https://doi.org/10.1140/epjs/s11734-025-02017-9
Regular Article
Casimir versus Helmholtz fluctuation-induced force in the Nagle–Kardar model: exact results
1
Institute of Mechanics, Bulgarian Academy of Sciences, Akad. G. Bonchev St. bl. 4, 1113, Sofia, Bulgaria
2
Institute of Solid State Physics, Bulgarian Academy of Sciences, 72, Tzarigradsko Chaussee, Blvd., 1784, Sofia, Bulgaria
3
Department of Physics and Astronomy, University of California, 90095, Los Angeles, CA, USA
a
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Received:
29
March
2025
Accepted:
10
October
2025
Published online:
19
October
2025
When used to describe finite systems, conjugate statistical-mechanical ensembles are not equivalent. This has physical implications for the behavior of the fluctuation-induced forces pertinent to the different ensembles. Here, we study the Nagle–Kardar model within the grand-canonical ensemble (GCE) and the canonical ensemble (CE) (with conserved total magnetization) for periodic boundary conditions (PBC). We focus on two fluctuation-induced forces: the Casimir force (CF) in the GCE and the Helmholtz force (HF) in the CE. In the infinite system limit, the model exhibits a critical line, which ends at a tricritical point. Unexpectedly, the critical Casimir force (CCF) is repulsive near the critical line and tricritical point, decaying rapidly upon departure from those two regions and becoming attractive. This violates the widely accepted “boundary condition rule,” which presumes that the CCF is attractive for equivalent boundary conditions (BC) and repulsive for conflicting BC. For the HF, we find that it also changes sign as a function of temperature and magnetization. We conclude that CCF and HF have a behavior quite different from each other as a function of the tunable parameters (temperature, magnetic field, or magnetization) of the model. This dependence allows for the control of the sign of these forces, as well as their magnitude.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
