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arxiv: 2509.26274 · v3 · pith:KUO7VUAMnew · submitted 2025-09-30 · 🧮 math.NA · cond-mat.str-el· cs.NA

Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics

Andreas A. Buchheit , Jonathan K. Busse , Torsten Ke{\ss}ler , Filipp N. Rybakov This is my paper

Pith reviewed 2026-05-18 12:07 UTC · model grok-4.3

classification 🧮 math.NA cond-mat.str-elcs.NA
keywords periodic boundary conditionslong-range interactionszeta functionsmicromagneticsRiesz potentialsdipolar interactionsnumerical summationdemagnetization field
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The pith

Infinite lattice sums for dipolar and power-law interactions can be computed exactly using a small direct sum plus zeta function derivative corrections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to compute exact infinite sums for long-range interactions like dipolar forces and Riesz potentials between bodies in periodic arrays without truncating the lattice. By adding a correction based on derivatives of generalized zeta functions to a small direct sum, the method achieves machine precision efficiently. This addresses a key challenge in micromagnetics simulations under periodic boundary conditions, where truncation errors have been common. A superexponentially convergent algorithm for evaluating the zeta functions and related special functions like incomplete Bessel functions makes this practical.

Core claim

For general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. The resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes.

What carries the argument

zeta expansion correction term involving derivatives of generalized zeta functions

Load-bearing premise

The required derivatives of generalized zeta functions admit a superexponentially convergent algorithm that remains practical and stable for the geometries and derivative orders needed in micromagnetics.

What would settle it

Direct comparison of the zeta-corrected sum against an extremely large truncated summation for a specific periodic cuboidal geometry, verifying agreement to machine precision.

Figures

Figures reproduced from arXiv: 2509.26274 by Andreas A. Buchheit, Filipp N. Rybakov, Jonathan K. Busse, Torsten Ke{\ss}ler.

Figure 1
Figure 1. Figure 1: Sketch illustrating a 3D system of of Nx × Ny × Nz = 5 × 3 × 4 interacting cuboids repeated infinitely in x direction (a), and both in x and y direction (b). The green cuboid is influenced by the red cuboid indicated by the arrow, as well as by the total influence of all opaque red cuboids complex shapes by representing them as composed of simplices such as tetrahedrons. At the same time, it is challenging… view at source ↗
Figure 2
Figure 2. Figure 2: Relative error between the three-dimensional potential U (0,0,2)(r) centered at r = (1, 1, 1)/2 for the lattice at Z2 × {0} as obtained by the zeta representation in Theorem 2.3 in comparison with direct summation over the truncated grid {−Ncut, . . . , Ncut} 2× {0} as dots a function of the the truncation length integers 1 ≤ Ncut ≤ 10. The error scaling CνN −ν cut for some fitted parameter Cν is shown as … view at source ↗
Figure 3
Figure 3. Figure 3: Cuboid Ω = [1/(2N), 1/(2N)]3 and its infinite repeti￾tions z + Ω along the two-dimensional lattice embedded in three dimensions L = Z2 × {0} for N = 2 (a), N = 5 (b) and N = 10 (c). On the other hand, for N → ∞, the cuboid is small compared to the lattice spacing, such that it interacts with its repetitions as a point dipole. The magnetic field contribution from the copies is expressed through a lattice su… view at source ↗
Figure 4
Figure 4. Figure 4: Difference between demagnetization factor Dz for 3D cuboids Ω = [−1/(2N), 1/(2N)]3 interacting with a lattice Λ = Z2 , computed from our method, and the known asymptotic formula. The asymptotic behavior is correctly reproduced, as well as the corner cases (Dz = 1 for N = 1 and Dz → 1/3 for N → ∞). The next order correction from the known asymptotics as N → ∞ scales as N −7 (obtained by a fit). 4. Computati… view at source ↗
Figure 5
Figure 5. Figure 5: In what follows, we discuss the procedures employed in Algorithm 1 and [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Regions of Algorithm 1. The white line indicates the split, where above the dashed white line we employ the reciprocal values. The purple region where the upper bound of the incomplete Bessel function is smaller than 10−16 is shown for ν = 0. and the statement follows from the symmetry Ks(z) = K−s(z) [31, Sec. 7.2.2, Eq. (14)]. □ For small vector arguments, we may now use the series expansion in the second… view at source ↗
Figure 6
Figure 6. Figure 6: Cuboid Ω centered in the Cartesian coordinate sys￾tem (x, y, z). We consider a cuboid-shaped source (domain Ω) of size a × b × c placed in the center of the Cartesian coordinate system, see [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

We address the efficient computation of power-law-based interaction potentials of homogeneous $d$-dimensional bodies with an infinite $n$-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields and related quantities can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with sufficiently large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a zeta-expansion technique for computing exact infinite lattice sums of power-law (Riesz) and dipolar interactions for homogeneous bodies under 1D or 2D periodic boundary conditions. The central construction supplements a small direct sum over nearby images with a correction assembled from derivatives of generalized zeta functions and incomplete Bessel functions; a superexponentially convergent algorithm is supplied for these special functions. The resulting scheme is asserted to reach machine precision at a cost comparable to truncation, with benchmarks against analytic formulas and direct summation for large exponents, plus new corrections to the demagnetization-field asymptotics and tabulated high-precision reference values for micromagnetic solvers.

Significance. If the convergence and numerical-stability claims hold, the work supplies a practical, controllable-precision alternative to truncation for long-range periodic sums in micromagnetics and molecular dynamics. The explicit identification of new asymptotic corrections and the release of high-precision benchmark tables constitute concrete, reusable contributions to the field. The generality to arbitrary cuboids and to both dipolar and Riesz potentials broadens the potential impact beyond the immediate application.

major comments (2)
  1. [§3] §3 (algorithm for generalized zeta derivatives): the claim that the representation converges exponentially in derivative order and reaches machine precision is load-bearing for the central result, yet no a-priori bounds on conditioning, round-off propagation, or stability of the recurrence relations are given for derivative orders 10–30 in 3D cuboids with 1D/2D periodicity; this directly addresses the weakest assumption identified in the stress test.
  2. [§5] §5 (numerical benchmarks): while full precision is reported against analytic formulas and direct summation, the tables do not list the derivative orders employed, the lattice spacings tested, or any observed condition numbers, preventing independent verification that the method remains stable for the geometries and Riesz exponents required in micromagnetics.
minor comments (2)
  1. The notation for the incomplete Bessel functions and the precise definition of the generalized zeta function could be collected in a single preliminary subsection to improve readability for readers outside the immediate special-functions community.
  2. Figure captions for the convergence plots should explicitly state the Riesz exponent, periodicity dimensions, and derivative order used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding numerical stability analysis and the completeness of benchmark reporting are well taken, and we address them point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (algorithm for generalized zeta derivatives): the claim that the representation converges exponentially in derivative order and reaches machine precision is load-bearing for the central result, yet no a-priori bounds on conditioning, round-off propagation, or stability of the recurrence relations are given for derivative orders 10–30 in 3D cuboids with 1D/2D periodicity; this directly addresses the weakest assumption identified in the stress test.

    Authors: We acknowledge that the manuscript does not supply a-priori bounds on conditioning or round-off propagation for the recurrence relations at high derivative orders. The central construction relies on the observed superexponential convergence, which in practice limits the required orders to moderate values (typically below 20) for machine precision. Extensive numerical experiments confirm stability in double precision across the relevant geometries and periodicity settings. In the revised manuscript we will add a dedicated paragraph in §3 summarizing these stability observations, including representative condition-number estimates for derivative orders up to 30, while noting that a complete theoretical error analysis lies beyond the present scope. revision: partial

  2. Referee: [§5] §5 (numerical benchmarks): while full precision is reported against analytic formulas and direct summation, the tables do not list the derivative orders employed, the lattice spacings tested, or any observed condition numbers, preventing independent verification that the method remains stable for the geometries and Riesz exponents required in micromagnetics.

    Authors: We agree that the benchmark tables would benefit from explicit reporting of the derivative orders, lattice spacings, and observed condition numbers. This information will be added to all tables in §5, together with a short note on the parameter ranges chosen to match typical micromagnetics applications. These updates will allow readers to reproduce and verify the stability claims directly. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via standard zeta-function identities

full rationale

The paper obtains the exact periodic lattice sum by writing it as a small direct sum plus a correction built from derivatives of generalized zeta functions (and incomplete Bessel functions). This representation follows from classical Poisson-summation or Mellin-transform techniques applied to power-law kernels; the zeta functions themselves are independent special functions whose properties are not defined in terms of the target micromagnetic sum. No parameter is fitted to the output quantity, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The claimed superexponential convergence of the special-function algorithm is an independent numerical claim that does not presuppose the final result. Consequently the derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and efficient computability of derivatives of generalized zeta functions together with the exponential convergence of the resulting series for the geometries considered.

axioms (1)
  • standard math Generalized zeta functions and their derivatives admit a superexponentially convergent evaluation algorithm for the required arguments and orders.
    Invoked to justify the correction term and its practical cost; stated in the abstract as the basis for machine-precision results.

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