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<p>Scaling analysis and experimentation have been advanced by the recent arrival of the <i>finite-similitude</i> theory with the proven existence of an unlimited number of similitude rules. The rules facilitate the exact transfer of information across any number of scaled experiments and additionally have been shown to capture to high accuracy behaviours in many scientific fields including fracture mechanics, fluid dynamics, and electromagnetism. The theory applies to all classical physics and has recently been linked to an extended form of dimensional analysis, which in principle facilitates the applicability of the approach to all quantitative disciplines. Despite the extensive breadth of applicability of the new scaling approach, no work has been published on the application of the similitude rules to stochastic dynamical systems. The focus of this paper is on establishing whether it is possible to scale stochastic systems under the constraints imposed by the new finite-similitude theory. Pathological functions that are nowhere differentiable and constrained by It<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10665_2025_10458_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(\hat{\textrm{o}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mtext>o</mtext> <mo stretchy="false">^</mo> </mover> </math></EquationSource> </InlineEquation> or Stratonovich calculus and associated stochastic differential equations (SDEs) are investigated under the new scaling theory. It is confirmed that scaling is possible with SDEs taking on a standard form on a scaling space that is a projection of the trial space, where the scaled system resides. Noisy mechanical and electrical systems are examined to showcase the reach and benefits of the new approach. Its efficacy is demonstrated using stochastic simulations where the trial-scale projections successfully replicated the statistical behaviour of full-scale systems.</p>

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The scaling of stochastic physical systems with applications in electrodynamics

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月14日

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  • Keith Davey,
  • Raul Ochoa-Cabrero,
  • Zainab Safaa Ali,
  • Jiahe Xu

摘要

Scaling analysis and experimentation have been advanced by the recent arrival of the finite-similitude theory with the proven existence of an unlimited number of similitude rules. The rules facilitate the exact transfer of information across any number of scaled experiments and additionally have been shown to capture to high accuracy behaviours in many scientific fields including fracture mechanics, fluid dynamics, and electromagnetism. The theory applies to all classical physics and has recently been linked to an extended form of dimensional analysis, which in principle facilitates the applicability of the approach to all quantitative disciplines. Despite the extensive breadth of applicability of the new scaling approach, no work has been published on the application of the similitude rules to stochastic dynamical systems. The focus of this paper is on establishing whether it is possible to scale stochastic systems under the constraints imposed by the new finite-similitude theory. Pathological functions that are nowhere differentiable and constrained by It \(\hat{\textrm{o}}\) o ^ or Stratonovich calculus and associated stochastic differential equations (SDEs) are investigated under the new scaling theory. It is confirmed that scaling is possible with SDEs taking on a standard form on a scaling space that is a projection of the trial space, where the scaled system resides. Noisy mechanical and electrical systems are examined to showcase the reach and benefits of the new approach. Its efficacy is demonstrated using stochastic simulations where the trial-scale projections successfully replicated the statistical behaviour of full-scale systems.