Theory of Equilibrium States of Quantum Systems

Oscar E. Lanford, Derek W. Robinson

Jørgen Rammer

Quantum field theory is the application of quantum mechanics to systems with infinitely many degrees of freedom. This 2007 textbook presents quantum field theoretical applications to systems out of equilibrium. It introduces the real-time approach to non-equilibrium statistical mechanics and the quantum field theory of non-equilibrium states in general. It offers two ways of learning how to study non-equilibrium states of many-body systems: the mathematical canonical way and an easy intuitive way using Feynman diagrams. The latter provides an easy introduction to the powerful functional methods of field theory, and the use of Feynman diagrams to study classical stochastic dynamics is considered in detail. The developed real-time technique is applied to study numerous phenomena in many-body systems. Complete with numerous exercises to aid self-study, this textbook is suitable for graduate students in statistical mechanics and condensed matter physics.

Fabio Anza, Cameron Hahn

Projected ensembles---the collections of conditional pure states induced on a system by measuring an entangled environment---have become central objects in quantum science, underlying deep thermalization, quantum state designs, the emergence of classicality, and exhibiting interesting phase transitions. Here we develop a general and exact theory of their dynamics, yielding a systematic approach to their dynamics and equilibrium and non-equilibrium stationary states. We derive an exact continuity equation on quantum state space for the projected ensemble, together with microscopic kinetic equations and analytical expressions for the probability flux and source terms generated by the system-environment interaction. The resulting dynamics admits a classical representation: isolated systems obey Hamiltonian transport and Liouville's theorem, while open systems are described by a kinetic theory of probability transport. The stationary continuity equation provides a general characterization of equilibrium and non-equilibrium stationary projected ensembles, which can be analyzed through the method of characteristics. The theory provides an analytical framework for studying the emergence, structure, and timescales of equilibrium and non-equilibrium stationary projected ensembles, with applications ranging from deep thermalization and the emergence of classicality to random quantum-state generation and quantum device benchmarking.

Pierre M. V. Résibois

H.Y. Chu, Z. Ye, F. C. Khanna +1 more

In this paper we study the dynamical properties of a dissipative two-state fermionic system embedded in a phonon bath at finite temperature. The non-equilibrium thermo field dynamics, a real time formalism of thermal quantum field theory, is employed to formulate time dependent energy renormalization and derive transport equations. The renormalization procedures derived in this paper replace the on-shell energy definitions in stationary situations and thus can be used in any situations. The procedure reduces to that of equilibrium situations when there is no time dependence. We show that in a system where there are many coupled stages (with multi-components) quantum fluctuations among these states cannot be eliminated so that the long time limit of non-equilibrium theory does not exactly reproduce an equilibrium theory. The procedures and conclusions obtained also apply to more general systems with multi-components.

Gérard G. Emch, Chuang Liu