A probabilistic evolution approach trilogy, part 3: temporal variation of state variable expectation values from Liouville equation perspective Original Paper

First Online: 19 October 2012 Received: 09 August 2012 Accepted: 02 September 2012 DOI :
10.1007/s10910-012-0081-z

Cite this article as: Demiralp, M. & Tunga, B. J Math Chem (2013) 51: 1198. doi:10.1007/s10910-012-0081-z
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Abstract This is the third and therefore the final part of a trilogy on probabilistic evolution approach. The work presented here focuses on the probabilistic evolution determination for the state variables of a many particle system from classical mechanical point of view. Probabilistic evolution involves the expected value evolutions for all natural number Kronecker powers of the state variables, positions and momenta. We use the phase space distribution of the Liouville equation perspective to construct the expected values of the state variables’ Kronecker powers to define unknown temporal functions. The infinite number homogeneous linear ODEs with an infinite constant coefficient matrix are constructed by following the same steps as in the previous two works on quantum mechanics. The only difference is in the definitions of the expected values here. We also focus on a system of many harmonic oscillators to illustrate the block triangularity.

Keywords Probabilistic evolution Expected value dynamics Evolution matrix Phase space distribution Elastic spring forces

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Authors and Affiliations 1. Istanbul Teknik Üniversitesi Bilişim Enstitüsü Istanbul Turkey