Well-Posedness of the Iterative Boltzmann Inversion
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The iterative Boltzmann inversion is a fixed point iteration to determine an effective pair potential for an ensemble of identical particles in thermal equilibrium from the corresponding radial distribution function. Although the method is reported to work reasonably well in practice, it still lacks a rigorous convergence analysis. In this paper we provide some first steps towards such an analysis, and we show under quite general assumptions that the associated fixed point operator is Lipschitz continuous (in fact, differentiable) in a suitable neighborhood of the true pair potential, assuming that such a potential exists. In other words, the iterative Boltzmann inversion is well-defined in the sense that if the kth iterate of the scheme is sufficiently close to the true pair potential then the \(k+1\)st iterate is an admissible pair potential, which again belongs to the domain of the fixed point operator. On our way we establish important properties of the cavity distribution function and provide a proof of a statement formulated by Groeneveld concerning the rate of decay at infinity of the Ursell function associated with a Lennard-Jones type potential.
KeywordsStatistical mechanics Cluster expansion Grand canonical ensemble Radial distribution function Cavity distribution function Fréchet derivative
Mathematics Subject Classification82B21 82B80
The results of this paper have first been presented at the Oberwolfach Mini-Workshop Cluster Expansions: From Combinatorics to Analysis through Probability (February 2017). The author is indebted to Roberto Fernández, Sabine Jansen, and Dimitrios Tsagkarogiannis for the invitation and the opportunity to contribute this presentation. During the workshop David C. Brydges, Aldo Procacci, and Daniel Ueltschi provided arguments which have considerably simplified our original proof of Theorem 5.2. This input and many discussions with further participants of this workshop are gratefully acknowledged.
- 3.Groeneveld, J.: Rigorous bounds for the equation of state and pair correlation function of classical many-particle systems. In: Bak, T.A. (ed.) Statistical Mechanics. Foundations and Applications, pp. 110–145. Benjamin, New York (1967)Google Scholar
- 4.Groeneveld, J.: Estimation methods for Mayer’s graphical expansions. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 229–259. Academic Press, New York (1967)Google Scholar
- 5.Hanke, M.: Fréchet differentiability of molecular distribution functions I. \(L^\infty \) analysis. Lett. Math. Phys. (2017). https://doi.org/10.1007/s11005-017-1009-0
- 6.Hanke, M.: Fréchet differentiability of molecular distribution functions II. The Ursell function. Lett. Math. Phys. (2017). https://doi.org/10.1007/s11005-017-1010-7
- 16.Stell, G.: Cluster expansions for classical systems in equilibrium. In: Fritsch, H.L., Lebowitz, J.L. (eds.) The Equilibrium Theory of Classical Fluids, pp. II-171–II-266. Benjamin, New York (1964)Google Scholar