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

Demiralp M., Demiralp E., Hernandez-Garcia L.

JOURNAL OF MATHEMATICAL CHEMISTRY, vol.50, no.4, pp.850-869, 2012 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 4
  • Publication Date: 2012
  • Doi Number: 10.1007/s10910-011-9929-x
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.850-869
  • Istanbul Technical University Affiliated: Yes


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.