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The European Physical Journal Plus

Feynman–Vernon influence functional approach for the damped driven oscillator in RLC circuit

L.S. Lima, L. G. de Almeida Arruda

Published Mar 27, 2023• 0 citations
Publisher
DOI
10.1140/epjp/s13360-023-03846-0

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Feynman-Vernon influence functional approach for the damped driven oscillator in RLC circuit

L.S. Lima, L. G. de Almeida Arruda

Abstract The Feynman-Vernon influence functional formalism which is adequate in the study of the dissipative dynamics, is used to describe weakly coupled circuits. The classical behavior of the system formed by resistance, electric and magnetic polarizations is characterized by the influence functional approach whose behavior although is linear in some sense, is not representable by systems of perfect oscillators. In addition, linear combinations of complex q-exponentials are used in representation of nonlinear circuits in which the solution of the correspondent nonlinear differentials equations are mapped in modified differentials equations for a new deformed variable.

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This chapter demonstrates the equivalence between Feynman–Vernon influence functional approach and Keldysh diagram technique developed to describe non-equilibrium phenomena in quantum many-body systems with interactions. Path integrals appearing in both quantum mechanics and quantum field theory may be viewed as a formal representation of the exact infinite series of Feynman diagrams. In other words, path integral and diagram techniques are formally equivalent to each other. The chapter discusses a single degree of freedom, introducing interactions with a bath of harmonic oscillators. In a way, to employ the Caldeira–Leggett model of a bath of harmonic oscillators can be considered just as a physical trick helping to derive the influence functional in the case of linear Ohmic damping. Gaussian influence functionals are exact provided the environment consists of an arbitrary number of harmonic oscillators coupled linearly to the system of interest q .

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Abstract The Feynman-Vernon influence functional formalism which is adequate in the study of the dissipative dynamics, is used to describe weakly coupled circuits. The classical behavior of the system formed by resistance, electric and magnetic polarizations is characterized by the influence functional approach whose behavior although is linear in some sense, is not representable by systems of perfect oscillators. In addition, linear combinations of complex q-exponentials are used in representation of nonlinear circuits in which the solution of the correspondent nonlinear differentials equations are mapped in modified differentials equations for a new deformed variable.
The change of the von Neumann entropy of a set of harmonic oscillators initially in thermal equilibrium and interacting linearly with an externally driven quantum system is computed by adapting the Feynman–Vernon influence functional formalism. This quantum entropy production has the form of the expectation value of three functionals of the forward and backward paths describing the system history in the Feynman–Vernon theory. In the classical limit of Kramers–Langevin dynamics (Caldeira–Leggett model) these functionals combine to three terms, where the first is the entropy production functional of stochastic thermodynamics, the classical work done by the system on the environment in units of k B T, and the second and the third other functionals which have no analogue in stochastic thermodynamics.
This chapter demonstrates the equivalence between Feynman–Vernon influence functional approach and Keldysh diagram technique developed to describe non-equilibrium phenomena in quantum many-body systems with interactions. Path integrals appearing in both quantum mechanics and quantum field theory may be viewed as a formal representation of the exact infinite series of Feynman diagrams. In other words, path integral and diagram techniques are formally equivalent to each other. The chapter discusses a single degree of freedom, introducing interactions with a bath of harmonic oscillators. In a way, to employ the Caldeira–Leggett model of a bath of harmonic oscillators can be considered just as a physical trick helping to derive the influence functional in the case of linear Ohmic damping. Gaussian influence functionals are exact provided the environment consists of an arbitrary number of harmonic oscillators coupled linearly to the system of interest q .