{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T15:17:43Z","timestamp":1790522263689,"version":"4.1.0"},"reference-count":66,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2014,12,31]],"date-time":"2014-12-31T00:00:00Z","timestamp":1419984000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Nanothermodynamics extends standard thermodynamics to facilitate finite-size effects on the scale of nanometers. A key ingredient is Hill\u2019s subdivision potential that accommodates the non-extensive energy of independent small systems, similar to how Gibbs\u2019 chemical potential accommodates distinct particles. Nanothermodynamics is essential for characterizing the thermal equilibrium distribution of independently relaxing regions inside bulk samples, as is found for the primary response of most materials using various experimental techniques. The subdivision potential ensures strict adherence to the laws of thermodynamics: total energy is conserved by including an instantaneous contribution from the entropy of local configurations, and total entropy remains maximized by coupling to a thermal bath. A unique feature of nanothermodynamics is the completely-open nanocanonical ensemble. Another feature is that particles within each region become statistically indistinguishable, which avoids non-extensive entropy, and mimics quantum-mechanical behavior. Applied to mean-field theory, nanothermodynamics gives a heterogeneous distribution of regions that yields stretched-exponential relaxation and super-Arrhenius activation. Applied to Monte Carlo simulations, there is a nonlinear correction to Boltzmann\u2019s factor that improves agreement between the Ising model and measured  non-classical critical scaling in magnetic materials. Nanothermodynamics also provides a fundamental mechanism for the 1\/f noise found in many materials.<\/jats:p>","DOI":"10.3390\/e17010052","type":"journal-article","created":{"date-parts":[[2014,12,31]],"date-time":"2014-12-31T08:49:33Z","timestamp":1420015773000},"page":"52-73","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":65,"title":["The Big World of Nanothermodynamics"],"prefix":"10.3390","volume":"17","author":[{"given":"Ralph","family":"Chamberlin","sequence":"first","affiliation":[{"name":"Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2014,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3182","DOI":"10.1063\/1.1732447","article-title":"Thermodynamics of small systems","volume":"36","author":"Hill","year":"1962","journal-title":"J. Chem. Phys"},{"key":"ref_2","unstructured":"Hill, T.L. (2013). Thermodynamics of Small Systems, Parts I & II, Dover Publications."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/0022-3093(82)90089-8","article-title":"The size of cooperatively rearranging regions at the glass transition","volume":"53","author":"Donth","year":"1982","journal-title":"J. Non-Cryst. Solids"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0370-1573(91)90074-V","article-title":"Phase transitions and heterophase fluctuations","volume":"208","author":"Yukalov","year":"1991","journal-title":"Phys. Rep"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/01411599808209287","article-title":"Experiments and theory of the nonexponential relaxation in liquids, glasses, polymers and crystals","volume":"65","author":"Chamberlin","year":"1998","journal-title":"Phase Transit"},{"key":"ref_6","first-page":"1","article-title":"Nature of the non-exponential primary relaxation in structural glass-formers probed by dynamically selective experiments","volume":"235\u2013237","author":"Chamberlin","year":"1998","journal-title":"J. Non-Cryst. Solids"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1146\/annurev.physchem.51.1.99","article-title":"Spatially heterogeneous dynamics in supercooled liquids","volume":"51","author":"Ediger","year":"2000","journal-title":"Annu. Rev. Phys. Chem"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"R703","DOI":"10.1088\/0953-8984\/14\/23\/201","article-title":"Heterogeneous dynamics in liquids: Fluctuations in space and time","volume":"14","author":"Richert","year":"2002","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1038\/35042534","article-title":"Mean-field cluster model for the critical behaviour of ferromagnets","volume":"408","author":"Chamberlin","year":"2000","journal-title":"Nature"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1021\/nl010010d","article-title":"Perspective: Nanothermodynamics","volume":"1","author":"Hill","year":"2001","journal-title":"Nano Lett"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1021\/nl010027w","article-title":"A different approach to nanothermodynamics","volume":"1","author":"Hill","year":"2001","journal-title":"Nano Lett"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2520","DOI":"10.1103\/PhysRevLett.82.2520","article-title":"Mesoscopic mean-field theory for supercooled liquids and the glass transition","volume":"82","author":"Chamberlin","year":"1999","journal-title":"Phys. Rev. Lett"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0375-9601(03)01038-7","article-title":"Critical behavior from Landau theory in nanothermodynamic equilibrium","volume":"315","author":"Chamberlin","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"154503","DOI":"10.1063\/1.2354471","article-title":"A free-energy landscape picture and Landau theory for the dynamics of disordered materials","volume":"125","author":"Javaheri","year":"2006","journal-title":"J. Chem. Phys"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1140\/epjb\/e2009-00053-3","article-title":"Fluctuation-theory constraint for extensive entropy in Monte-Carlo simulations","volume":"67","author":"Chamberlin","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjb\/e2009-00278-0","article-title":"Beyond the Boltzmann factor for corrections to scaling in ferromagnetic materials and critical fluids","volume":"71","author":"Chamberlin","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5384","DOI":"10.1016\/j.physa.2012.06.016","article-title":"simulations including energy from an entropic force","volume":"391","author":"Chamberlin","year":"2012","journal-title":"Physica A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"012142","DOI":"10.1103\/PhysRevE.90.012142","article-title":"1\/f noise from the laws of thermodynamics for finite-size fluctuations","volume":"90","author":"Chamberlin","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_19","unstructured":"Kogan, S. (2008). Electronic Noise and Fluctuations in Solids, Cambridge University Press."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1103\/RevModPhys.53.497","article-title":"Low-frequency fluctuations in solids: 1\/f noise","volume":"53","author":"Dutta","year":"1981","journal-title":"Rev. Mod. Phys"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1103\/RevModPhys.60.537","article-title":"1\/f noise and other slow, nonexponential kinetics in condensed matter","volume":"60","author":"Weissman","year":"1988","journal-title":"Rev. Mod. Phys"},{"key":"ref_22","unstructured":"Milotti, E. (2002). 1\/f noise: A pedagogical review."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1103\/RevModPhys.86.361","article-title":"1\/f noise: Implications for solid-state quantum information","volume":"86","author":"Paladin","year":"2014","journal-title":"Rev. Mod. Phys"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1038\/258317a0","article-title":"\u20181\/f noise\u2019 in music and speech","volume":"258","author":"Voss","year":"1975","journal-title":"Nature"},{"key":"ref_25","first-page":"40","article-title":"The noise in natural phenomena","volume":"78","author":"West","year":"1990","journal-title":"Am. Sci"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/0378-4371(94)90502-9","article-title":"Statistical mechanics in biology: How ubiquitous are long-range correlations?","volume":"78","author":"Stanley","year":"1994","journal-title":"Physica A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1837","DOI":"10.1126\/science.7892611","article-title":"1\/f noise in human cognition","volume":"267","author":"Gilden","year":"1995","journal-title":"Science"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0304-8853(92)90336-M","article-title":"Spin-glass fluctuation statistics: Mesoscopic experiments in Cu Mn","volume":"114","author":"Weissman","year":"1992","journal-title":"J. Magn. Magn. Mater"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1073\/pnas.0705349105","article-title":"Noise in solid-state nanopores","volume":"105","author":"Smeets","year":"2008","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"050603","DOI":"10.1103\/PhysRevLett.103.050603","article-title":"Trajectory approach to two-state kinetics of single particles on sculpted energy landscapes","volume":"103","author":"Wu","year":"2008","journal-title":"Phys. Rev. Lett"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6681","DOI":"10.1103\/PhysRevB.18.6681","article-title":"Excess (1\/f) noise in metals","volume":"18","author":"Eberhard","year":"1978","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Paszkiewicz, T., and Rapcewicz, K. (1994). Die Kunst of Phonons, Springer.","DOI":"10.1007\/978-1-4615-2455-7"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1126\/science.274.5288.752","article-title":"Nonresonant spectral hole burning in the slow dielectric response of supercooled liquids","volume":"274","author":"Schiener","year":"1996","journal-title":"Science"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5134","DOI":"10.1103\/PhysRevLett.83.5134","article-title":"Nonresonant spectral hole burning in a spin glass","volume":"83","author":"Chamberlin","year":"1999","journal-title":"Phys. Rev. Lett"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01293604","article-title":"Finite size scaling analysis of Ising model block distribution functions","volume":"43","author":"Binder","year":"1981","journal-title":"Zeitschrift f\u00fcr Physik B Condensed Matter"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1356","DOI":"10.1063\/1.1529193","article-title":"Dielectric hole burning in the high frequency wing of supercooled glycerol","volume":"118","author":"Duvvuri","year":"2002","journal-title":"J. Chem. Phys"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"208","DOI":"10.1016\/S0022-3093(02)01460-6","article-title":"Comparative study of the NMR length scale of dynamic heterogeneities of three different glass formers","volume":"307\u2013310","author":"Reinsberg","year":"2002","journal-title":"J. Non-Cryst. Solids"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"R4060","DOI":"10.1103\/PhysRevE.58.R4060","article-title":"Nature of crossover from classical to Ising-like critical scaling behavior","volume":"58","author":"Luijten","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1103\/PhysRevB.59.1145","article-title":"Observation of isotropic dipolar to uniaxial dipolar crossover in gadolinium","volume":"59","author":"Srinath","year":"1999","journal-title":"Phys. Rev. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"336S","DOI":"10.1063\/1.2000459","article-title":"Paramagnetism of polycrystalline gadolinium, terbium, and dysprosium metals","volume":"32","author":"Arajs","year":"1961","journal-title":"J. Appl. Phys"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1092","DOI":"10.1103\/PhysRev.132.1092","article-title":"Magnetization and electrical resistivity of gadolinium single crystals","volume":"132","author":"Nigh","year":"1963","journal-title":"Phys. Rev"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"2728","DOI":"10.1103\/PhysRevLett.62.2728","article-title":"Anisotropy of the critical magnetic susceptibility of gadolinium","volume":"62","author":"Geldart","year":"1989","journal-title":"Phys. Rev. Lett"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0040-6031(97)00147-0","article-title":"Heat capacity of LaMnO3","volume":"299","author":"Satoh","year":"1997","journal-title":"Thermochim. Acta"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1016\/j.jssc.2008.06.046","article-title":"Nanoscale structural order from the atomic pair distribution function (PDF): There\u2019s plenty of room in the middle","volume":"181","author":"Billinge","year":"2008","journal-title":"J. Solid State Chem"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"177203","DOI":"10.1103\/PhysRevLett.94.177203","article-title":"Orbital correlations in the pseudocubic O and rhombohedral R phases of LaMnO3","volume":"94","author":"Qiu","year":"2005","journal-title":"Phys. Rev. Lett"},{"key":"ref_46","first-page":"793","article-title":"Accidental deviations of density and opalescence at the critical point of a single substance","volume":"17","author":"Ornstein","year":"1914","journal-title":"Proc. Acad. Sci. Amsterdam"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3020","DOI":"10.1103\/PhysRevLett.66.3020","article-title":"Nature of nonexponential loss of correlation above the glass transition investigated by multidimensional NMR","volume":"66","author":"Spiess","year":"1991","journal-title":"Phys. Rev. Lett"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2727","DOI":"10.1103\/PhysRevLett.81.2727","article-title":"Length scale of dynamic heterogeneities at the glass transition determined by multidimensional nuclear magnetic resonance","volume":"81","author":"Tracht","year":"1998","journal-title":"Phys. Rev. Lett"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1021\/nl020295+","article-title":"Fluctuations in energy in completely open small systems","volume":"2","author":"Hill","year":"2002","journal-title":"Nano Lett"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1103\/PhysRevB.3.1689","article-title":"Critical magnetic properties and exchange interaction in EuO","volume":"3","author":"Menyuk","year":"1971","journal-title":"Phys. Rev. B"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"5255","DOI":"10.1103\/PhysRevB.12.5255","article-title":"Faraday rotation near the Curie point of EuO","volume":"12","author":"Huang","year":"1975","journal-title":"Phys. Rev. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"855","DOI":"10.1143\/JPSJ.11.855","article-title":"Change of magnetic susceptibility of transition metals and alloys at their melting points","volume":"11","author":"Nakagawa","year":"1956","journal-title":"J. Phys. Soc. Jpn"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/0022-3697(65)90174-5","article-title":"Magnetic susceptibility of face-centered cubic cobalt just above the ferromagnetic Cure temperature","volume":"26","author":"Colvin","year":"1965","journal-title":"J. Phys. Chem. Solids"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1051\/jphysrad:0194400508015300","article-title":"Paramagn\u00e9tisme des \u00e9l\u00e9ments ferromagn\u00e9tiques","volume":"5","author":"Fallot","year":"1944","journal-title":"J. Phys. Radium"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1063\/1.1714136","article-title":"Paramagnetic behavior of nickel just above the ferromagnetic Curie temperature","volume":"36","author":"Arajs","year":"1965","journal-title":"J. Appl. Phys"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"2043","DOI":"10.1063\/1.470961","article-title":"Dynamics of glass-forming liquids. II. Detailed comparison of dielectric relaxation, dc-conductivity, and viscosity data","volume":"104","author":"Stickel","year":"1995","journal-title":"J. Chem. Phys"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0022-3093(98)00561-4","article-title":"Dielectric and far-infrared spectroscopy of glycerol","volume":"235\u2013237","author":"Schneider","year":"1998","journal-title":"J. Non-Cryst. Solids"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1006\/jcht.1994.1046","article-title":"Calorimetric study of d,l-propene carbonate: Observation of the \u03b2- as well as \u03b1-glass transition in the supercooled liquids","volume":"26","author":"Fujimori","year":"1994","journal-title":"J. Chem. Thermodyn"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/0038-1098(94)00754-3","article-title":"Discovery of a potentially homogeneous-nucleation-based crystallization around the glass transition temperature in salol","volume":"93","author":"Hikima","year":"1995","journal-title":"Solid State Comm"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"174518","DOI":"10.1103\/PhysRevB.68.174518","article-title":"Specific heat of liquid helium in zero gravity very near the lambda point","volume":"68","author":"Lipa","year":"2003","journal-title":"Phys. Rev. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.physd.2004.01.006","article-title":"What should a statistical mechanics satisfy to reflect nature?","volume":"193","author":"Tsallis","year":"2004","journal-title":"Physica D"},{"key":"ref_62","unstructured":"ter Haar, D., and Gregg, J.F. (1992). From Microphysics to Macrophysics, Springer. Chapter 3."},{"key":"ref_63","unstructured":"Collins, M.F. (1989). Magnetic Critical Scattering, Oxford University Press."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/BF02405146","article-title":"Critical phenomena studied via nuclear techniques","volume":"49","author":"Hohenemser","year":"1989","journal-title":"Hyperfine Interact"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Gemmer, J., Michel, M., and Mahler, G. (2009). Quantum Thermodynamics, Springer. [2nd ed].","DOI":"10.1007\/978-3-540-70510-9"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"4429","DOI":"10.1098\/rsta.2012.0162","article-title":"Environmental decoherence versus intrinsic decoherence","volume":"370","author":"Stamp","year":"2012","journal-title":"Phil. Trans. R. Soc. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/1\/52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:11:59Z","timestamp":1760217119000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/1\/52"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,31]]},"references-count":66,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1]]}},"alternative-id":["e17010052"],"URL":"https:\/\/doi.org\/10.3390\/e17010052","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12,31]]}}}