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Interacting Bose gas, the logistic law, and complex networks

Abstract

We discuss a mathematical link between the Quantum Statistical Mechanics and the logistic growth and decay processes. It is based on an observation that a certain nonlinear operator evolution equation, which we refer to as the Logistic Operator Equation (LOE), provides an extension of the standard model of noninteracting bosons. We discuss formal solutions (asymptotic formulas) for a special calibration of the LOE, which sets it in the number-theoretic framework. This trick, in the tradition of Julia and Bost-Connes, makes it possible for us to tap into the vast resources of classical mathematics and, in particular, to construct explicit solutions of the LOE via the Dirichlet series. The LOE is applicable to a range of modeling and simulation tasks, from characterization of interacting boson systems to simulation of some complex man-made networks. The theoretical results enable numerical simulations, which, in turn, shed light at the unique complexities of the rich and multifaceted models resulting from the LOE.

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References

  1. A. Sowa, “On an Eigenvalue Problem with a Reciprocal-Linear Term,” Waves Random and Complex Media 22(2), 186–206 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Laohakosol, “Dependence of Arithmetic Functions and Dirichlet Series,” Proc. Amer. Math. Soc. 115, 637–645 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Ostrowski, “Über Dirichletsche Reihen und algebraische Differentialgleichungen,” Math Z. 8, 241–298 (1920).

    Article  MATH  MathSciNet  Google Scholar 

  4. J.-B. Bost and A. Connes, “Hecke Algebras, Type III Factors and Phase Transitions with Spontaneous Symmetry Breaking in Number Theory,” Selecta Math. (N. S.) 1, 411–457 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Sowa, “Spectra of Nonlocally Bound Quantum Systems,” Russ. J. Math. Phys. 18(2), 227–241 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  6. K. Hurose, “The Relation between the Speed of Demand Saturation and the Dynamism of the Labour Market,” Struct. Change and Econom. Dynam. 20, 151–159 (2009).

    Article  ADS  Google Scholar 

  7. R.W. Dimand and J. Geanakoplos (eds.), Celebrating Irving Fisher: The Legacy of a Great Economist (Blackwell Publishing, Malden MA, 2005).

    Google Scholar 

  8. K. Martinás, “Neumannian Economy in Multi-Agent Approach. Investigation of Stability and Instability in Economic Growth,” Interdiscip. Description of Complex Syst. 2(1), 70–78 (2004).

    ADS  Google Scholar 

  9. E. Smith, and D. K. Foley, “Classical Thermodynamics and Economic General Equilibrium Theory,” J. Econom. Dynam. Control 32, 7–65 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Sousa and T. Domingos, “Equilibrium Econophysics: A Unified Formalism for Neoclassical Economics and Equilibrium Thermodynamics,” Physica A 371, 492–512 (2006).

    Article  ADS  MathSciNet  Google Scholar 

  11. M. Ausloos, “Gas-Kinetic Theory and Boltzmann Equation of Share Price within an Equilibrium Market Hypothesis and Ad Hoc Strategy,” Physica A 284, 385–392 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  12. F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo, A. Vespignani, and D. R. White, “Economic Networks: The New Challenges,” Science 325, 422–425 (2009).

    ADS  MATH  MathSciNet  Google Scholar 

  13. F. Vega-Redondo, Complex Social Networks (Cambridge University Press, Cambridge, New York, 2007).

    Book  MATH  Google Scholar 

  14. V. P. Maslov, “Solution of the Gibbs Paradox Using the Notion of Entropy as a Function of the Fractal Dimension,” Russ. J. Math. Phys. 17, 288–306 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” J. Political Economy 81, 637–654 (1973).

    Article  MATH  Google Scholar 

  16. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory (second edition: Springer-Verlag: New York, Berlin, 1990).

    Book  MATH  Google Scholar 

  17. J. Heath, “Methodological Individualism,” The Stanford Encyclopedia of Philosophy (Spring 2011 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/spr2011/entries/methodologicalindividualism/.

  18. J. M. Keynes, The General Theory of Employment, Interest and Money (Macmillan, London, 1936).

    Google Scholar 

  19. R. W. Dimand, Fisher, Keynes, and the Corridor of Stability (in: R.W. Dimand and J. Geanakoplos (eds.), Celebrating Irving Fisher: The Legacy of a Great Economist, Blackwell Publishing, Malden, MA 2005).

  20. A. Baker, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/sum2011/entries/simplicity/.

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Sowa, A. Interacting Bose gas, the logistic law, and complex networks. Russ. J. Math. Phys. 22, 98–111 (2015). https://doi.org/10.1134/S1061920815010112

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  • DOI: https://doi.org/10.1134/S1061920815010112

Keywords

  • Exogenous Variable
  • Dirichlet Series
  • Prime Decomposition
  • Logistic Decay
  • Consecutive Word