The derivation of macroscopic irreversible dynamics of nonequilibrium systems from microscopic equations was recently revisited from the point of view of infinitely extended quantum systems. Here we have briefly reviewed the $C^*$ algebra and its application to equilibrium systems as well as introduced some recent results on NESS. In addition, we have demonstrated the derivation of Landauer formula rigorously for quadratic systems but using a more physical presentation.
Quantum statistical mechanics in infinitely extended systems (C∗ algebraic approach)
The derivation of macroscopic irreversible dynamics of nonequilibrium systems from microscopic equations was recently revisited from the point of view of infinitely extended quantum systems. Here we have briefly reviewed the C∗ algebra and its application to equilibrium systems as well as introduced some recent results on NESS. In addition, we have demonstrated the derivation of Landauer formula rigorously for quadratic systems but using a more physical presentation.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
The purpose of this work is to discuss recent progress in deriving the fundamental laws of thermodynamics (0th, 1st and 2nd-law) from nonequilibrium quantum statistical mechanics. Basic thermodynamic notions are clarified and different reversible and irreversible thermodynamic processes are studied from the point of view of quantum statistical mechanics. Special emphasis is put on new adiabatic theorems for steady states close to and far from equilibrium, and on investigating cyclic thermodynamic processes using an extension of Floquet theory.
This is a lecture note of Professor Shuichi Tasaki. In the last part of this manuscript, SA and FB demonstrate some recent results as well. As far as SA's knowledge, regular courses were given every two years for graduate students at Waseda university, and SA attended the course twice (probably in 2005 and 2007 or in 2006 and 2008). Some topics were added in the lectures of the later year, probably Professor Tasaki modified his lectures as he was discovering new results. A similar but intensive short course was given once as at Kyoto university. SA thinks Professor Tasaki had a unique philosophy of physics, which enabled him to give very attractive lectures that unfortunately it is not possible to attend any more. In Professor Tasaki's lectures, both beautiful mathematical views and clear physical intuition were always given together, which SA believe, is quite difficult. As many people know Professor Tasaki was very strong in mathematics, but he never forgot to think of physical interpretations. Professor Tasaki was writing a lecture note of this course based on the note taken by a student of Kyoto university. Unfortunately, Prof. Tasaki passed away before he finished writing the manuscript. This lecture note is based on the incomplete Japanese note[1] and a note taken by SA at Waseda university. It seems that main stories of the lectures are the same, but Professor Tasaki gave different materials in detail, and use different notations. This manuscript mainly follows the note of SA. SA and FB decided to write this manuscript (1) to complete a note of Professor Tasaki's lectures, and (2) to give an access for non-Japanese speaker to his lecture note. However, they are slow writer, and they could not finish it before the deadline of this volume. They could finish only parts before nonequilibrium states (only some of the results will be shown in this version), and some recent results about Landauer formula and sufficient condition on interaction for the existence of unique steady states. They hope to complete the lecture note part, and upload it on Arxiv in the future.
The purpose of this work is to discuss recent progress in deriving the fundamental laws of thermodynamics (0 th, 1 st and 2 nd-law) from nonequilibrium quantum statistical mechanics. Basic thermodynamic notions are clarified and different reversible and irreversible thermodynamic processes are studied from the point of view of quantum statistical mechanics. Special emphasis is put on new adiabatic theorems for steady states close to and far from equilibrium, and on investigating cyclic thermodynamic processes using an extension of Floquet theory. This work is based on the author’s doctoral thesis, ETH-Diss 16187.