Density-operator evolution: Complete positivity and the Keldysh real-time expansion

We study the reduced time-evolution of open quantum systems by combining quantum-information and statistical field theory. Inspired by prior work [EPL 102, 60001 (2013) and Phys. Rev. Lett. 111, 050402 (2013)] we establish the explicit structure guaranteeing the complete positivity (CP) and trace-preservation (TP) of the real-time evolution expansion in terms of the microscopic system-environment coupling. This reveals a fundamental two-stage structure of the coupling expansion: Whereas the first stage defines the dissipative timescales of the system --before having integrated out the environment completely-- the second stage sums up elementary physical processes described by CP superoperators. This allows us to establish the nontrivial relation between the (Nakajima-Zwanzig) memory-kernel superoperator for the density operator and novel memory-kernel operators that generate the Kraus operators of an operator-sum. Importantly, this operational approach can be implemented in the existing Keldysh real-time technique and allows approximations for general time-nonlocal quantum master equations to be systematically compared and developed while keeping the CP and TP structure explicit. Our considerations build on the result that a Kraus operator for a physical measurement process on the environment can be obtained by 'cutting' a group of Keldysh real-time diagrams 'in half'. This naturally leads to Kraus operators lifted to the system plus environment which have a diagrammatic expansion in terms of time-nonlocal memory-kernel operators. These lifted Kraus operators obey coupled time-evolution equations which constitute an unraveling of the original Schr\"odinger equation for system plus environment. Whereas both equations lead to the same reduced dynamics, only the former explicitly encodes the operator-sum structure of the coupling expansion.Comment: Submission to SciPost Physics, 49 pages including 6 appendices, 13 figures. Significant improvement of introduction and conclusion, added discussions, fixed typos, no results change

Range separation: The divide between local structures and field theories

This work presents parallel histories of the development of two modern theories of condensed matter: the theory of electron structure in quantum mechanics, and the theory of liquid structure in statistical mechanics. Comparison shows that key revelations in both are not only remarkably similar, but even follow along a common thread of controversy that marks progress from antiquity through to the present. This theme appears as a creative tension between two competing philosophies, that of short range structure (atomistic models) on the one hand, and long range structure (continuum or density functional models) on the other. The timeline and technical content are designed to build up a set of key relations as guideposts for using density functional theories together with atomistic simulation.Comment: Expanded version of a 30 minute talk delivered at the 2018 TSRC workshop on Ions in Solution, to appear in the March, 2019 issue of Substantia (https://riviste.fupress.net/index.php/subs/index

The large deviation approach to statistical mechanics

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text, figures and appendices added, many references cut, close to published versio