shall be concerned with the simplest case the zeroth-order conductance of electrolytes in an infinitely dilute solution. We shall describe this situation by assuming that the ions—which are so far from each other that their mutual interaction may be completely neglected—have a very large mass with respect to the solvent molecules we are then confronted with a typical Brownian motion problem. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

Experiments involving anisotropy of phosphorescence or of the absorption of the triplet state rely upon the same principles as the measurement of fluorescence anisotropy. All are based upon the photoselection of molecules by polarized light and the randomization of polarization due to Brownian motion occurring on the time scale of the excited state. Anisotropy is defined as... 

Diffusion is defined as the random translational motion of molecules or ions that is driven by internal thermal energy - the so-called Brownian motion. The mean movement of a water molecule due to diffusion amounts to several tenth of micrometres during 100 ms. Magnetic resonance is capable of monitoring the diffusion processes of molecules and therefore reveals information about microscopic tissue compartments and structural anisotropy. Especially in stroke patients diffusion sensitive imaging has been reported to be a powerful tool for an improved characterization of ischemic tissue. 

A mathematical expression that defines how an ensemble average fluctuates with time. An example is the analysis of Brownian motion, where one may seek to understand the nature of the frictional force. If one considers T as a time interval that is very small on the macroscopic scale, but large on the microscopic scale, then... 

Brownian motion of a single molecular species results in a Lorentzian spectrum defined by the relationship ... 

There are some very special characteristics that must be considered as regards colloidal particle behavior size and shape, surface area, and surface charge density. The Brownian motion of particles is a much-studied field. The fractal nature of surface roughness has recently been shown to be of importance (Birdi, 1993). Recent applications have been reported where nanocolloids have been employed. Therefore, some terms are needed to be defined at this stage. The definitions generally employed are as follows. Surface is a term used when one considers the dividing phase between... 

In this section, we begin the description of Brownian motion in terms of stochastic process. Here, we establish the link between stochastic processes and diffusion equations by giving expressions for the drift velocity and diffusivity of a stochastic process whose probability distribution obeys a desired diffusion equation. The drift velocity vector and diffusivity tensor are defined here as statistical properties of a stochastic process, which are proportional to the first and second moments of random changes in coordinates over a short time period, respectively. In Section VILA, we describe Brownian motion as a random walk of the soft generalized coordinates, and in Section VII.B as a constrained random walk of the Cartesian bead positions. 

Using this definition of we may generalize the diffusion equation for the distribution /( ) on the /-dimensional constraint surface to an equivalent diffusion equation for a distribution V Q) in the 3A-dimensional unconstrained space. We consider a model in which a system of 3N coordinates undergoes Brownian motion in the full unconstrained space under the influence of the mobility, defined above, as described by a diffusion equation... 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

In what follows, we distinguish between the drift velocity V associated with a random variable A , which is defined by Eq. (2.223), and the corresponding drift coefficient that appears explicitly in a corresponding SDE for A , which will be denoted by A (A), and which is found to be equal to V only in the case of an Ito SDE. The values of the generalized and Cartesian drift velocities required to force each type of SDE to mimic constrained Brownian motion are determined in what follows by requiring that the resulting drift velocities have the values obtained in Section VII. 

In this section, we have considered four possible ways of formulating and interpreting a set of SDE to describe Brownian motion, and tried to clarify the relationships among them. Because each interpretation may be defined as the At 0 limit of a discrete Markov processes, this discussion of SDEs provides a useful starting point for the discussion of possible simulation algorithms. 

There have been several other theoretical approaches towards defining the time dependence of quenching by both long-range energy (or electron) transfer and diffusion. Some of these are discussed in later chapters. For instance, improvements to the description of Brownian motion as diffusion are discussed in Chap. 11. 

In Figs. 4.1 and 4.2, the broken lines do not represent the sample paths of the process X(t), but join the outcoming states of the system observed at a discrete set of times f, t2,.. . , tn. To understand the behavior of X(t), it is necessary to know the transition probability. In Fig. 4.3 are given numerical simulations of a Wiener process W(t) (Brownian motion) and a Cauchy process C(t), both supposed one dimensional, stationary, and homogeneous. Their transitions functions are defined... 

In order to take particle-particle interactions into account, a stability ratio W is included which relates the collision kernel /So to the aggregation kernel /3agg. The stability ratio W depends on the interaction potential aggregation rate without to the rate with interactions additional to the omnipresent van der Waals forces. For Brownian motion as dominant reason for collisions, the stability ratio W can be calculated according to Eq. (6) taken from Fuchs [ 10]. In case of shear as aggregation mechanism, the force dip/dr relative to the friction force should rather be considered instead of the ratio of interaction energy relative to thermal energy. 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Settling of particles less than 0.5 pm is slowed by Brownian motion (random motion of small particles from thermal effects) in the water. Conversely, large sand-sized particles are not affected by viscous forces and typically generate a frontal pressure or wake as they sink. Thus, Stokes law can only apply to particles with Reynolds numbers (Re) that are less than unity. The particle Reynolds number according to Allen (1985) is defined as follows ... 

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