Multi-particle distribution function

Reduction of the number of independent parameters in the evolution of an indignant system proceeds continuously as a result of parameter synchronization and the determination of the correlative bonds between them. However, taking into account the difference between partial processes and their respective relaxation times, separate states of evolution can be noted, every one of which is described by its own number of independent parameters changing through transition from one state to another. For example, after the finishing time r(/V from the start of a single-component system, a so-called kinetic state of a process takes place [11,12] (ro and V are characteristic size of a particle and heat rate of their moving, respectively is an indication of particle interaction time under collision, in the order of 10" -10s). In this state a state of a system is fully determined by a partial distribution function tU] that rules by the temporary evolution of a system. Multi-partial distribution function and as a result a full one represent a function of (Oil... 

The second assumption has far-reaching consequences if the solute dynamics is not coupled to the structural relaxation of the polymer, the problem becomes much easier-instead of solving a formidable dynamic multi-body problem one describes the behavior and properties of the solute with a time-independent single-particle distribution function p(r), thus reducing the problem to that of an ideal gas subjected to an external field stationary in time. 

In quantum theory, identical particles must be indistinguishable in order for the theory to predict results that agree with experimental observations. Consequently, as shown in Section 8.1, the wave functions for a multi-particle system must be symmetric or antisymmetric with respect to the interchange of any pair of particles. If the wave functions are not either symmetric or antisymmetric, then the probability densities for the distribution of the particles over space are dependent on how the particles are labeled, a property that is inconsistent with indistinguishability. It turns out that these wave functions must be further restricted to be either symmetric or antisymmetric, but not both, depending on the identity of the particles. 

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X ", U ", and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X " 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have... 

In particular, the ensemble of a new phase during phase separation proceeding in a multi-component system belongs to such systems. In this case, the turbidity spectrum method provides the determination of the mass-volume new-phase particle concentration, as well as the degree of phase transition this can be used to construct the molecular-mass polymer distribution function (the method of spectroturbidimetric titration for polymer solutions—subsection 3.2.3) and for a phase analysis (identification) of the pheise separation type in polymer. systems (paragraph 3.6.2.5, sections 6.2 and 6.4). 

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