Particle dynamics stochastic formulation

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

In this chapter size effects in encounter and reaction dynamics are evaluated using a stochastic approach. In Section IIA a Hamiltonian formulation of the Fokker-Planck equation (FPE) is develojjed, the form of which is invariant to coordinate transformations. Theories of encounter dynamics have historically concentrated on the case of hard spheres. However, the treatment presented in this chapter is for the more realistic case in which the particles interact via a central potential K(/ ), and it will be shown that for sufficiently strong attractive forces, this actually leads to a simplification of the encounter problem and many useful formulas can be derived. These reduce to those for hard spheres, such as Eqs. (1.1) and (1.2), when appropriate limits are taken. A procedure is presented in Section IIB by which coordinates such as the center of mass and the orientational degrees of freedom, which are often characterized by thermal distributions, can be eliminated. In the case of two particles the problem is reduced to relative motion on the one-dimensional coordinate R, but with an effective potential (1 ) given by K(l ) — 2fcTln R. For sufficiently attractive K(/ ), a transition state appears in (/ X this feature that is exploited throughout the work presented. The steady-state encounter rate, defined by the flux of particles across this transition state, is evaluated in Section IIC. 

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