Distribution function, particle

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... 

The one-particle distribution function fp specifies both the total particle density p and momentum density pu, where u is the average fluid velocity. 

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 ... 

If three-body collisions are neglected, which is permitted at sufficiently low densities, all the interactions take place between pairs of particles the two-particle distribution function will, therefore, satisfy Liouville s equation for two interacting particles. For /<2)(f + s) we may write Eq. (1-121) ... 

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... 

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

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. 

These two expressions are exact they allow us in principle to calculate the N-particle distribution function at time t (to the first order in the external field) if its initial value is known. This will be our starting point for analyzing electrolytes both at equilibrium and out of equilibrium. 

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... 

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. 

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... 

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. 

G. Nienhuis. Theory of quantum corrections to the equation of state and the particle distribution function. J. Math. Phys., 11 239, 1970. 

The situation is similar to the one encountered in the kinetic theory of dilute plasmas. 510 To lowest order in the density+) the one-particle distribution function of the electrons obeys the Vlasov equation. The next order approximation consists of two coupled equations for the one-particle and two-particle distribution functions. On the other hand, in the kinetic theory of gases Bogolyubovft) has proposed an... 

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... 

The three-particle distribution function g3(r,s) can be expressed in a series of Legendre polynomials [63]. Then expressing the Legendre polynomials in terms of spherical harmonics, we can write the expression for g3(r,s) as... 

The BE was found intuitively. Here, the hypothesis of molecular chaos, that is, the assumption that any pair of particles enters the collision process uncorrelated (statistical independence) is the most important one. Only this assumption allows the formulation of a closed equation for the single-particle distribution function. 

Equation (1.24) is very similar to that of the single-particle distribution function of classical statistical mechanics. In the limit h—>0 we get the first equation of the BBGKY hierarchy. 

Yu. L. Klimontovich, Dissipative equations for many particle distribution functions (in Russian), Ukrain. Fiz. Zh. 139, 689 (1983). 

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. 

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