Interrelations between different canonical descriptions of dissipative systems

TL;DR

This paper compares various canonical models of dissipative systems, analyzing their interrelations, especially focusing on the Bateman model and approaches without explicit environmental variables, to understand their physical interpretations.

Contribution

It clarifies the connections between different canonical descriptions of dissipative systems and explores how additional degrees of freedom can be eliminated through constraints.

Findings

Constraints allow elimination of extra variables in the Bateman model.

Different choices of constraints affect the physical interpretation.

The models are linked via expanding coordinate transformations.

Abstract

There are many approaches for the description of dissipative systems coupled to some kind of environment. This environment can be described in different ways; only effective models will be considered here. In the Bateman model, the environment is represented by one additional degree of freedom and the corresponding momentum. In two other canonical approaches, no environmental degree of freedom appears explicitly but the canonical variables are connected with the physical ones via non-canonical transformations. The link between the Bateman approach and those without additional variables is achieved via comparison with a canonical approach using expanding coordinates since, in this case, both Hamiltonians are constants of motion. This leads to constraints that allow for the elimination of the additional degree of freedom in the Bateman approach. These constraints are not unique. Several choices are studied explicitly and the consequences for the physical interpretation of the additional variable in the Bateman model are discussed.