Estimates of general Mayer graphs. II. Long range behaviour of graphs with two root points occuring in the theory of ionized systems
We find the asymptotic behavior of general Mayer 2-graphs (Mayer graphs with two root points), which occur in the theory of ionized systems. This problem arises when one wants to compute corrections to the Debye length for large values of the plasma parameter. For a given 2-graph \(\Gamma\) (r) with Debye-Hückel lines \(e^{-r}/r\), we prove the inequalities \(C_ mr^{-\lambda}e^{-\lambda r}\leq \Gamma (r)\leq \Gamma (r_ 0)C_ Mr^{3k-1}e^{-\lambda r},\) for any \(r\geq r_ 0\), where \(C_ m\) and \(C_ M\) are positive and finite constants which depend only on \(\Gamma\). These bounds are finite whenever \(\Gamma\) (r) is not infinite everywhere. The integers l,k, and \(\lambda\) denote, respectively, the number of lines of the graph \(\Gamma\), its number of field points, and its local line connectivity (the maximum number of chains linking the root points, which have no line in common). From this result, we deduce that the simple irreducible 2-graphs dominant at large distances decay exponentially like \(e^{-r}\) and have an isthmus between the root points (an isthmus is a line whose deletion separates the graph into two disjoint components, each one containing a root point). We prove also that 2-graphs that have a number of lines \(l>3k+\lambda\) are infinite. We exhibit simple, irreducible prototypes satisfying this condition, for any \(k\geq 6\). This implies that the Abe-Meeron theory of ionized gases as applied to a classical plasma is not free from divergences. Finally, we extend the preceding results to 2-graphs with lines \(f_ L=(e^{- r}/r)^{k_ L}\), with \(k_ L\) real positive. We prove that they still decay exponentially like \(e^{-\lambda r}\), where \(\lambda\) is now the maximal flow in a network associated to \(\Gamma\) by assigning the capacity \(k_ L\) to each line L. [See also the joint review for parts I, III and IV below.]
scientific article; zbMATH DE number 3916325