Particle 394 Subject

It is left to reader to verify that, under Lee .s discrete mechanics, both free particles and particles subjected to a constant force, behave in essentially the sa e way as they do under continuous equations of motion. Moreover, the time intervals At = t-i i — ti are all equal. While the spatial behavior for non-constant forces (ex particles in a harmonic oscillator V potential) also remains essentially... 

K r,f,r) = e r e " ° r) which neglects all quantum effects arising from the noncommutativity of the operators and v. In order to appreciate the nature of the approximation, let us consider the case where the energy potential v(r) = v + A r, with v and A constant quantities. Although the QMP for a particle subjected to a constant force is one of the few cases explicitly known [32], for our convenience we adopt the following exact alternative representation of the propagator for a particle moving in a linear potential [see Appendix A, eq. (A.8)]... 

Borkent BM, Arora M, Ohl CD, Jong ND, Versluis M, Lohse D, Morch KA, Klaseboer E, Khoo BC (2008) The acceleration of solid particles subjected to cavitation nucleation. J Fluid Mech 610 157-182... 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

For small particles, subject to noncontinuum effects but not to compressibility, Re is very low see Eq. (10-52). In this case, nonradiative heat transfer occurs purely by conduction. This situation has been examined theoretically in the near-free-molecule limit (SI4) and in the near-continuum limit (T8). The following equation interpolates between these limits for a sphere in a motionless gas ... 

Kotzick, R., U. Panne, and R. Niessner, Changes in Condensation Properties of Ultrafine Carbon Particles Subjected to Oxidation by Ozone, J. Aerosol Sci, 28, 725-735 (1997). 

Pugnaloni, L.A., Ettelaie, R., Dickinson, E. (2005). Brownian dynamics simulation of adsorbed layers of interacting particles subjected to large extensional deformation. Journal of Colloid and Interface Science, 287, 401 114. 

The one-dimensional particle-in-a-box problem is that of a single particle subject to the following potential-energy function ... 

A free particle is a particle subject to no forces, so that V = 0 everywhere. For a free particle moving in one dimension, the Schrodinger equation is (1.122) and its solution is [Eq. (1.98)]... 

Next consider the same Brownian particle, subject to an additional constant force, say a gravitational field Mg in the direction of — X. If we write My for the friction of the particle in the surrounding fluid, it will now receive an average drift velocity —g/y. This is superimposed on the Brownian... 

Exercise. Argue that an overdamped particle subject to an external force with potential U X) is described by ... 

Consider a Brownian particle subject to a force F(X) depending on the position. The obvious generalization of the Fokker-Planck equation (3.5)... 

Exercise. Find the probability distribution of a free particle subject to dichotomic kicks x = p, p = (/) = two-valued Markov process, when x and p are given initially. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

GEMC utilizes two simulation subsystems ( boxes ) though physically separate, the two boxes are thermodynamically coupled through the MC algorithm, which allows them to exchange both volume and particles subject to the constraint that the total volume and number of particles remain fixed. Implementing these updates (in a way that respects detailed balance) ensures that the two systems will come to equilibrium at a common temperature, pressure, and chemical potential. The temperature is fixed explicitly in the MC procedure but the procedure itself selects the chemical potential and pressure that will secure equilibrium. 

The anisotropy in g(ri, r2) may be determined by the use of akinematic argument. Consider a bulk of particle subjected to a mean shear flow. The radial distribution function, which is spherical in equilibrium, becomes distorted into an ellipsoidal distribution as a result of the presence of the mean shear. Hence, in order for g(n, r2) to exhibit an anisotropy, g (ri, r2) should depend not only on ap, r, and r2 but also on Tc, vi, and v2. For dimensional homogeneity, g can only be a function of ap, k U2i/Tcl/2, and U x/Tc. For a small deformation rate (or when the magnitude of U2i is small relative to Tc1/2), it is assumed that g(ri, r2) takes the form [Jenkins and Savage, 1983]... 

The dynamic response of a magnetic particle subject to an applied field follows from the Gilbert equation of motion [7]... 

To study electrophoresis of particles subject to an external electric field, one needs to know the electrical potential, fluid flow and ion fluxes around the particle. In this section, we first present the fundamental electrokinetic equations for electrophoresis of colloidal particles. Previous studies on the electrophoresis of a single particle will then be reviewed, and important results will be stated. 

Figure 8-39 Flat Plane Projection of the Location of a Colloidal Particle Subject to Brownian Movement. Source From H. Schubert, Food Particle Technology. Part 1 Properties of Particles and Particulate Food Systems, J. Food Eng., Vol. 6, pp. 1-32,1987, Elsevier Applied Science Publishers, Ltd.

This general technique was employed to simulate the motion of a mono-layer of identical spherical particles subjected to a simple shear. Confinement to a monolayer represents tremendous economies of computer time compared with a three-dimensional simulation. Hopefully, these highly specialized monolayer results will provide comparable insights into the physics of three-dimensional suspensions. Periodic boundary conditions were used in the simulation, and the method of Evans (1979) was incorporated to reproduce the imposed shear. 

For micrometer-sized particles subject to steric- or lift-hyperlayer-FFF, the driving forces are higher (10 14 to 10 8 N per particle) but are not balanced by back diffusion as in the normal FFF mode. Steric- and lift-hyperlayer-FFF provide powerful means for the investigation of hydrodynamic lift forces [79]. Here, retention times have been measured for well-characterized particles such as latex spheres under widely varying conditions, and the hydrodynamic lift force FL has been determined. 

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