A -particle distribution function

Fixed pressure boundary conditions are implemented by assigning the equilibrium distribution functions, computed with zero velocity and specified density at the reservoirs, to the distribution functions. Periodic boundary conditions are applied in the span-wise directions. A particle distribution function bounce-back scheme57 is used at the walls to obtain no-slip boundary condition. 

At this point, it is convenient to introduce the functional derivative. The A-particle distribution function can be written as a A " order functional derivative, so together with a graphical interpretation of the action of a functional derivative, this is the easiest method of derivation. Thefunctional F... 

The A-particle distribution function gives the probability of finding at time t the tV ions in the positions ri,..., rjv regardless of the momenta and positions of... 

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. 

Flere g(r) = G(r) + 1 is called a radial distribution function, since n g(r) is the conditional probability that a particle will be found at fif there is another at tire origin. For strongly interacting systems, one can also introduce the potential of the mean force w(r) tln-ough the relation g(r) = exp(-pm(r)). Both g(r) and w(r) are also functions of temperature T and density n... 

Modeling the pore size in terms of a probability distribution function enables a mathematical description of the pore characteristics. The narrower the pore size distribution, the more likely the absoluteness of retention. The particle-size distribution represented by the rectangular block is the more securely retained, by sieve capture, the narrower the pore-size distribution. 

Second-Order Integral Equations for Associating Fluids As mentioned above in Sec. II A, the second-order theory consists of simultaneous evaluation of the one-particle (density profile) and two-particle distribution functions. Consequently, the theory yields a much more detailed description of the interfacial phenomena. In the case of confined simple fluids, the PY2 and HNC2 approaches are able to describe surface phase transitions, such as wetting and layering transitions, in particular see, e.g.. Ref. 84. 

On a hexagonal lattice, for example, the two-particle distribution function, is... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Assuming that our LG is in a local equilibrium, it is reasonable to expect that the one-particle distribution functions should depend only on the macroscopic parameters u x,t) and p x,t) and their derivatives [wolf86c]. While there is no reason to believe that this dependence should be a particularly simple one, it is reasonable to expect that both u and p are slowly varying functions of x and t. Moreover, in the subsonic limit, we can assume that li << 1. 

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 ... 

This binary collision approximation thus gives rise to a two-particle distribution function whose velocities change, due to the two-body force F12 in the time interval s, according to Newton s law, and whose positions change by the appropriate increments due to the particles velocities. 

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... 

Section II deals with the general formalism of Prigogine and his co-workers. Starting from the Liouville equation, we derive an exact transport equation for the one-particle distribution function of an arbitrary fluid subject to a weak external field. This equation is valid in the so-called "thermodynamic limit , i.e. when the number of particles N —> oo, the volume of the system 2-> oo, with Nj 2 = C finite. As a by-product, we obtain very easily a formulation for the equilibrium pair distribution function of the fluid as well as a general expression for the conductivity tensor. 

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. 

As is shown in Eqns. 2-48 and 2-49, the probability density W(e) of electron energy states in the reductant or oxidant particles is represented as a normal distribution function (Gaussian distribution) centered at the most probable electron level (See Fig. 2-39.) as expressed in Eqns. 8-10 and 8-11 ... 

A particle distribution weighting function p y y,y, At) represents the mean time a particle resides at a position y during its move from y to y in a time step At, and it is given by... 

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. 

Criticize or defend the following proposition The data give the time required for particles to fall 20 cm, making it easy to convert time to sedimentation velocity for each point. Equation (11) may then be used to convert the velocity into the radius of an equivalent sphere. The resulting graph of W versus radius is a cumulative distribution function similar to that shown in Figure 1.18b. 

Suppose a polydisperse system is investigated experimentally by measuring the number of particles in a set of different classes of diameter or molecular weight. Suppose further that these data are believed to follow a normal distribution function. To test this hypothesis rigorously, the chi-squared test from statistics should be applied. A simple graphical examination of the hypothesis can be conducted by plotting the cumulative distribution data on probability paper as a rapid, preliminary way to evaluate whether the data conform to the requirements of the normal distribution. 

Now return to the system of N particles, the distribution function for which, pN, changes frequently because of a large number of successive instantaneous collisions. Each collision causes the N-particle distribution function to change, because of the change of position and velocity of the two hard spheres which collide. The effect of all these collisions is additive and each instantaneously alters the distribution, pN. The Liouville equation for hard spheres is... 

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... 

Particle Size Distribution. Methods for determining the particle size of pigments should provide not only the mean particle size but also the complete particle size distribution. Thus, it is preferable to use methods that give not only the parameters of a particular distribution function, but also allow direct measurement of the true particle size distribution. Closely specified standard methods of determination exist for particle sizes > 1 pm. During recent years several methods have been developed for the particle size region of particular relevance to pigments (< 1 pm). The following methods are used ... 

Unfortunately, this expansion cannot be used as a basis for the development of approximate methods since - unlike the superposition approximation -in the case of considerable spatial correlation, neglect of the forms b(m m > mo leads to the correlation functions not satisfying the proper boundary conditions and increase of mo does not lead to the convergence of results. A comparison of the two kinds of expansion of the many-particle distribution function demonstrates that the superposition approximation even for small mo corresponds to the choice in the additive expansion of b 0 with any m. Therefore, in terms of the latter expansion the many-particle correlation forms are not neglected in the superposition approximations but are no longer independent. 

However, Waite s approach has several shortcomings (first discussed by Kotomin and Kuzovkov [14, 15]). First of all, it contradicts a universal principle of statistical description itself the particle distribution functions (in particular, many-particle densities) have to be defined independently of the kinetic process, but it is only the physical process which determines the actual form of kinetic equations which are aimed to describe the system s time development. This means that when considering the diffusion-controlled particle recombination (there is no source), the actual mechanism of how particles were created - whether or not correlated in geminate pairs - is not important these are concentrations and joint densities which uniquely determine the decay kinetics. Moreover, even the knowledge of the coordinates of all the particles involved in the reaction (which permits us to find an infinite hierarchy of correlation functions = 2,...,oo, and thus is... 

Due to the rapid decrease in the process probability with increase of the distance between the reagents, it should be expected that reaction (13) will result in electron transfer primarily to the particle A which is nearest to the excited donor particle D. In this case, the condition n < N is satisfied for reaction (13), where n is the concentration of the particles D and N is that of the particles A, and with the random initial distribution of the particles, A, the distribution function over the distances in the pairs D A formed, will have the same form [see Chap. 4, eqn. (13)] as with the non-paired random distribution under the conditions when n IV. In such a situation the kinetics of backward recombination of the particles in the pairs D A [reaction (12)] will be described by eqn. (24) of Chap.4 which coincides with eqn. (35) of Chap. 4 for electron tunneling reactions under a non-paired random distribution of the acceptor particles. Therefore, in the case of the pairwise recombination via electron tunneling considered here, the same methods of determining the parameters ve and ae can be applied as those described in the previous section for the case of the non-pair distribution. However, examples of the reliable determination of the parameters ve and ae for the case of the pairwise recombination using this method are still unknown to us. 

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