{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T21:58:28Z","timestamp":1761429508004,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T00:00:00Z","timestamp":1734480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"\u201cGruppo Nazionale per la Fisica Matematica\u201d (GNFM)","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"European Union\u2014Next Generation EU","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"PNRR, National Center for HPC, Big Data and Quantum Computing\u2014M4C2\u2014I 1.4","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"CALL PRIN 2022 PNRR","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"PRIN","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"CALL PRIN 2022","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]},{"name":"\u201cIstituto Nazionale di Fisica Nucleare\u201d (INFN)","award":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"],"award-info":[{"award-number":["CN00000013","CUP D93C22000430001","2022XLBLRX","CUP D53D23006020006","P2022KHFNB","CUP D53D23018910001","P2022MXCJ2","CUP D53D23018940001","2022MKB7MM","CUP D53D23005900006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Multi-stable behavior at the microscopic length-scale is fundamental for phase transformation phenomena observed in many materials. These phenomena can be driven not only by external mechanical forces but are also crucially influenced by disorder and thermal fluctuations. Disorder, arising from structural defects or fluctuations in external stimuli, disrupts the homogeneity of the material and can significantly alter the system\u2019s response, often leading to the suppression of cooperativity in the phase transition. Temperature can further introduce novel effects, modifying energy barriers and transition rates. The study of the effects of fluctuations requires the use of a framework that naturally incorporates the interaction of the system with the environment, such as Statistical Mechanics to account for the role of temperature. In the case of complex phenomena induced by disorder, advanced methods such as the replica method (to derive analytical formulas) or refined numerical methods based, for instance, on Monte Carlo techniques, may be needed. In particular, employing models that incorporate the main features of the physical system under investigation and allow for analytical results that can be compared with experimental data is of paramount importance for describing many realistic physical phenomena, which are often studied while neglecting the critical effect of randomness or by utilizing numerical techniques. Additionally, it is fundamental to efficiently derive the macroscopic material behavior from microscale properties, rather than relying solely on phenomenological approaches. In this perspective, we focus on a paradigmatic model that includes both nearest-neighbor interactions with multi-stable (elastic) energy terms and linear long-range interactions, capable of ensuring the presence of an ordered phase. Specifically, to study the effect of environmental noise on the control of the system, we include random fluctuation in external forces. We numerically analyze, on a small-size system, how the interplay of temperature and disorder can significantly alter the system\u2019s phase transition behavior. Moreover, by mapping the model onto a modified version of the Random Field Ising Model, we utilize the replica method approach in the thermodynamic limit to justify the numerical results through analytical insights.<\/jats:p>","DOI":"10.3390\/e26121109","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T11:17:22Z","timestamp":1734520642000},"page":"1109","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Effects of Temperature and Random Forces in Phase Transformation of Multi-Stable Systems"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5499-2530","authenticated-orcid":false,"given":"Giuseppe","family":"Florio","sequence":"first","affiliation":[{"name":"Department of Civil, Environmental, Land, Building Engineering and Chemistry (DICATECh), Polytechnic University of Bari, Via Orabona 4, 70125 Bari, Italy"},{"name":"INFN, Section of Bari, 70126 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4023-5384","authenticated-orcid":false,"given":"Stefano","family":"Giordano","sequence":"additional","affiliation":[{"name":"University of Lille, CNRS, Centrale Lille, Universit\u00e9 Polytechnique Hauts-de-France, UMR 8520-IEMN-Institut d\u2019\u00c9lectronique de Micro\u00e9lectronique et de Nanotechnologie, F-59000 Lille, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6771-5495","authenticated-orcid":false,"given":"Giuseppe","family":"Puglisi","sequence":"additional","affiliation":[{"name":"Department of Civil, Environmental, Land, Building Engineering and Chemistry (DICATECh), Polytechnic University of Bari, Via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"025010","DOI":"10.1088\/2515-7639\/ab6e46","article-title":"Giant fluctuations in the flow of fluidised soft glassy materials: An elasto-plastic modelling approach","volume":"3","author":"Bertin","year":"2020","journal-title":"J. Phys. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1038\/s41699-017-0009-3","article-title":"Mechanics of thermally fluctuating membranes","volume":"1","author":"Los","year":"2017","journal-title":"Npj 2D Mater. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1038\/s41467-020-14996-5","article-title":"Soft three-dimensional network materials with rational bio-mimetic designs","volume":"11","author":"Yan","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"065003","DOI":"10.1103\/PhysRevE.106.065003","article-title":"Fluctuation-based fracture mechanics of heterogeneous materials","volume":"106","author":"Mulla","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"207203","DOI":"10.1103\/PhysRevLett.127.207203","article-title":"Experimental Realization of the 1D Random Field Ising Model","volume":"127","author":"Bingham","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1038\/s41467-017-01612-2","article-title":"Observation of replica symmetry breaking in disordered nonlinear wave propagation","volume":"8","author":"Pierangeli","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"M\u00e9zard, M., Parisi, G., and Virasoro, M.A. (1987). Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications, World Scientific.","DOI":"10.1142\/0271"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1007\/BF01009444","article-title":"Random External Fields","volume":"34","author":"Imry","year":"1984","journal-title":"J. Stat. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3580","DOI":"10.1103\/PhysRevB.19.3580","article-title":"Influence of quenched impurities on first-order phase transitions","volume":"19","author":"Imry","year":"1979","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1038\/35065675","article-title":"Crackling noise","volume":"410","author":"Sethna","year":"2001","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/S0370-1573(98)00008-8","article-title":"Collective transport in random media: From superconductors to earthquakes","volume":"301","author":"Fisher","year":"1998","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1729","DOI":"10.1103\/PhysRevE.61.1729","article-title":"Replica field theory for a polymer in random media","volume":"61","author":"Goldschmidt","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1103\/PhysRevLett.56.889","article-title":"Dynamic Scaling of Growing Interfaces","volume":"56","author":"Kardar","year":"1986","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1051\/jphys:0198800490120200900","article-title":"Statistics of a polymer in a random potential, with implications for a nonlinear interfacial growth model","volume":"49","author":"Cates","year":"1988","journal-title":"J. Phys. France"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Porter, D.A., Easterling, K.E., and Sherif, M. (2009). Phase Transformations in Metals and Alloys, CRC Press.","DOI":"10.1201\/9781439883570"},{"key":"ref_16","first-page":"555","article-title":"The electrical resistivity due to structural defects","volume":"24","author":"Martin","year":"1971","journal-title":"Philos. Mag. J. Theor. Exp. Appl. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1016\/j.matchar.2019.02.034","article-title":"Solid solution strengthening theories of high-entropy alloys","volume":"151","author":"LaRosa","year":"2019","journal-title":"Mater. Charact."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1002\/adem.200300567","article-title":"Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes","volume":"6","author":"Yeh","year":"2004","journal-title":"Adv. Eng. Mater."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1492","DOI":"10.1103\/PhysRev.109.1492","article-title":"Absence of diffusion in certain random lattices","volume":"109","author":"Anderson","year":"1958","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1103\/RevModPhys.58.801","article-title":"Spin glasses: Experimental facts, theoretical concepts, and open questions","volume":"58","author":"Binder","year":"1986","journal-title":"Rev. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zallen, R. (1998). The Physics of Amorphous Solids, Wiley-VCH Verlag GmbH & Co. KGaA.","DOI":"10.1002\/9783527617968"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1126\/science.276.5315.1109","article-title":"Reversible unfolding of individual titin immunoglobulin domains by AFM","volume":"276","author":"Rief","year":"1997","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1126\/science.1219021","article-title":"The protein-folding problem, 50 years on","volume":"338","author":"Dill","year":"2012","journal-title":"Science"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1126\/science.271.5250.795","article-title":"Overstretching B-DNA: The elastic response of individual double-stranded and single-stranded DNA molecules","volume":"271","author":"Smith","year":"1996","journal-title":"Science"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1016\/S0006-3495(01)76067-5","article-title":"Force-induced melting of the DNA double helix. 1. Thermodynamic analysis","volume":"80","author":"Rouzina","year":"2001","journal-title":"Biophys. J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Florio, G., and Puglisi, G. (2019). Unveiling the influence of device stiffness in single macromolecule unfolding. Sci. Rep., 9.","DOI":"10.1038\/s41598-019-41330-x"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"8680","DOI":"10.1039\/C9SM01376H","article-title":"The influence of device handles in single-molecule experiments","volume":"15","author":"Bellino","year":"2019","journal-title":"Soft Matter"},{"key":"ref_28","first-page":"100009","article-title":"On the competition between interface energy and temperature in phase transition phenomena","volume":"2","author":"Bellino","year":"2020","journal-title":"Appl. Eng. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"033227","DOI":"10.1103\/PhysRevResearch.2.033227","article-title":"Role of temperature in the decohesion of an elastic chain thetered to a substrate by onsite breakable links","volume":"2","author":"Florio","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"445001","DOI":"10.1088\/1751-8121\/ac2a07","article-title":"Temperature controlled decohesion regimes of an elastic chain adhering to a fixed substrate by softening and breakable bonds","volume":"54","author":"Cannizzo","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1140\/epjp\/s13360-022-02790-9","article-title":"Thermal control of nucleation and propagation transition stresses in discrete lattices with non-local interactions and non-convex energy","volume":"137","author":"Cannizzo","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.actbio.2022.11.046","article-title":"A predictive model for the thermomechanical overstretching transition of double stranded DNA","volume":"157","author":"Florio","year":"2023","journal-title":"Acta Biomater."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"20230130","DOI":"10.1098\/rsif.2023.0130","article-title":"Cooperative melting in double-stranded peptide chains through local mechanical interactions","volume":"20","author":"Bellino","year":"2023","journal-title":"J. R. Soc. Interface"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1519","DOI":"10.1103\/PhysRevB.15.1519","article-title":"Random-field instability of the ferromagnetic state","volume":"15","author":"Schneider","year":"1977","journal-title":"Phys. Rev. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/BF01020615","article-title":"On the mean-field Ising model in a random external field","volume":"41","author":"Salinas","year":"1985","journal-title":"J. Stat. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"074002","DOI":"10.7566\/JPSJ.83.074002","article-title":"Mean-Field Theory Is Exact For the Random-Field Model with Long-Range Interactions","volume":"83","author":"Tsuda","year":"2014","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00220-024-04986-9","article-title":"Features of a Spin Glass in the Random Field Ising Model","volume":"405","author":"Chatterjee","year":"2024","journal-title":"Commun. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00220-014-2269-5","article-title":"Absence of Replica Symmetry Breaking in the Random Field Ising Model","volume":"337","author":"Chatterjee","year":"2015","journal-title":"Commun. Math. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0378-4371(90)90316-K","article-title":"Directed polymers in a random medium","volume":"163","author":"Derrida","year":"1990","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"207208","DOI":"10.1103\/PhysRevLett.104.207208","article-title":"Elusive spin-glass phase in the random field Ising model","volume":"104","author":"Krzakala","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"042003","DOI":"10.1088\/1751-8113\/44\/4\/042003","article-title":"No spin glass phase in ferromagnetic random-field random-temperature scalar Ginzburg-Landau model","volume":"44","author":"Krzakala","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1821","DOI":"10.1103\/PhysRevLett.67.1821","article-title":"Ordering and Phase Transitions in Random-Field Ising Systems","volume":"67","author":"Maritan","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/BF01224901","article-title":"Phase Transition in the 3d Random Field Ising Model","volume":"116","author":"Bricmont","year":"1988","journal-title":"Commun. Math. Phys."},{"key":"ref_44","first-page":"383","article-title":"Dynamics versus replicas in the random field Ising model","volume":"327","author":"Brezin","year":"1999","journal-title":"Comptes Rendus l\u2019Acad\u00e9mie Sci.-Ser. IIB-Mech.-Phys.-Astron."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"7524","DOI":"10.1073\/pnas.84.21.7524","article-title":"Spin glasses and the statistical mechanics of protein folding","volume":"84","author":"Brygelson","year":"1987","journal-title":"Proc. Nati. Acad. Sci. USA"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"5202","DOI":"10.1103\/PhysRevE.60.5202","article-title":"Depinning transition and thermal fluctuations in the random-field Ising model","volume":"60","author":"Roters","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2060","DOI":"10.1016\/j.jmps.2006.04.006","article-title":"Hysteresis in multi-stable lattices with non-local interactions","volume":"54","author":"Puglisi","year":"2006","journal-title":"J. Mech. Phys. Solids"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Young, A.P. (1997). Out of Equilibrium Dynamics in Spin-Glasses and Other Glassy Systems. Spin Glasses and Random Fields, World Scientific.","DOI":"10.1142\/3517"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1109\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:55:05Z","timestamp":1760115305000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/12\/1109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,18]]},"references-count":48,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["e26121109"],"URL":"https:\/\/doi.org\/10.3390\/e26121109","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,12,18]]}}}