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A probabilistic evolution approach trilogy. III: Temporal variation of state variable expectation values from Liouville equation perspective. (English) Zbl 1365.82022

Summary: This is the third and therefore the final part of a trilogy on probabilistic evolution approach (see Zbl 1365.82020 and Zbl 1365.82021). 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.

MSC:

82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
81S25 Quantum stochastic calculus
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
37N05 Dynamical systems in classical and celestial mechanics
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

Software:

RODAS; MuPAD
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References:

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