{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T21:28:29Z","timestamp":1760909309230,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T00:00:00Z","timestamp":1735171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003336","name":"Bulgarian National Science Fund","doi-asserted-by":"publisher","award":["KP-06-H72\/5"],"award-info":[{"award-number":["KP-06-H72\/5"]}],"id":[{"id":"10.13039\/501100003336","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>For finite systems the thermodynamic potentials are not equivalent. Thus, in any of them the fluctuations induce different forces: Casimir force (CF) in the grand canonical ensemble (GCE) and Helmholtz force (HF) in the canonical (CE) one with fixed magnetization. Here, we consider the HF within the Nagle\u2013Kardar model with periodic boundary conditions. The model represents a one-dimensional Ising chain with both nearest nearest-neighbor short-ranged and long-ranged interactions present in it. The nearest-neighbor interactions of strength Js could be either ferromagnetic (Js&gt;0), or antiferromagnetic (Js&lt;0). The long-ranged equivalent-neighbor ferromagnetic interactions are of strength Jl\/N&gt;0. In the thermodynamic limit the model exhibits in the (Ks=\u03b2Js,\u00a0Kl=\u03b2Jl) plane a second-order phase transition line 2Kl=exp\u22122Ks, with Ks&gt;\u2212ln(3)\/4, which ends at a tricritical point (Kl=3\/2,\u00a0Ks=\u2212ln(3)\/4). The obtained results demonstrate that the temperature behavior of the HF differs essentially from the one of the CF. Furthermore, we show that the CE and GCE are not equivalent, even in the thermodynamic limit. Finally, in the {Ks,Kl,m} phase space we determine the regions of stable states of the system.<\/jats:p>","DOI":"10.3390\/sym17010022","type":"journal-article","created":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T19:33:07Z","timestamp":1735241587000},"page":"22","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Fluctuation Induced Force in the Nagle\u2013Kardar Model in Ensemble with Conserved Magnetization"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4762-617X","authenticated-orcid":false,"given":"Daniel","family":"Dantchev","sequence":"first","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Academic Georgy Bonchev St. Building 4, 1113 Sofia, Bulgaria"},{"name":"Max-Planck-Institut f\u00fcr Intelligente Systeme, Heisenbergstrasse 3, D-70569 Stuttgart, Germany"},{"name":"Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA"}]},{"given":"Nicholay S.","family":"Tonchev","sequence":"additional","affiliation":[{"name":"Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria"}]},{"given":"Joseph","family":"Rudnick","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,26]]},"reference":[{"key":"ref_1","first-page":"793","article-title":"On the Attraction Between Two Perfectly Conducting Plates","volume":"51","author":"Casimir","year":"1948","journal-title":"Proc. K. Ned. Akad. Wet."},{"key":"ref_2","first-page":"207","article-title":"Ph\u00e9nom\u00e8nes aux parois dans un m\u00e9lange binaire critique","volume":"287","author":"Fisher","year":"1978","journal-title":"C. R. Seances Acad. Sci. Paris Ser. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1103\/RevModPhys.90.045001","article-title":"Collective behavior of colloids due to critical Casimir interactions","volume":"90","author":"Dietrich","year":"2018","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2022.12.004","article-title":"Critical Casimir effect: Exact results","volume":"1005","author":"Dantchev","year":"2023","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3212","DOI":"10.1039\/D3SM01408H","article-title":"Critical Casimir forces in soft matter","volume":"20","author":"Gambassi","year":"2024","journal-title":"Soft Matter"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Dantchev, D. (2024). On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results. Entropy, 26.","DOI":"10.3390\/e26060499"},{"key":"ref_7","first-page":"146","article-title":"Finite-size Scaling","volume":"Volume 8","author":"Domb","year":"1983","journal-title":"Phase Transitions and Critical Phenomena"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Privman, V. (1990). Finite-size scaling theory. Finite Size Scaling and Numerical Simulations of Statistical Systems, World Scientific.","DOI":"10.1142\/1011"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Brankov, J.G., Dantchev, D.M., and Tonchev, N.S. (2000). The Theory of Critical Phenomena in Finite-Size Systems\u2014Scaling and Quantum Effects, World Scientific.","DOI":"10.1142\/9789812813435"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1103\/PhysRevLett.83.1187","article-title":"Critical Fluctuation-Induced Thinning of 4He Films near the Superfluid Transition","volume":"83","author":"Garcia","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"086101","DOI":"10.1103\/PhysRevLett.88.086101","article-title":"Critical Casimir Effect near the 3He\u20134He Tricritical Point","volume":"88","author":"Garcia","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"075301","DOI":"10.1103\/PhysRevLett.97.075301","article-title":"Critical Casimir Force in 4He Films: Confirmation of Finite-Size Scaling","volume":"97","author":"Ganshin","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1038\/nature06443","article-title":"Direct measurement of critical Casimir forces","volume":"451","author":"Hertlein","year":"2008","journal-title":"Nature"},{"key":"ref_14","first-page":"271","article-title":"Tunable critical Casimir forces counteract Casimir\u2013Lifshitz attraction","volume":"19","author":"Schmidt","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"L042103","DOI":"10.1103\/PhysRevE.106.L042103","article-title":"Exact expressions for the partition function of the one-dimensional Ising model in the fixed-M ensemble","volume":"106","author":"Dantchev","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Dantchev, D.M., Tonchev, N.S., and Rudnick, J. (2023). Casimir versus Helmholtz forces: Exact results. Ann. Phys., 459.","DOI":"10.1016\/j.aop.2023.169533"},{"key":"ref_17","unstructured":"Dantchev, D., and Tonchev, N. (2024). A Brief Survey of Fluctuation-induced Interactions in Micro- and Nano-systems and One Exactly Solvable Model as Example. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"169647","DOI":"10.1016\/j.aop.2024.169647","article-title":"Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions","volume":"464","author":"Dantchev","year":"2024","journal-title":"Ann. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1406","DOI":"10.1103\/PhysRev.130.1406","article-title":"Ising model with a long-range interaction in presence of residual short-range interactions","volume":"130","author":"Baker","year":"1963","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1103\/PhysRevA.2.2124","article-title":"Ising chain with competing interactions","volume":"2","author":"Nagle","year":"1970","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1103\/PhysRevB.28.244","article-title":"Crossover to equivalent-neighbor multicritical behavior in arbitrary dimensions","volume":"28","author":"Kardar","year":"1983","journal-title":"Phys. Rev. B"},{"key":"ref_22","first-page":"345","article-title":"Statistical mechanics and dynamics of long-range interacting systems","volume":"4","author":"Patelli","year":"2013","journal-title":"Riv. Mat. Della Univ. Parma"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Campa, A., Dauxois, T., Fanelli, D., and Ruffo, S. (2014). Physics of Long-Range Interacting Systems, Oxford University Press. [1st ed.].","DOI":"10.1093\/acprof:oso\/9780199581931.001.0001"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1741018","DOI":"10.1142\/S0217751X17410184","article-title":"The world of long-range interactions: A bird\u2019s eye view","volume":"32","author":"Gupta","year":"2017","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_25","unstructured":"Husimi, K. (1953, January 14\u201324). Statistical mechanics of condensation. Proceedings of the International Conference on Theoretical Physics, Kyoto, Tokyo."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1088\/0370-1298\/67\/3\/306","article-title":"The Mayer theory of condensation tested against a simple model of imperfect gas","volume":"A 67","author":"Temperley","year":"1954","journal-title":"Proc. Phys. Soc. Lond."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4261","DOI":"10.1103\/PhysRevB.6.4261","article-title":"Spin model with antiferromagnetic and ferromagnetic interactions","volume":"6","author":"Hoye","year":"1972","journal-title":"Phys. Rev. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7638","DOI":"10.1103\/PhysRevB.37.7638","article-title":"Cayley-tree Ising-model with antiferromagnetic nearest-neighbor and ferromagnetic equivalent-neighbor interactions","volume":"37","author":"Kaufman","year":"1988","journal-title":"Phys. Rev. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"210","DOI":"10.5488\/CMP.5.210","article-title":"One-dimensional lattice gas","volume":"5","author":"Vieira","year":"1995","journal-title":"Condens. Matter Phys."},{"key":"ref_30","first-page":"1597","article-title":"Ferrimagnetism in a disordered Ising-model","volume":"4","author":"Paladin","year":"1994","journal-title":"J. Phys. I"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0304-8853(98)00362-X","article-title":"One-dimensional Ising model with long-range and random short-range interactions","volume":"192","author":"Vieira","year":"1999","journal-title":"J. Magn. Magn. Mater."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s100510051131","article-title":"One-dimensional Ising-like systems: An analytical investigation of the static and dynamic properties, applied to spin-crossover relaxation","volume":"15","author":"Boukheddaden","year":"2000","journal-title":"Eur. Phys. J. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"240604","DOI":"10.1103\/PhysRevLett.95.240604","article-title":"Breaking of ergodicity and long relaxation times in systems with long-range interactions","volume":"95","author":"Mukamel","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mukamel, D. (2009). Notes on the Statistical Mechanics of Systems with Long-Range Interactions. arXiv.","DOI":"10.1063\/1.2839123"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.physrep.2009.07.001","article-title":"Statistical mechanics and dynamics of solvable models with long-range interactions","volume":"480","author":"Campa","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"4389","DOI":"10.1016\/j.physa.2010.02.024","article-title":"Thermodynamics and dynamics of systems with long-range interactions","volume":"389","author":"Bouchet","year":"2010","journal-title":"Phys. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"50008","DOI":"10.1209\/0295-5075\/97\/50008","article-title":"Mean-field models with short-range correlations","volume":"97","author":"Ostilli","year":"2012","journal-title":"EPL"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"032120","DOI":"10.1103\/PhysRevE.92.032120","article-title":"Phase diagrams of a spin-1 Ising system with competing short- and long-range interactions","volume":"92","author":"Salmon","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1103\/PhysRevLett.51.1210","article-title":"Competing Criticality of Short- and Infinite-Range Interactions on the Cayley Tree","volume":"51","author":"Kardar","year":"1983","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"055002","DOI":"10.1088\/1751-8113\/48\/5\/055002","article-title":"Shared information in classical mean-field models","volume":"48","author":"Cohen","year":"2015","journal-title":"J. Phys. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2250100","DOI":"10.1142\/S0217984922501007","article-title":"Cooling a long-range interacting system faster via applying an external magnetic field","volume":"36","author":"Li","year":"2022","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"344002","DOI":"10.1088\/1751-8121\/ab2baf","article-title":"Ising chains with competing interactions in the presence of long-range couplings","volume":"52","author":"Campa","year":"2019","journal-title":"J. Phys. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1140\/epjb\/s10051-022-00452-4","article-title":"Effect of the nearest-neighbor biquadratic interactions on the spin-1 Nagle-Kardar model","volume":"95","author":"Yang","year":"2022","journal-title":"Eur. Phys. J. B"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.physa.2016.07.057","article-title":"The spin-3\/2 Blume-Capel model with competing short- and long-range interactions","volume":"464","author":"Salmon","year":"2016","journal-title":"Phys. A-Stat. Mech. Its Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"968","DOI":"10.1007\/s10773-021-04719-x","article-title":"Phase Diagram of the Nagel-Kardar Model in the Microcanonical-Canonical Crossover","volume":"60","author":"Yao","year":"2021","journal-title":"Int. J. Theor. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s10955-021-02809-y","article-title":"Computation of Microcanonical Entropy at Fixed Magnetization Without Direct Counting","volume":"184","author":"Campa","year":"2021","journal-title":"J. Stat. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1016\/j.cjph.2024.03.027","article-title":"Ensemble inequivalence in an extended spin-1 Nagle-Kardar model","volume":"89","author":"Yang","year":"2024","journal-title":"Chin. J. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1088\/0305-4470\/21\/1\/027","article-title":"Crossover between short-term and long-range interactions in the one-dimensional Ising-model","volume":"21","author":"Kislinsky","year":"1988","journal-title":"J. Phys. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"031128","DOI":"10.1103\/PhysRevE.84.031128","article-title":"Instability of the mean-field states and generalization of phase separation in long-range interacting systems","volume":"84","author":"Mori","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"P10003","DOI":"10.1088\/1742-5468\/2013\/10\/P10003","article-title":"Phase transitions in systems with non-additive long-range interactions","volume":"2013","author":"Mori","year":"2013","journal-title":"J. Stat. Mech: Theory Exp."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"122011","DOI":"10.1063\/5.0132478","article-title":"Thermal fluctuations in metastable fluids","volume":"34","author":"Gallo","year":"2022","journal-title":"Phys. Fluids"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"045304","DOI":"10.1103\/PhysRevE.109.045304","article-title":"Metastable and unstable hydrodynamics in multiphase lattice Boltzmann","volume":"109","author":"Lulli","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_53","unstructured":"Abramowitz, M., and Stegun, I.A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"L062104","DOI":"10.1103\/PhysRevE.110.L062104","article-title":"Finite-size Nagle\u2013Kardar model: Casimir force","volume":"110","author":"Dantchev","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Kardar, M. (2007). Statistical Physics of Fields, Cambridge University Press.","DOI":"10.1017\/CBO9780511815881"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:00:35Z","timestamp":1760115635000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,26]]},"references-count":55,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010022"],"URL":"https:\/\/doi.org\/10.3390\/sym17010022","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,12,26]]}}}