Nonequilibrium statistical mechanics. I. Mathematical formalism. Theorem of N. S. Krylov

V. I. Gerasimenko, D. Ya. Petrina

Esteban Calzetta, B. L. Hu

Perhaps due to its technical complexity, oftentimes one sees in research papers on nonequilibrium quantum field theory (NEqQFT) more emphasis placed on the field-theoretical formalisms than the ideas these sophisticated techniques attempt to capture, or the issues such problems embody. All the more so, we need some basic understanding of the important issues and concepts in nonequilibrium statistical mechanics (NEqSM), and how they are manifested in the context of quantum field theory. Many important advances in this field came from asking such questions and finding out how to answer them in the language of quantum field theory. Because of this somewhat skewed existing emphasis in NEqQFT, and since we do not assume the reader to have had a formal course on NEqSM before, we shall give a brief summary of the basic concepts of NEqSM relevant to the field-theoretical processes discussed in this book. Many fine monographs and reviews written on this subject take a more formal mathematical approach. approach. Since our purpose here is to familiarize readers with these issues and their subtleties, rather than training them to work in the rich field of NEqSM (which includes in addition to the traditional subject matter such as the projection operator formalism and open system concepts, also current topics at their foundation, such as dynamical systems and quantum chaos), we choose to approach these topics in a more intuitive and physical way, sacrificing by necessity rigor and completeness.

Vitor Rocha Vieira

The finite-temperature real-time field theories for spin (1/2 are derived with use of a functional-integral approach, and their application to problems of equilibrium and nonequilibrium statistical mechanics discussed. These formalisms have a standard Wick's theorem and linked-graph theorem, leading to the use of Feynman diagrams. Similar to the problems with bosons or fermions, the calculation of equilibrium quantities requires the consideration of the imaginary-time branch, leading to a three-component formalism, instead of the usual two-component formalism, typical of real-time field theories.

Vojkan Jakšić, Claude‐Alain Pillet

Zhaoyu Fei, H. T. Quan

The calculation of work distributions in a quantum many-body system is of significant importance and also of formidable difficulty in the field of nonequilibrium quantum statistical mechanics. To solve this problem, inspired by the Schwinger-Keldysh formalism, we propose the contour-integral formulation for work statistics. Based on this contour integral, we show how to do the perturbation expansion of the characteristic function of work (CFW) and obtain the approximate expression of the CFW to the second order of the work parameter for an arbitrary system under a perturbative protocol. We also demonstrate the validity of fluctuation theorems by utilizing the Kubo-Martin-Schwinger condition. Finally, we use noninteracting identical particles in a forced harmonic potential as an example to demonstrate the powerfulness of our approach.

David Pierre Ruelle