Independent Brownian motions

In the special case of uncorrelated Brownian Motions, together with the substitution T = r — f, we hnd the ODE (7.6) given hy 

As a general solution of this differential equation we have  

plugging the boundary condition D,(0,z) = 0 in equation (9.5) leads to 

Together with the similar change of variables we find for the ODE (7.7) 

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Videomicroscopy offers possibilities that other techniques do not, such as the ability to visualize three-dimensional clusters in which cluster membership is determined by the particle dynamic properties, e.g., whether the particles are fast- or slow-moving(33). Videomicroscopy also allows measurement of quantities not directly accessible from other techniques, such as the van Hove self- and distinct distribution functions (x, t) and Gj(x, t) of the particle displacements. The Gs(x, t) gives the likelihood that a particle will have a displacements during a time t Gd(x, t) gives the likelihood that if one particle is at the origin at time 0 then a different particle will be at s at time t. A dilute bidisperse system in which particles of two different sizes all perform independent Brownian motion would have d.Gs(x, t) that was a sum of two Gaussians, a field correlation function t)... 

Theoretical models of the film viscosity lead to values about 10 times smaller than those often observed [113, 114]. It may be that the experimental phenomenology is not that supposed in derivations such as those of Eqs. rV-20 and IV-22. Alternatively, it may be that virtually all of the measured surface viscosity is developed in the substrate through its interactions with the film (note Fig. IV-3). Recent hydrodynamic calculations of shape transitions in lipid domains by Stone and McConnell indicate that the transition rate depends only on the subphase viscosity [115]. Brownian motion of lipid monolayer domains also follow a fluid mechanical model wherein the mobility is independent of film viscosity but depends on the viscosity of the subphase [116]. This contrasts with the supposition that there is little coupling between the monolayer and the subphase [117] complete explanation of the film viscosity remains unresolved. 

Independent bacterial motion is a true movement of translation and must be distinguished from the quivering or back-and-forth motion exhibited by very small particles suspended in a liquid. This latter type of motion is called Brownian movement and is caused by the bombardment of the bacteria by the molecules of the suspending fluid. 

An interesting aspect of Eq. (4.24) is that, even though F (t) and C (/) must represent genuine Brownian motions, / (/)may represent non-Brownian, even cyclic, motions of the antenna in the rod frame. Motion of the coordinates o>R t) and 2R(t) has been assumed to be statistically independent of the rod motions, but the nature of their trajectories has not yet been specified. 

The content of diffusion equation (2.175) for such a model is, moreover, independent of our choice of a system of 3,N coordinates for the unconstrained space. Constrained Brownian motion may thus be described by a model with a mobility and an effective potential /eff in any system of 3N coordinates for... 

This law, independently found by Einstein (1905) and the Polish physicist Smolu-chowski when they studied the Brownian motion of small particles, became an important clue for proving the real existence of atoms and molecules as discrete items of mass which move randomly. 

The formulation of the Fokker-Planck equation is due to Fokker s and Planck s independent works on the description of the Brownian motion of particles [17, 18]. Commonly, an N variables equation of the type... 

What is the size of the overlapping volume The complete independence of the constant in Equation 1 from the degree of polymerization shows that the overlapping volume always consists of the same portion of the volume of the polymer coil (12). This can be easily understood by assuming that two polymer coils are able to migrate nearly unhindered through each other. Then the mean depth of permeation and, therefore, the time of overlapping is determined only by the statistics of the free Brownian motion. Equation 1 is based on this assumption. 

In the pulmonary region, air velocities are too low to impact particles small enough to reach that region, and the mechanisms of deposition are sedimentation and Brownian diffusion. The efficiency of both processes depends on the length of the respiratory cycle, which determines the stay time in the lung. If the cycle is 15 breaths/min, the stay time is of the order of a second. Table 7.1 shows the distance fallen in one second and the root mean square distance travelled by Brownian diffusion in one second by unit density particles (Fuchs, 1964). Sedimentation velocity is proportional to particle density, but Brownian motion is independent of density. Table 7.1 shows that sedimentation of unit density particles is more effective in causing deposition than Brownian diffusion when dp exceeds 1 pm, whereas the reverse is true if dp is less than 0.5 pm. For this reason, it is appropriate to use the aerodynamic diameter dA equal to pj dp when this exceeds 1 pm, but the actual diameter for submicrometre particles. 

Lazer Zee microcapillary electrophoresis, and the Pen Kem Model 3000, which measures in a automated manner according to the Brownian motion of the particles. The results all showed the same constant electrophoretic mobility over the pH range 5-10, independent of the type of surface group. 

When aerosols are in a flow configuration, diffusion by Brownian motion can take place, causing deposition to surfaces, independent of inertial forces. The rate of deposition depends on the flow rate, the particle diffusivity, the gradient in particle concentration, and the geometry of the collecting obstacle. The diffusion processes are the key to the effectiveness of gas filters, as we shall see later. 

Suspensions and true solutions do not exhibit Brownian movement. It was seen that Brownian motion was independent of the nature of the colloidal particles but was more rapid when the particles are smaller and the solution is less viscous. 

In 1851, Stokes derived Eq. (4.1) from the model of solid spherical particles falling independently through a homogeneous liquid without Brownian motion, slippage, and wall effects. Slippage is an inconstant rate of fall wall effects refer to axial orientation in the outermost planes of fluid in contact with a surface, and the differential velocity of flow in the outermost and innermost planes of a fluid in a confining tube ... 

Under the assumption that the target and incoming floccules engage in Brownian motion independently,18 D = Dm + Dn, and the rate coefficient becomes... 

As pointed out earlier, the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Bom repulsion forces are included in the analysis of the relative motion between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the intermolecular potential, modelled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix I of (1). The Brownian motions of the two particles are no longer independent because of the interaction force between the two. It is, therefore, necessary to describe the relative motion between the two particles in order to predict the rate of collision and of subsequent coagulation. 

For non-ideal polymer solutions where there are interactions between the polymer molecules, Einstein and Debye showed independently that if the solute is uniformly distributed throughout the solution, no light is scattered by the solution because light scattered by one particle will interfere destructively with light scattered by the neighbouring particle. Random Brownian motion causes fluctuations in concentration, the extent of fluctuations is inversely proportional to the osmotic pressure developed by the concentration difference. It is found that... 

Equation 6.33 states that the root-mean-square displacement is proportional to the square root of the number of jumps. For very large values of n, the net displacement of any one atom is extremely small compared to the total distance it travels. It turns out, that the diffusion coefficient is related to this root-mean-square displacement. It was shown independently by Albert Einstein (1879-1955) and Marian von Smoluchowski (1872-1917) that, for Brownian motion of small particles suspended in a liquid, the root-mean-square displacement, is equal to V(2Dt), where t is the time... 

Brownian motion in an oscillator potential. We consider a Brownian motion of Langevin type with a damping constant /S independent of position, so that the residual random acceleration behaves as white noise, independent of both position and velocity. The Langevin equation is... 

Brownian Motion of a Rigid Body. Many molecules of interest are extended in three dimensions with three principal moments of inertia of comparable size. Gyroscopic forces complicate the discussion of Brownian motion for such bodies and a number of workers have developed convenient formal treatments without much physical novelty emerging. Steele has given a clear treatment of this problem, but obtains tractable expressions only for spherical-top molecules, which have the same moment of inertia I about all axes through the molecular mass centre, so that the Euler equations of motion fall apart in independent variables. 

In general, 0) will be given by the time-independent particle density p(r). For Brownian motion, the function cftir, t) will obey Pick s law, where the spatial derivatives refer to the coordinate r. Consequently, we may write... 

If the energy barrier to aggregation is removed (e.g., by adding excess electrolyte) then aggregation is diffusion controlled only Brownian motion of independent droplets or particles is present. For a monodisperse suspension of spheres, Smoluchowski developed an equation for this rapid coagulation ... 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

Figure 11 shows the relative-viscosity-concentration behavior for a variety of hard-sphere suspensions of uniform-size glass beads. Even though the particle size was varied substantially (0.1 to 440 xm), the relative viscosity is independent of the particle size. However, when the particle diameter was small ( 1 fJLm), the relative viscosity was calculated at high shear rates, so that the effect of Brownian motion was negligible. Figure 8 shows that becomes independent of the particle size at high shear stress (or shear rate). 

A mathematical model of a nanoparticles growth during evaporation of a micron size droplet in a low pressure aerosol reactor is developed. The main factor is found to be evaporating cooling of droplets which affects formation of supersaturated solution in the droplet. The rate of cooling can reach 2T0 K/s. The final radius of nanoparticles was found to be independent on the precursor radius. Manifestation of Lifshitz-Slezov instability is illustrated by experimental data. Effects of Brownian motion of nanoparticles inside the droplet are discussed. 

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

Formula (483) was first obtained by Albert Einstein (1879-1955) in 1905 and bears his name. Independently of Einstein, the theory of the Brownian motion was developed by Marian von Smoluchowski (1872-1917) in 1905-1906. The expression obtained by him agrees with formula (483) with a constant multiplier equal to one. 

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