Classical and Quantum Stochastic Models of Resistive and Memristive Circuits

TL;DR

This paper develops stochastic models for resistive and memristive circuits in phase space, incorporating dissipative components and extending to quantum analogues, with novel symplectic noise concepts and preservation of Poisson structures.

Contribution

It introduces a canonical stochastic framework for classical and quantum circuits with resistors and memristors, including symplectic noise and Poisson structure preservation.

Findings

Dilation of stochastic models preserving Poisson brackets.

Introduction of symplectic noise with conjugate properties.

Quantum analogue of the stochastic circuit models.

Abstract

The purpose of this paper is to examine stochastic Markovian models for circuits in phase space for which the drift term is equivalent to the standard circuit equations. In particular we include dissipative components corresponding to both a resistor and a memristor in series. We obtain a dilation of the problem for which is canonical in the sense that the underlying Poisson Brackets structure is preserved under the stochastic flow. We do this first of all for standard Wiener noise, but also treat the problem using a new concept of symplectic noise where the Poisson structure is extended to the noise as well as the circuit variables, and in particular where we have canonically conjugate noises. Finally we construct a dilation which describes the quantum mechanical analogue.