Well-Posedness of the Iterative Boltzmann Inversion

Hanke, Martin

doi:10.1007/s10955-017-1944-2

Cited by 15 publications

(35 citation statements)

References 13 publications

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“…for some C ą 0 which only depends on the upper bound q of }w k } L 1 pR 3 q , cf. [5]. Rewriting c as f˚pe´cq by virtue of (A.1) and (A.6), we conclude from (A.8) that…”

mentioning

confidence: 78%

“…Since the cavity distribution function in L 8 pR`q depends locally Lipschitz continuously on the pair potential in L 8 ̺ (see Proposition 3.1 in [5]) it follows that…”

Section: Well-posedness Of the Ihnc And Hncn Schemesmentioning

confidence: 99%

“…Soper, who redeveloped this scheme in [28], gave some heuristic arguments to support this observation. However, a rigorous convergence analysis is still lacking; see [5] for some preliminary results in this direction. A certain shortcoming of IBI is that it may require quite a few iterations to determine a sufficiently accurate potential.…”

Section: )mentioning

confidence: 99%

“…Appendix: The Wiener lemma. For ̺ defined in (2.3) we have shown in [5] that the space L 8 ̺ pR 3 q of all functions f : R 3 Ñ R, for which }f } L 8 ̺ pR 3 q " ess sup RPR 3 ̺p|R|qˇˇf pRqˇˇă 8 constitutes a Banach algebra with respect to convolution † . We can extend L 8 ̺ pR 3 q to a Banach algebra W ̺ with unit element e (given by the delta distribution at the origin), using the canonical norm }λe`f } W̺ " |λ|`γ}f } L 8 ̺ pR 3 q , λ P R, f P L 8 ̺ pR 3 q , where γ ą 0 is a small enough constant to make the norm of W ̺ submultiplicative.…”

Section: Numerical Discretizationmentioning

confidence: 99%

“…Since }w} L 1 pR 3 q has been shown to be less than 1, Corollary 4.3 in [5] allows to conclude that the series W Σ :" 8 ÿ n"1 W n (A.4) † Throughout this appendix we only consider functions of three variables, whether they be radial functions, or not. If f P L 8 ̺ pR 3 q is radially symmetric, then its representation defined in R`belongs to the Banach space L 8 ̺ introduced in (2.4).…”

Section: Numerical Discretizationmentioning

confidence: 99%

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A generalized Newton iteration for computing the solution of the inverse Henderson problem

Delbary

Hanke

Ivanizki

2020

Inverse Problems in Science and Engineering

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We develop a generalized Newton scheme IHNC for the construction of effective pair potentials for systems of interacting point-like particles. The construction is made in such a way that the distribution of the particles matches a given radial distribution function. The IHNC iteration uses the hypernetted-chain integral equation for an approximate evaluation of the inverse of the Jacobian of the forward operator.In contrast to the full Newton method realized in the Inverse Monte Carlo (IMC) scheme, the IHNC algorithm requires only a single molecular dynamics computation of the radial distribution function per iteration step, and no further expensive cross-correlations. Numerical experiments are shown to demonstrate that the method is as efficient as the IMC scheme, and that it easily allows to incorporate thermodynamical constraints.

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“…for some C ą 0 which only depends on the upper bound q of }w k } L 1 pR 3 q , cf. [5]. Rewriting c as f˚pe´cq by virtue of (A.1) and (A.6), we conclude from (A.8) that…”

mentioning

confidence: 78%

“…Since the cavity distribution function in L 8 pR`q depends locally Lipschitz continuously on the pair potential in L 8 ̺ (see Proposition 3.1 in [5]) it follows that…”

Section: Well-posedness Of the Ihnc And Hncn Schemesmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Numerical Discretizationmentioning

confidence: 99%

Section: Numerical Discretizationmentioning

confidence: 99%

See 3 more Smart Citations

A generalized Newton iteration for computing the solution of the inverse Henderson problem

Delbary

Hanke

Ivanizki

2020

Inverse Problems in Science and Engineering

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A variational framework for the inverse Henderson problem of statistical mechanics

Frommer

Hanke

2022

Lett Math Phys

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The inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. For a canonical ensemble in a bounded domain, it has been observed that this pair potential minimizes a corresponding convex relative entropy functional, and that the Newton iteration for minimizing this functional coincides with the so-called inverse Monte Carlo (IMC) iterative scheme. In this paper, we show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by Georgii and Zessin and a proper formulation of the IMC iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials, including, for example, Lennard-Jones-type potentials. It is further shown that the pressure is strictly convex as a function of the pair potential and the chemical potential, and that the specific relative entropy at fixed density is a strictly convex function of the pair potential. At a given reference potential and a corresponding density in the gas phase, we determine the gradient and the Hessian of the specific relative entropy, and we prove that the Hessian extends to a symmetric positive semidefinite quadratic functional in the space of square integrable perturbations of this potential.

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Cluster Expansions for GIBBS Point Processes

Jansen

2019

Adv. Appl. Probab.

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We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood-Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández and Procacci (2010). In addition, we provide formulas for double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

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Well-Posedness of the Iterative Boltzmann Inversion

Cited by 15 publications

References 13 publications

A generalized Newton iteration for computing the solution of the inverse Henderson problem

A generalized Newton iteration for computing the solution of the inverse Henderson problem

A variational framework for the inverse Henderson problem of statistical mechanics

Cluster Expansions for GIBBS Point Processes

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