Designing the mesoscopic approach of an autonomous linear dynamical system by a quantum formulation

Abstract

[EN] This paper is an attempt to translate the quantum formulation from physics to general systems modelled by dynamical systems. Their quantum formulation provides a mesoscopic approach due to the stochastic nature of quantum theory. A quantum formulation needs a previous Hamiltonian. A first order Hamiltonian was provided in past works by following Dirac’s generalized dynamics. The corresponding quantum approach, given by a first order Schrodinger equation, was provided from the Hamiltonian found, by applying the corresponding quantization rules. The split of the wave function in its amplitude and phase provides the probability conservation law for the square amplitude and the HamiltonJacobi equation for the phase. However, this last equation lacks the stochastic term, in opposition to the stochastic term appearing for the current Schrodinger equation corresponding to the physics laws. Thus, the approach presented in past works is unsuitable to obtain a mesoscopic approach for dynamical systems. The hypothesis here presented considers the existence of a second order Hamiltonian from the base that the physics laws can be defined with a similar structure. The existence of a Hamiltonian like this is proved to be always possible for the case of an autonomous linear dynamical system. In the beginning the Schrodinger equation is written for this case, as well as its timeindependent version. This case does present a mesoscopic approach, which is developed and its stochastic term is stressed. The general solution of the Schrodinger equation is found and two application cases are presented. In the conclusions section some ways are sketched about how to generalize the formalism to nonlinear dynamical systems.