Hamiltonian structure for dispersive and dissipative dynamical systems

TL;DR

This paper introduces a Hamiltonian framework for modeling dispersive and dissipative media, coupling the system to auxiliary heat bath fields to exactly reproduce dissipative dynamics and derive physical quantities.

Contribution

It develops a Hamiltonian theory for dispersive and dissipative systems using auxiliary fields, providing exact reproduction of dissipative evolution and explicit expressions for physical quantities.

Findings

Exact Hamiltonian representation of dissipative systems

Closed-form expressions for energy and momentum densities

Approximate formulas for time-averaged energy and stress tensors

Abstract

We develop a Hamiltonian theory of a time dispersive and dissipative inhomogeneous medium, as described by a linear response equation respecting causality and power dissipation. The proposed Hamiltonian couples the given system to auxiliary fields, in the universal form of a so-called canonical heat bath. After integrating out the heat bath the original dissipative evolution is exactly reproduced. Furthermore, we show that the dynamics associated to a minimal Hamiltonian are essentially unique, up to a natural class of isomorphisms. Using this formalism, we obtain closed form expressions for the energy density, energy flux, momentum density, and stress tensor involving the auxiliary fields, from which we derive an approximate, ``Brillouin-type,'' formula for the time averaged energy density and stress tensor associated to an almost mono-chromatic wave.