## Computing equilibrium measures with power law kernels

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Timon S. Gutleb, José A. Carrillo and Sheehan Olver
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## Abstract:

We introduce a method to numerically compute equilibrium measures for problems with attractive-repulsive power law kernels of the form $K(x-y) = \frac {|x-y|^\alpha }{\alpha }-\frac {|x-y|^\beta }{\beta }$ using recursively generated banded and approximately banded operators acting on expansions in ultraspherical polynomial bases. The proposed method reduces what is naïvely a difficult to approach optimization problem over a measure space to a straightforward optimization problem over one or two variables fixing the support of the equilibrium measure. The structure and rapid convergence properties of the obtained operators result in high computational efficiency in the individual optimization steps. We discuss stability and convergence of the method under a Tikhonov regularization and use an implementation to showcase comparisons with analytically known solutions as well as discrete particle simulations. Finally, we numerically explore open questions with respect to existence and uniqueness of equilibrium measures as well as gap forming behaviour in parameter ranges of interest for power law kernels, where the support of the equilibrium measure splits into two intervals.## References

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## Additional Information

**Timon S. Gutleb**- Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
- MR Author ID: 1384319
- ORCID: 0000-0002-8239-2372
- Email: t.gutleb18@imperial.ac.uk
**José A. Carrillo**- Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- ORCID: 0000-0001-8819-4660
- Email: carrillo@maths.ox.ac.uk
**Sheehan Olver**- Affiliation: Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
- MR Author ID: 783322
- ORCID: 0000-0001-6920-0826
- Email: s.olver@imperial.ac.uk
- Received by editor(s): October 30, 2020
- Received by editor(s) in revised form: August 28, 2021
- Published electronically: June 14, 2022
- Additional Notes: The second author was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363). The third author was supported by the Leverhulme Trust Research Project Grant RPG-2019-144. The second and third authors were also supported by the Engineering and Physical Sciences Research Council (EPSRC) grant EP/T022132/1.

The first author is the corresponding author - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2247-2281 - MSC (2020): Primary 65N35; Secondary 65R20, 65K10
- DOI: https://doi.org/10.1090/mcom/3740
- MathSciNet review: 4451462