Distribution functions single-particle

A novel approach (proposed by Skinner and Wolynes, 1978) for non-diffusive particle motion starts with an irreversible Liouville equation for a reduced phase space distribution function (single particle)... 

To describe the behavior of the multicomponent reactive granular material Lathouwers and Bellan started out from the work of Simonin [77] defining a single particle distribution function of particle type i such... 

To describe a chemical process in a fluid bed reactor Lathouwers and Bellan [61] introduced an extended form of the single-particle distribution function of particle phase i such that r, c, Y, t) with Y = u>c,T is the probable... 

In the derivation of the Boltzmann equation it is assumed that the distribution function changes only in consequence of completed collisions, i.e., the effect of partial collisions is neglected. We shall, therefore, consider the single-particle distribution function averaged23 over a time r, which will (later) be taken large compared with a collision time ... 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

We consider a simple reaction composed of only a single elementary step of reacting particles that obey the Boltzmann distribution function. Then, the reaction rate, v, is given in Eqn. 7-13 [Rysselberghe, 1963] ... 

Consider a collection of particles with energies ei, 62, S3,Si, each corresponding to a single quantum state, starting with the lowest lying state and proceedith to the ith state. The number of particles in each state can be written symbolically as Ao, Ai, A72, As,... A, The Boltzmann distribution function relating the relative occupancy of two states is written as ... 

Volume-normalized extinction is plotted in Fig. 11.2 as a function of photon energy for several polydispersions of MgO spheres both scales are logarithmic. For comparison of bulk and small-particle properties the bulk absorption coefficient a = Airk/X is included. Some single-particle features, such as ripple structure, are effaced by the distribution of radii. The information contained in these curves is not assimilated at a glance they require careful study. 

In the second half of this article, we discuss dynamic properties of stiff-chain liquid-crystalline polymers in solution. If the position and orientation of a stiff or semiflexible chain in a solution is specified by its center of mass and end-to-end vector, respectively, the translational and rotational motions of the whole chain can be described in terms of the time-dependent single-particle distribution function f(r, a t), where r and a are the position vector of the center of mass and the unit vector parallel to the end-to-end vector of the chain, respectively, and t is time, (a should be distinguished from the unit tangent vector to the chain contour appearing in the previous sections, except for rodlike polymers.) Since this distribution function cannot describe internal motions of the chain, our discussion below is restricted to such global chain dynamics as translational and rotational diffusion and zero-shear viscosity. 

As a measure of the relaxation of the single particle distribution function we define a function A co /) by... 

This function is the analogue of U2 introduced in the study of independent particle dynamics. The significance of Eqs. (48) and (50) is that the relaxation goes as a first order of p for both the single and two-particle density functions. In contrast, in the independent particle dynamics case the two-particle distribution function went to zero at a faster rate than did the single-particle distribution. A further, and more detailed comparison of the two types of dynamics must, therefore, be made in terms of three and... 

We can now calculate the rate of decay of initial correlations by making use of Eq. (69). The single-particle distribution function has the asymptotic behavior. 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

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