Brownian motion, definition

Molecular diffusion (or self-diffusion) is the process by which molecules show a net migration, most commonly from areas of high to low concentration, as a result of their thermal vibration, or Brownian motion. The majority of reactive transport models are designed to simulate the distribution of reactions in groundwater flows and, as such, the accounting for molecular diffusion is lumped with hydrodynamic dispersion, in the definition of the dispersivity. 

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... 

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... 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

Let the convective-diffusion-controlled region be defined as those sets of conditions (Pe, R, A/kT) for which the rate may he calculated to within 10% by ignoring London forces. Similarly, let the London-force-controlled region be those sets of conditions for which the rate may be calculated to within 10% by ignoring Brownian motion. These definitions suggest a method for determining limits for the regions. 

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

However, J. J. Thomson did not irrefutably establish the particulate nature of matter. It remained until 1909 for Jean Perrin to provide the definitive evidence for atoms, which he did by measuring the motion of microscopic pollen particles suspended in water. His detailed observations of this Brownian motion (named after the botanist Robert Brown) could be explained if it were assumed they were being buffeted about by moving atoms. His observations convinced the scientific community of the validity of the atomic model. Of course, they had been using the... 

When we use a time step of constant interval At (t = i At), the diffusion process can be expressed as a simple sequence of A% . This random walk model is convenient to express the normal Brownian motion [9,10]. The definition of Brownian motion,... 

However, this proposition is not valid for all biological systems because the diffusion occurs in an inhomogeneous space where various kinds of structures are interfering with the diffusing molecules. In a precise definition, the diffusion in a biological system is not true Brownian motion. 

This aspect of diffusion in inhomogeneous space is called anomalous diffusion [13-18] and ordinal diffusion which can be expressed as a Brownian motion is called normal diffusion or Euclid diffusion [14]. The definition of... 

The surface excess can be defined in various ways. Actually, there is no true dividing plane, but rather an AW interface that is not sharp, since molecules have a finite size and moreover exhibit Brownian motion. Flence the interface extends over a layer of some molecular diameters. In the derivation of Eq. (10.2), the position of the dividing plane has been chosen so that the surface excess of the solvent is zero. In Figure 10.5 the concentration of the solute is depicted as a function of the distance from the dividing plane (z). In Figure 10.5a, there is no adsorption the two hatched areas on either side of the dividing plane are equal. (Because of the definition... 

Assuming the particles undergo Brownian motion, the velocity is by definition an isotropic random variable described by a real Gaussian process of zero mean. Hence all the odd-ordered velocity correlation functions are zero, and the factorization property for a real Gaussian process can be used to determine the even ordered velocity correlations in terms of the second order velocity correlation function. Noting in particular that... 

The atomic theory of matter, which was conjectured on qualitative empirical grounds as early as the sixth century BC, was shown to be consistent with increasing experimental and theoretical developments since the seventeenth century AD, and definitely proven by the quantitative explanation of the Brownian motion by Einstein and Perrin early in the twentieth century [1], It then took no more than a century between the first measurements of the electron properties in 1896 and of the proton properties in 1919 and the explosion of the number of so-called elementary particles - and their antiparticles - observed in modern accelerators to several hundred (most of which are very short lived and some, not even isolated). Today, the standard model assumes all particles to be built from three groups of four basic fermions - some endowed with exotic characteristics - interacting through four basic forces mediated by bosons - usually with zero charge and mass and with integer spin [2],... 

The quantitative laws of chemical combination provide clear pointers to the molecular theory of matter, which increases progressively in vividness and realism with the application of Newton s laws to the motions of the particles. The interpretation of phenomena such as the pressure and viscosity of gases and the Brownian motion, and the assignment of definite magnitudes to molecular speeds, masses, and diameters render it clear that a continual interchange of energies must occur between the molecules of a material system, a circumstance which lies at the basis of temperature equilibrium and determines what in ordinary experience is called the flow of heat. It is responsible indeed for far more than this, and a large part of physical chemistry follows from the conception of the chaotic motion of the molecules. This matter must now be examined more deeply. 

Consider a problem on definition of collision frequency of small spherical particles executing Brownian motion in a quiescent liquid. In Section 8.2, Brownian motion was considered as diffusion with a effective diffusion factor. It was supposed that suspension is sufficiently diluted, so it is possible to consider only the pair interactions of particles. To simplify the problem, consider a bi-disperse system of particles, that is, a suspension consisting of particles of two types particles of radius ai and particles of radius a2. In this formulation, the problem was first considered by Smolukhowski [59]. 

As discussed in Chapter 3, each bead of the Rouse chain, under ceaseless collisions with fast moving small molecules and/or microstructural segments, is undergoing Brownian motion. At the same time the bead at Rn is affected by both the tensile forces on the nth and (n — l)th bonds. According to the same definition used in Eqs. (6.4) and (6.5), the total force asserted on the nth bead by the springs is expressed as... 

Molecular theory asserts that all matter is composed of molecules, with molecules made up of one or more atoms. What evidence do we have for the existence of molecules That is, why do we believe that matter is ultimately composed of lumps, rather than being continuous on all scales (For a review of the nineteenth-century debate on the discrete vs. continuous universe, see Nye [4].) One piece of evidence is the law of definite proportions the elements of the periodic table combine in discrete amounts to form compounds. Another piece of evidence is obtained by shining X rays on a crystalline solid the resulting diffraction pattern is an array of discrete points, not a continuous spectrum. More evidence is provided by Brownian motion see Figure 1.2. 

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