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A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation

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Abstract

In this paper we describe a probabilistic framework for describing dynamical systems. The approach is inspired by quantum dynamical expectation dynamics. Specifically, an abstract evolution operator corresponding to the Hamiltonian in quantum dynamics is constructed. The evolution of this operator defining PDE’s solution is isomorphic to the functional structure of the wave function as long as its initial form permits. This operator enables us to use one of the most important probabilistic concepts, namely expectations. The expectation dynamics are governed by equations which are constructed via commutator algebra. Based on inspiration from quantum dynamics, we have used both the independent variables and the symmetric forms of their derivatives. For construction of the expectation dynamics, the algebraic independent variable operators which multiply their operands by the corresponding independent variable suffice. In our descriptions, we remain at the conceptual level in a self-consistent manner. The phenomenological implications and the tremendous potential of this approach for scientific discovery and advancement is described in the companion to this paper.

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Authors and Affiliations

  1. İstanbul Teknik Üniversitesi Bilişim Enstitüsü, Maslak, 34469, Istanbul, Turkey

    Metin Demiralp

  2. Department of Psychology, University of Michigan, 1012 East Hall, 530 Church Street, Ann Arbor, MI, 41809-1043, USA

    Emre Demiralp

  3. 1096 BIRB, 2360 Bonisteel St., Ann Arbor, MI, 48105-2108, USA

    Luis Hernandez-Garcia

Authors
  1. Metin Demiralp
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  2. Emre Demiralp
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  3. Luis Hernandez-Garcia
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Correspondence to Metin Demiralp.

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Demiralp, M., Demiralp, E. & Hernandez-Garcia, L. A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation. J Math Chem 50, 850–869 (2012). https://doi.org/10.1007/s10910-011-9929-x

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  • DOI: https://doi.org/10.1007/s10910-011-9929-x

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