Abstract
Chapter 1 showed that looking at part of a system naturally gave rise to memories. This happens also in the quite different context of the study of the central problem of statistical mechanics, viz. understanding how reversibility in the microscopic equations of motion gives rise to irreversibility in the macroscopic equations of motion. Memories arise as an intermediate feature in this voyage from the microscopic to the macroscopic levels of description. Projection operators are particularly designed to perform this passage. After learning the basic concept from the Zwanzig technique of looking at only the diagonal part of a density matrix, the chapter analyzed four examples constituting a mix for simple and involved systems, the final one of which was assigned as an exercise, with solution provided in the literature: a time-dependent complex quantity, the Bloch vector appearing in the control of dynamic localization, the modification of a memory in a lattice due to reservoir interactions, and the open quantum trimer. Attempting the associated exercises should familiarize the reader with analytic calculational details of the projection technique.
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Notes
- 1.
I knew Bob Zwanzig and had the good fortune, when I was a graduate student about to finish my Ph.D., to spend some time in his office at the University of Maryland discussing various aspects of statistical mechanics. I remember his words well. Their sense was that he had introduced projections only to clarify the calculational aspect, and to cut through the conceptual cacophony of the derivation of the Master equation.
- 2.
Although what we are discussing here may not be characterized by great profundity, it is true that many serious practitioners of science, not trained in this particular small area of nonequilibrium quantum statistical mechanics, are not aware of this peculiar way that an evolution of the probabilities of a quantum system can be closed in the probabilities. Indeed, I was present at an event at the Eastern Theoretical Physics Conference held in Rochester in the 1970s when a brilliant theoretical physicist, famous for multiple contributions to our science, shook his head most vehemently at a suggestion made by a junior colleague that such could happen. The latter had been exposed to the projection technique. The former had not. For a small fee along with an oath of secrecy, I might consider naming the parties privately to anyone sufficiently interested.
- 3.
The reader is warned there are special situations in which the initial driving term can change the character of the evolution drastically. This may not be usual but an experimentally relevant realization is discussed in detail for transient grating experiments in the second part of Chap. 5.
- 4.
I have always wanted to use the two adjectives, simple and complex, together in this manner that surely must be considered legitimate.
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Kenkre, V.M.(. (2021). Zwanzig’s Projection Operators: How They Yield Memories. In: Memory Functions, Projection Operators, and the Defect Technique. Lecture Notes in Physics, vol 982. Springer, Cham. https://doi.org/10.1007/978-3-030-68667-3_2
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