Particle distribution evolution

The evolution of the one-particle distribution function fs can be described by the Boltzmann equation... 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

In this section we shall explain somewhat the results which we have just presented. We are interested this time in the evolution equation for the one-particle distribution function. We write down the virial series expansion of the transport equation and we recall that every contribution to this equation is proportional to V n+d, where n is the number of particles which are involved... 

Because particles of different sizes are distributed throughout the bulk randomly, developing an exact model that couples diffusion to particle size evolution is daunting. However, a mean-field approximation is reasonable because diffusion near a spherical sink (see Section 13.4.2) has a short transient and a steady state characterized by steep concentration gradients near the surface. The particles act as independent sinks in contact with a mean-field as in Fig. 15.2. 

The constructed system of equations is a closed one. It is solved with the preset initial conditions 6j (r — 0), 0 jg(, t — 0), 6i (2, t = 0). The system of equations makes it possible to describe arbitrary distributions of particles on a surface and their evolution in time. The only shortcoming is the large dimension. The minimal fragment of a lattice on which a process with cyclic boundary conditions should be described is 4 x 4. It is, therefore, natural to raise the question of approximating the description of particle distribution to lower the dimension of the system of equations. In this connection, it is reasonable to consider simpler point-like models. 

The next steps are the following Step 1 Passage to the entropy representation and specification of the dissipative thermodynamic forces and the dissipative potential E. Step 2 Specification of the thermodynamic potential o. Step 3 Recasting of the equation governing the time evolution of the np-particle distribution function/ p into a Liouville equation corresponding to the time evolution of np particles (or p quasi-particles, Up > iip —see the point 4 below) that then represent the governing equations of direct molecular simulations. 

The length of the first quasi-stationary stage and relaxation time reaching the third equilibrium stage increases as a nontrivial power-type function of degrees of freedom N, namely x NlJ. This time scale is also confirmed by observing temporal evolution of single-particle distribution of momenta. The distributions taken at the same scaled time (i.e., t/NlJ = constant) are well-superposed irrespective of values of N, while a trivial time scale x N is numerically excluded. 

In this section the population balance modeling approach established by Randolph [95], Randolph and Larson [96], Himmelblau and Bischoff [35], and Ramkrishna [93, 94] is outlined. The population balance model is considered a concept for describing the evolution of populations of countable entities like bubble, drops and particles. In particular, in multiphase reactive flow the dispersed phase is treated as a population of particles distributed not only in physical space (i.e., in the ambient continuous phase) but also in an abstract property space [37, 95]. In the terminology of Hulburt and Katz [37], one refers to the spatial coordinates as external coordinates and the property coordinates as internal coordinates. The joint space of internal and external coordinates is referred to as the particle phase space. In this case the quantity of basic interest is a density function like the average number of particles per unit volume of the particle state space. The population balance may thus be considered an equation for the number density and regarded as a number balance for particles of a particular state. 

As soon as the movements of different species have to be distinguished from each other, however, the mutual correlation of the molecules in singlefile systems makes it impossible to predict the evolution of the particle distributions by differential equations. Eor this reason, the time dependence of the tracer exchange in single-file systems has thus far only been investigated by Monte Carlo simulations [1,55-57]. 

Our understanding of diffusion and reaction in single-file systems is impaired by the lack of a comprehensive analytical theory. The traditional way of analytically treating the evolution of particle distributions by differential equations is prevented by the correlation of the movement of distant particles. One may respond to this restriction by considering joint probabilities covering the occupancy and further suitable quantities with respect to each individual site. These joint probabilities may be shown to be subject to master equations. 

Particles distributed according to their mass (or volume) are frequently encountered in applications. The size reduction of solid materials is an example of such a breakage process. The evolution of drop size distributions... 

Wavefunctions describing time-dependent states are solutions to Schrodinger s time-dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors. 

The LBM is similar to the LGA in that one performs simulations for populations of computational particles on a lattice. It differs from the LGA in that one computes the time evolution of particle distribution functions. These particle distribution functions are a discretized version of the particle distribution function that is used in Boltzmann s kinetic theory of dilute gases. There are, however, several important differences. First, the Boltzmann distribution function is a function of three continuous spatial coordinates, three continuous velocity components, and time. In the LBM, the velocity space is truncated to a finite number of directions. One popular lattice uses 15 lattice velocities, including the rest state. The dimensionless velocity vectors are shown in Fig. 66. The length of the lattice vectors is chosen so that, in one time step, the population of particles having that velocity will propagate to the nearest lattice point along the direction of the lattice vector. If one denotes the distribution function for direction i by fi x,t), the fluid density, p, and fluid velocity, u, are given by... 

The particle size evolution Equations 18.19-18.22 are a set of three coupled partial 2D integro-differential equations in radius and time. No analytical solution has been found for such systems and the numerical solution of such a set of equations is an extremely challenging task. One efficient method is to discretize these equations with respect to radius and convert the evolution equations into a set of coupled ordinary differential equations for each radius r i). That is, the continuous distribution can be broken down into G discrete groups of particles as iUushated in Figure 18.2 [34],... 

As the density of the fluid is increased above values for which a linear term in the expansion of equation (5.1) is adequate (crudely above values for which a third virial coefficient is adequate to describe the compression factor of a gas), the basis of even the formal kinetic theory is in doubt. In essence, the difficulty arises because it becomes necessary, at higher densities, to consider the distribution function, in configuration and momentum space, of pairs, triplets etc. of molecules in order to formulate an equation for the evolution of the single-particle distribution function. Such an equation would be the generalization to higher densities of the Boltzmann equation, discussed in Chapter 4 (Ferziger Kaper 1972 Dorfman van Beijeren 1977). 

Xiao, Y. C., Wang, K. Y., Chung, T. S., and Ta, J. N. (2006). Evolution of nano-particle distribution during the fabrication of mixed matrix Ti02-polyimide hollow fiber membranes. Chem. Eng. Set 61, 6228. 

Better description of particle distributions in the present atmosphere will not in itself suffice to predict future distributions the evolution of ice particles in upper tropospheric conditions involves physical and chemical processes in temperature and humidity regimes that have not been quantitatively explored in the laboratory. Particle surface effects are important in most of the phenomena we have discussed, suggesting that traditional laboratory studies of processes... 

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