{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:41:19Z","timestamp":1760244079993,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2008,10,8]],"date-time":"2008-10-08T00:00:00Z","timestamp":1223424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The goal in this paper is to show how many high-frequency electromagnetic metrology areas can be understood and formulated in terms of entropy evolution, production, and fluctuations. This may be important in nanotechnology where an understanding of fluctuations of thermal and electromagnetic energy and the effects of nonequilibrium are particularly important. The approach used here is based on a new derivation of an entropy evolution equation using an exact Liouville-based statistical-mechanical theory rooted in the Robertson-Zwanzig-Mori formulations. The analysis begins by developing an exact equation for entropy rate in terms of time correlations of the microscopic entropy rate. This equation is an exact fluctuation-dissipation relationship. We then define the entropy and its production for electromagnetic driving, both in the time and frequency domains, and apply this to study dielectric and magnetic material measurements, magnetic relaxation, cavity resonance, noise, measuring Boltzmann\u2019s constant, and power measurements.<\/jats:p>","DOI":"10.3390\/e10040411","type":"journal-article","created":{"date-parts":[[2008,10,8]],"date-time":"2008-10-08T10:38:08Z","timestamp":1223462288000},"page":"411-429","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Electromagnetic Nanoscale Metrology Based on Entropy Production and Fluctuations"],"prefix":"10.3390","volume":"10","author":[{"given":"James","family":"Baker-Jarvis","sequence":"first","affiliation":[{"name":"National Institute of Standards and Technology, 325 Broadway, MS 818.01, Boulder, CO, USA"}]}],"member":"1968","published-online":{"date-parts":[[2008,10,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1103\/RevModPhys.61.981","article-title":"The origin of increasing entropy","volume":"61(4)","author":"Mackey","year":"1989","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(86)90005-0","article-title":"Dissipation in many-body systems: A geometric approach based on information theory","volume":"131","author":"Balian","year":"1986","journal-title":"Physics Reports"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6007","DOI":"10.1063\/1.1316000","article-title":"Positive definiteness of the entropy evolution in the Robertson formalism","volume":"114","author":"Nettleton","year":"2001","journal-title":"J. Chem. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4794","DOI":"10.1103\/PhysRevA.38.4794","article-title":"Microscopic and macroscopic entropy","volume":"38(9)","author":"Lindgren","year":"1988","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1088\/0368-3281\/2\/1\/330","article-title":"Non-equilibrium thermodynamics","volume":"2","author":"deGroot","year":"1961","journal-title":"J. Nucl. Energy: Part C"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"066613","DOI":"10.1103\/PhysRevE.72.066613","article-title":"Time-dependent entropy evolution in microscopic and macroscopic electromagnetic relaxation","volume":"72","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_7","unstructured":"deGroot, R. S., and Mazur, P. (1984). Nonequilibrium Thermodynamics, Dover."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1088\/0034-4885\/29\/1\/306","article-title":"The fluctuation-dissipation theorem","volume":"29","author":"Kubo","year":"1966","journal-title":"Repts Prog. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Robertson, B. Equations of motion in nonequilibrium statistical mechanics. Phys. Rev., 144, 151\u2013161. (April 1966).","DOI":"10.1103\/PhysRev.144.151"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1063\/1.1731409","article-title":"Ensemble method in the theory of irreversibility","volume":"33(5)","author":"Zwanzig","year":"1960","journal-title":"J. Chem. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1143\/PTP.33.423","article-title":"Transport, collective motion, and brownian motion","volume":"33","author":"Mori","year":"1965","journal-title":"Prog. Theor. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Grabert, H. (1982). Projection-operator techniques in nonequilibrium statistical mechanics, Springer-Verlag.","DOI":"10.1007\/BFb0044591"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/0378-4371(79)90065-7","article-title":"Nonlinear transport processes: Hydrodynamics","volume":"99A","author":"Oppenheim","year":"1979","journal-title":"Physica"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zubarev, D. N. (1974). Nonequilibrium Statistical Thermodynamics, Consultants Bureau.","DOI":"10.21236\/AD0784411"},{"key":"ref_15","unstructured":"Zubarev, D. N., Morozov, V., and Ropke, G. (1996). Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory, John Wiley and Sons."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2572","DOI":"10.1016\/j.physleta.2007.11.072","article-title":"Influence of environment and entropy production of a nonequilibrium open system","volume":"372","author":"Yu","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_17","unstructured":"Balian, R. (2007). From Microphysics to Macrophysics, Springer."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Grandy, W. T., and Milonni, P. W. (1993). Physics and Probability: Essays in Honor of Edwin T. Jaynes, Cambridge University Press.","DOI":"10.1017\/CBO9780511524448"},{"key":"ref_19","first-page":"056127\u20131:14","article-title":"Dynamic constitutive relations for polarization and magnetization","volume":"64","author":"Kabos","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_20","first-page":"036615\u20131\u201313","article-title":"Nonequilibrium electromagnetics: Local and macroscopic fields using statistical mechanics","volume":"70","author":"Kabos","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Robertson, B. Equations of motion of nuclear magnetism. Phys. Rev., 153, 391\u2013403. (January 1967).","DOI":"10.1103\/PhysRev.153.391"},{"key":"ref_22","first-page":"302","article-title":"Solution to boundary value problems using the method of maximum entropy","volume":"3","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_23","first-page":"133","article-title":"Approximating solutions to linear and nonlinear differential equations by the method of maximum entropy","volume":"5","author":"Alameddine","year":"1989","journal-title":"NMPDE"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1063\/1.528276","article-title":"Solving differential equations by a maximum entropy-minimum norm method with applications to Fokker-Planck equations","volume":"30","author":"Alameddine","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_25","unstructured":"Reichl, L. E. (1980). A modern course in statistical physics, University of Texas, Austin Press."},{"key":"ref_26","unstructured":"Brosseau, C. (1998). Fundamentals of polarization optics: A Statistical Optics Approach, Wiley."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1103\/RevModPhys.50.221","article-title":"General properties of entropy","volume":"50","author":"Wehrl","year":"1978","journal-title":"Rev. Mod. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Atwal, G. S., and Ashcroft, N. W. (2002). Phys. Rev. B, 65, 115109.","DOI":"10.1103\/PhysRevB.65.115109"},{"key":"ref_29","unstructured":"Robertson, B. (1978). The Maximum Entropy Formalism, M.I.T. Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"056612","DOI":"10.1103\/PhysRevE.75.056612","article-title":"Dielectric polarization equations and relaxation times","volume":"75","author":"Janezic","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Robertson, B., and Mitchell, W. C. Equations of motion in nonequilibrium statistical mechanics. III: Open systems. J. Math. Phys., 12, 563\u2013568. (March 1971).","DOI":"10.1063\/1.1665621"},{"key":"ref_32","unstructured":"Bertotti, G. (1998). Hysteresis in Magnetism, Academic Press."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Slonczewski, J. C. (1996). J. Magn. Magn. Mater., 159, L1.","DOI":"10.1016\/0304-8853(96)00062-5"},{"key":"ref_34","unstructured":"Landau, L. D., and Lifshitz, E. M. (1987). Electromagnetic of Continuous Media, Addison-Wesley."},{"key":"ref_35","unstructured":"Jackson, J. D. (1999). Classical Electrodynamics, John Wiley and Sons. [3rd ed.]."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Berne, B. J. (1971). Time-Dependent Propeties of Condensed Media, Academic Press. Chapter 9.","DOI":"10.1016\/B978-0-12-245658-9.50010-9"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1148","DOI":"10.1063\/1.1352694","article-title":"White-noise magnetization fluctuations in magnetoresistive heads","volume":"78","author":"Smith","year":"2001","journal-title":"Appl. Phys. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1109\/22.750234","article-title":"Suggestions for revised definitions of noise quantities, including quantum effects","volume":"47","author":"Kerr","year":"1999","journal-title":"IEEE Trans. Microwave Theory Techniques"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1103\/PhysRev.83.34","article-title":"Irreversibilty and generalized noise","volume":"83","author":"Callen","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1007\/BF01008729","article-title":"Nonlinear generalized Langevin equations","volume":"9","author":"Zwanzig","year":"1973","journal-title":"J. Stat. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Joulin, K., Mulet, J. P., Marquier, F., Carminati, R., and Greffet, J. J. (2007). Surf. Sci. Rep., 57, 59.","DOI":"10.1016\/j.surfrep.2004.12.002"},{"key":"ref_42","unstructured":"Rytov, S. M., Kravtsav, Y. A., and Tatarski, V. I. (1978). Principles of Statistical Radiophysics, Springer Verlag."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/411\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:21:02Z","timestamp":1760221262000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/411"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,10,8]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2008,12]]}},"alternative-id":["e10040411"],"URL":"https:\/\/doi.org\/10.3390\/e10040411","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2008,10,8]]}}}