Transport processes Brownian motion

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. 

The inhalation airflow comes to a rest in the alveolar region. In still air, the collision of gas molecules with each other results in Brownian motion. The same happens with sufficiently small particles (which can be seen when the dust particles in a nonventilated room are hit by a sunbeam). For very small or ultrafine particles (when the particle size is similar to the mean free path length of the air molecules), the motion is not determined by the flow alone but also by the random walk called diffusion. The diffusion process is always associated with a net mass transport of particles from a region of high particle concentration to regions of lower concentration in accordance with the laws of statistical... 

Transport control of flocculation is realized in an especially direct way in the process known as diffusion-limited cluster-cluster aggregation5 (aggregation as used in this term means flocculation in the present chapter). In this process, which is straightforward to simulate and visualize on a computer, particles undergo Brownian motion (i.e., diffusion) until they come together in close proximity, after which they coalesce instantaneously and irreversibly to form floccules (or clusters ). The clusters then diffuse until they contact one another and combine to form larger clusters, and so on, until gravitational... 

Equation (8.2) can be shown to apply equivalently to either a continuous concentration field or the position probability density of a single particle undergoing Brownian motion [174], This equation is used to model transport processes in a wide range of natural phenomena from population distribution in ecology [146] to pollutant distribution in groundwater [30], One of the earliest (and still important) applications to transport within cells and tissues is to describe the transport of oxygen from microvessels to the sites of oxidative metabolism in cells. 

Particles also may stick to solid surfaces after diffusing through a stagnant air boundary layer above the solid. Diffusive transport results from the random motion of particles, often called Brownian motion (see Section 2.2.5, Fig. 2-11). This process may occur for particles that are too small to deposit effectively by impaction. 

The relationship between transport and fluctuations alluded to above is most easily introduced by using Brownian motion as an example. Brownian motion is clearly a random or stochastic process, and if we follow a particular particle, initially located at the origin, we may consider its position x t) at time t as a random variable (for simplicity, here we assume that the particle may move along one dimension only). In the section above, we considered Brownian motion as a random walk and found that the average squared distance traveled by a particle is proportional to time. Here we reinvestigate this process by using the diffusion equation. In fact, an argument due to Einstein reveals that... 

Microbial colonization of a solid—liquid interface may occur in the following sequence [69]. First, there is the transport to the cell surface. The next step is the initial adhesion, which is mainly a physicochemical process. Adhesion can be reversible or irreversible. Irreversibly adhering bacteria exhibit no Brownian motion and cannot be removed unless by a strong shear force. Adhesion is followed by firm attachment, which is reached by forming strong finks between the cells and the solid surface. The final sequence is the surface colonization. 

A precise determination of the frictional coefficient C in terms of the intermolecular potential and the radial distribution function at present constitutes the principal unresolved problem of the Brownian motion approach to liquid transport processes. It has been suggested by Kirkwood that an analysis of the molecular basis of self-diffusion might be a fruitful approach. The diffusion constant so calculated would be related to the frictional coefficient through the Einstein equation, Eq. 46. 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process dfiN)/dt = 0 at every point in phase space, where f N) is the instantaneous phase density of the N molecules. In principle, at least, it should thus be possible to express the steady-state pressure tensor and the mass and heat currents in terms of ensemble averages constructed without preliminary time-averaging in the replica systems. Thus it is desirable to examine the possibility of obtaining solutions to the reduced Liouville equation, Eq. 8, without preliminary timeaveraging. 

A common property of all cells Is motility, the ability to move in a specified direction. Many cell processes exhibit some type of movement at either the molecular or the cellular level all movements result from the application of a force. In Brownian motion, for Instance, thermal energy constantly buffets molecules and organelles in random directions and for very short distances. On the other hand, materials within a cell are transported in specific directions and for longer distances. This type of movement results from the mechanical work carried out by proteins that function as motors. We first briefly describe the types and general properties of molecular motors and then look at how one type of motor protein generates force for movement. 

A characteristic of the CDE travel time distribution is that the variance of the travel times grows linearly with travel distance z. This is equivalent to the particle location distribution, which grows linearly with time for a Brownian motion process. As such, it is essential in the derivation of (12) that the hydrodynamic dispersion can be described as a diffusion process, i.e. on average, all solute particles are subjected to the same forces and the transport time is sufficiently large so that the incremental microscopic particle displacements are no longer statistically correlated. As a corollary, the CDE process cannot be valid for small soil volumes where the travel times are too small as compared to the mixing time, or to describe transport close to interfaces. 

In the foregoing discussion of the Brownian motion method, the ensemble averages are all constructed from an ensemble of replica systems of the subset of h molecules, the behavior of each replica having been time-smoothed over the interval r. However, in a steady-state transport process = 0 at every point in... 

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