A pair problem

For some time it was believed that the virial expansion allows to describe the critical point, but this hope failed. The point is that at high densities typical for the liquid phase, the most important terms of the expansion are those which describe the formation of a large cluster of interacting particles (the many-particle effect) whereas an approach based on the system s treatment as an ensemble of interacting pairs fails. From the point view of such a pair approach, formation of the ordered crystalline structure is a complete puzzle. 

In the chemical kinetics a pair problem is the first preliminary step in studying the many-particle problem its results cannot be overestimated and 

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At first sight, the problem just discussed seems to be symmetric, i.e., should be true also for a problem of the catching a single toper by a group of policemen - Fig. 1.13 that is all we need is to solve a pair problem. However, it is not the case except that the toper is immobile, Da = 0 (the toper is dead-drunk). In the latter particular case the problem indeed could be analytically solved at long time the probability that the toper did not meet the policemen is proportional to exp(-kta) with the exponent a equal to the classical value of unity for d 2 but a = 1/2 if d = 1. 

Up to now we neglected dynamical interaction of particles. In a pair problem it requires the use of the potential U(r) = Uab( ) specifying the A-B interaction in an ensemble of different particles interaction of similar particles described by additional potentials C/aa ), bb( ), could be essential. However, incorporation of such dynamic interactions makes a problem unsolvable analytically for any diffusion coefficients, analogously to the situation known in statistical physics of condensed matter. 

Equation (4.1.38) is obviously constructed in such a way that it correctly reproduces the solution of a pair problem. 

Despite the fact that formalism of the standard chemical kinetics (Chapter 2) was widely and successfully used in interpreting actual experimental data [70], it is not well justified theoretically in fact, in its derivation the solution of a pair problem with non-screened potential U (r) = — e2/(er) is used. However, in the statistical physics of a system of charged particles the so-called Coulomb catastrophes [75] have been known for a long time and they have arisen just because of the neglect of the essentially many-particle charge screening effects. An attempt [76] to use the screened Coulomb interaction characterized by the phenomenological parameter - the Debye radius Rd [75] does not solve the problem since K(oo) has been still traditionally calculated in the same pair approximation. 

This BLAST Quickstart chapter illustrates the use of the principal BLAST programs to solve problems that arise in the analysis of protein and nucleotide sequences. Each section provides a succinct description of a protocol with two problems that serve as practical examples. Relevant theory is given when it affects the selection of a search strategy or search parameter, however, the emphasis is on the procedure itself. The sections follow closely the structure of the BLAST QuickStart Mini-Course found at www.ncbi.nlm.nih.gov/Class/minicourses. The BLAST QuickStart is one of 10 2-h format Mini-Courses offered by NCBI on campus at the National Institutes of Health and at locations around the country to over 4000 students a year. The courses use a paired problems approach in which the first of two similar problems or problem sets is solved by the instructor during the first hour on a computer linked to a projection system, while the students watch in the second hour, the students tackle the second problem, or set of problems at their... 

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