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Jeffrey Commons, Ying-Jen Yang(杨颖任), and Hong Qian(钱纮). 2025: Duality symmetry, two entropy functions, and an eigenvalue problem in generalized Gibbs' theory, Chinese Physics B, 34(8): 080201. doi: 10.1088/1674-1056/adbed7
Citation: Jeffrey Commons, Ying-Jen Yang(杨颖任), and Hong Qian(钱纮). 2025: Duality symmetry, two entropy functions, and an eigenvalue problem in generalized Gibbs' theory, Chinese Physics B, 34(8): 080201. doi: 10.1088/1674-1056/adbed7
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Duality symmetry, two entropy functions, and an eigenvalue problem in generalized Gibbs' theory

  • 1 Department of Physics, University of Washington, Seattle, WA 98195-1560, USA;

  • 2 Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA;

  • 3 Laufer Center for Physical and Quantitative Biology, Stony Brook University, New York 11794, USA

  • Received Date: 30/06/2024
    Accepted Date: 02/03/2025

  • PACS: 02.50.Cw; 05.70.-a; 82.60.-s; 89.70.-a

  • We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between the Gibbs entropy $\varphi_{V,N}(E)$ as a function of mean internal energy $E$ and Massieu's free entropy $\varPsi_{V,N}(\beta)$ as a function of inverse temperature $\beta$. The duality in terms of Legendre-Fenchel transform tells us that Gibbs' thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample mean values what Shannon's information entropy is to the LLN for empirical counting frequencies in independent and identically distributed data. Proceeding with the same mathematical logic, we identify the energy of the state $\{u_i\}$ as the conjugate variable to the counting of statistical occurrence $\{m_i\}$ and find a Hamilton-Jacobi equation for the Shannon entropy analogous to an equation of state in thermodynamics. An eigenvalue problem that is reminiscent of certain features in quantum mechanics arises in the entropy theory of statistical counting frequencies of Markov correlated data.
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  • Smith E 2020 Entropy 22 1137

    Google Scholar Pub Med

    Qian H 2022 J. Chem. Theory Comput. 18 6421

    Google Scholar Pub Med

    Qian H 2024 Entropy 26 1091

    Google Scholar Pub Med

    Ge H and Qian H 2016 Phys. Rev. E 94 052150

    Google Scholar Pub Med

    Ge H and Qian H 2017 J. Stat. Phys. 166 190

    Google Scholar Pub Med

    Hill T L 1963 Thermodynamics of Small Systems (New York: Dover)

    Google Scholar Pub Med

    Lu Z and Qian H 2022 Phys. Rev. Lett. 128 150603

    Google Scholar Pub Med

    Oono Y 1989 Progr. Theor. Phys. Supp. 99 165

    Google Scholar Pub Med

    Dembo A and Zeitouni O 1998 Large Deviations Techniques and Applications, 2nd Ed. (New York: Springer)

    Google Scholar Pub Med

    Chibbaro S, Rondoni L and Vulpiani A 2014 Reductionism, Emergence and Levels of Reality (New York: Springer)

    Google Scholar Pub Med

    Angelini E and Qian H 2023 J. Phys. Chem. B 127 2552

    Google Scholar Pub Med

    Rockafellar R T 1996 Convex Analysis (NJ: Princeton Univ. Press)

    Google Scholar Pub Med

    Miao B, Qian H and Wu Y S 2024 arXiv:2406.02405

    Google Scholar Pub Med

    Shannon C E and Weaver W 1949 The Mathematical Theory of Communication (Champaign: Univ. Ill. Press)

    Google Scholar Pub Med

    Hobson A 1969 J. Stat. Phys. 1 383

    Google Scholar Pub Med

    Shore J E and Johnson R W 1980 IEEE Trans. Inf. Th. 26 26

    Google Scholar Pub Med

    Guggenheim E A 1933 Modern Thermodynamics by the Methods of Willard Gibbs (New York: Methuen & Co.)

    Google Scholar Pub Med

    Callen H B 1991 Thermodynamics and an Introduction to Thermostatistics, 2nd Ed. (New York: John Wiley & Sons)

    Google Scholar Pub Med

    Huang K 1963 Statistical Mechanics (New York: John Wiley & Sons)

    Google Scholar Pub Med

    Kirkwood J G 1935 J. Chem. Phys. 3 300

    Google Scholar Pub Med

    Yang Y J and Qian H 2020 Phys. Rev. E 101 022129

    Google Scholar Pub Med

    Barato A C and Chetrite R 2015 J. Stat. Phys. 160 1154

    Google Scholar Pub Med

    Baldi P and Piccioni M 1999 Stat. Probab. Lett. 41 107

    Google Scholar Pub Med

    Cohen J 1981 Proc. Amer. Math. Soc. 81 657

    Google Scholar Pub Med

    Yang C N and Lee T D 1952 Phys. Rev. 87 404

    Google Scholar Pub Med

    Anderson P W 1972 Science 177 393

    Google Scholar Pub Med

    Jaynes E T 2003 Probability Theory: The Logic of Science (Cambridge: Cambridge Univ. Press)

    Google Scholar Pub Med

    Ghosh K, Dixit P D, Agozzino L and Dill K A 2020 Annu. Rev. Phys. Chem. 71 213

    Google Scholar Pub Med

    Dill K A 2021 The maximum caliber principle for modeling stochastic dynamic processes

    Google Scholar Pub Med

    Salmon W C 1990 Four Decades of Scientific Explanation (Pittsburgh: Univ. Pittsburgh Press)

    Google Scholar Pub Med

    Laughlin R B 2006 A Different Universe: Reinventing Physics from the Bottom Down (New York: Basic Books)

    Google Scholar Pub Med

    Schellman J A 1997 Biophys. Chem. 64 7

    Google Scholar Pub Med

    Hoffmann B 2011 The Strange Story of the Quantum (New York: Dover)

    Google Scholar Pub Med

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Duality symmetry, two entropy functions, and an eigenvalue problem in generalized Gibbs' theory

  • Received Date: 30/06/2024
  • Accepted Date: 02/03/2025

Abstract: We generalize the convex duality symmetry in Gibbs' statistical ensemble formulation, between the Gibbs entropy $\varphi_{V,N}(E)$ as a function of mean internal energy $E$ and Massieu's free entropy $\varPsi_{V,N}(\beta)$ as a function of inverse temperature $\beta$. The duality in terms of Legendre-Fenchel transform tells us that Gibbs' thermodynamic entropy is to the law of large numbers (LLN) for arithmetic sample mean values what Shannon's information entropy is to the LLN for empirical counting frequencies in independent and identically distributed data. Proceeding with the same mathematical logic, we identify the energy of the state $\{u_i\}$ as the conjugate variable to the counting of statistical occurrence $\{m_i\}$ and find a Hamilton-Jacobi equation for the Shannon entropy analogous to an equation of state in thermodynamics. An eigenvalue problem that is reminiscent of certain features in quantum mechanics arises in the entropy theory of statistical counting frequencies of Markov correlated data.

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