Mathematical analysis of the complete iterative inversion method — I

Abstract

A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas.The complete iterative inversion method is a technique employed in an attempt to recover the pair potential from the second virial coefficient. Implicit in the complete iterative inversion method is the requirement that various mathematical expressions are meaningful: improper integrals converge, derivatives exist, etc. We provide a mathematical framework in which all these implicit assumptions are valid. We show that the complete iterative inversion method cannot recover the pair potential even if the target potential and the initial estimate are infinitely differentiable.Keywords Second virial coefficient · Spherical intermolecular potential · Equations of state for a gas · Integral equation · Iterative inversion. Mathematics Subject Classification (2000)45G (Nonlinear integral equations), 47J06 (Nonlinear ill-posed problems), 82B40 (Kinetic theory of gases).

Mathematical Analysis of the Complete Iterative Inversion Method II

“…Therefore, in order to advance our investigation, we shall henceforth consider only those functions B = B ϕ (T ) which are the second virial coefficient of some admissible potential ϕ as in (1.9). For the remainder of this work we use the Boltzmann temperature scale defined as in [4] byT = kT where T denotes the temperature in degrees kelvin. This has the practical effect of setting k = 1 in (1.2), (1.3), (1.9), (1.11), and (1.16).…”

“…It follows (see the final paragraph of [4]) that lim n→∞ ϕ n (r ) = ϕ 0 (r ) > ϕ(r ) (1.14) for all r > r 0 , thus violating (1.13). That is, the claim for the CIIM fails even if the target and initial potentials are C ∞ functions on (0, ∞).…”

“…This suggests a purely mathematical inversion problem: Can ϕ be reconstructed from (1.2) if the values of B(T ) are known for T > 0? In a previous work [4], we reviewed research on this question by other investigators before making some modest contributions of our own. We will briefly summarize these results for the convenience of readers.…”

“…8) It has been shown (see Theorem 3.1 in [4]) that the second virial coefficient of an admissible potential ϕ, given by…”

“…In [4] it was shown that there exist admissible potentials ϕ 0 and ϕ, analytic and C ∞ , respectively, on (0, ∞) which have precisely the same second virial coefficient function and for which there exists r 0 > 0 such that ϕ 0 (r ) > ϕ(r ) for all r > r 0 . It follows (see the final paragraph of [4]) that lim n→∞ ϕ n (r ) = ϕ 0 (r ) > ϕ(r ) (1.14) for all r > r 0 , thus violating (1.13).…”