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PreprintCC0arXiv

Geometric Thermodynamics of Scallop Motion with Two Control Parameters

  • Hisao Hayakawa
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Abstract

According to Purcell's scallop theorem, reciprocal single-degree-of-freedom shape deformations cannot achieve net propulsion in a viscous fluid. We show that this limitation is bypassed by thermal fluctuations in a two-parameter driven potential landscape. Formulating the stochastic shape dynamics via a Smoluchowski equation with position-dependent mobility Meff(x), we utilize a generalized inverse operator to evaluate the slow-driving response. Cyclic modulation of the control parameters induces a non-zero Berry-Sinitsyn-Nemenman curvature F12(θ), resulting in directed geometric propulsion. Simultaneously, the non-adiabatic excess dissipation is dictated by a Riemannian thermodynamic metric gij(θ). Our results provide a unified geometric foundation that bridges hydrodynamic friction, stochastic mechanics, and thermodynamic trade-offs in micro-swimmers.

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Cite

BibTeXRIS
BibTeX
@misc{hayakawa2026geometric,
  title = {{Geometric Thermodynamics of Scallop Motion with Two Control Parameters}},
  author = {Hisao Hayakawa},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.24158},
  eprint = {2608.24158},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.24158},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Geometric Thermodynamics of Scallop Motion with Two Control Parameters
AU  - Hisao Hayakawa
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.24158
UR  - https://arxiv.org/abs/2608.24158
AB  - According to Purcell's scallop theorem, reciprocal single-degree-of-freedom shape deformations cannot achieve net propulsion in a viscous fluid. We show that this limitation is bypassed by thermal fluctuations in a two-parameter driven potential landscape. Formulating the stochastic shape dynamics via a Smoluchowski equation with position-dependent mobility M_eff(x), we utilize a generalized inverse operator to evaluate the slow-driving response. Cyclic modulation of the control parameters induces a non-zero Berry-Sinitsyn-Nemenman curvature F_12(θ), resulting in directed geometric propulsion. Simultaneously, the non-adiabatic excess dissipation is dictated by a Riemannian thermodynamic metric g_ij(θ). Our results provide a unified geometric foundation that bridges hydrodynamic friction, stochastic mechanics, and thermodynamic trade-offs in micro-swimmers.
ER  -