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Pseudospectral Methods and Critical Phenomena

  • Jake Skelton
  • Joseph Brader
  • Salomée Tschopp
  • Benjamin Goddard
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Abstract

We employ pseudospectral methods to solve the homogeneous Ornstein-Zernike (OZ) equation for a model fluid in the vicinity of the critical point. Focusing on the Mean-Spherical Approximation (MSA) as a closure to the OZ equation, we obtain numerical estimates for the critical exponents η, δ and γ for a system of hard-core Yukawa particles both in two and three dimensions. The three-dimensional MSA exponents are already well-known from an analytic solution and are recovered by our numerical methods. The two-dimensional exponents are a new output of this work. The pseudospectral method allows for rapid and highly accurate solution of liquid-state integral equation theories, and enables calculations on truely infinite domains, as needed for highly correlated states. In addition, we analyse the standard Picard iteration scheme and propose a variation of it which provides increased stability and speed of convergence.

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Cite

BibTeXRIS
BibTeX
@misc{skelton2026pseudospectral,
  title = {{Pseudospectral Methods and Critical Phenomena}},
  author = {Jake Skelton and Joseph Brader and Salomée Tschopp and Benjamin Goddard},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03627},
  eprint = {2609.03627},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03627},
}
RIS
TY  - UNPB
TI  - Pseudospectral Methods and Critical Phenomena
AU  - Jake Skelton
AU  - Joseph Brader
AU  - Salomée Tschopp
AU  - Benjamin Goddard
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03627
UR  - https://arxiv.org/abs/2609.03627
AB  - We employ pseudospectral methods to solve the homogeneous Ornstein-Zernike (OZ) equation for a model fluid in the vicinity of the critical point. Focusing on the Mean-Spherical Approximation (MSA) as a closure to the OZ equation, we obtain numerical estimates for the critical exponents η, δ and γ for a system of hard-core Yukawa particles both in two and three dimensions. The three-dimensional MSA exponents are already well-known from an analytic solution and are recovered by our numerical methods. The two-dimensional exponents are a new output of this work. The pseudospectral method allows for rapid and highly accurate solution of liquid-state integral equation theories, and enables calculations on truely infinite domains, as needed for highly correlated states. In addition, we analyse the standard Picard iteration scheme and propose a variation of it which provides increased stability and speed of convergence.
ER  -