Methods in Statistical Mechanics

MANUEL GADELLA; OSVALDO CIVITARESE

Springer

Año: 2020 p. 179

978-3-030-53657-2

PrefaceAmong the various branches of theoretical physics, the statistical mechanics ispresent in the analysis of practically all physical systems, independently of the scaleof them. Examples on the applications of statistical concepts are, for instance, thetheory of massive astrophysical objects, the models of elementary particle systemsat finite densities and temperatures, diverse types of classical and quantum fluids,the thermodynamics of real and ideal gases, etc. To these examples, we may add theextremely rich field of phase transitions, both quantum and classical.The most widely adopted models of hadrons and their interactions, of nucleonand nuclear structure, atoms and molecules, and ultimately of extended bodies (say,e.g., solids, liquids, gases) often rely on concepts like densities, temperature andvarious forms of statistical equilibrium.The conventional presentations of statistical mechanics usually resort to idealsystems without interactions, or to approximations like mean-field scenarios. Toconcepts like the statistical equilibrium and the correspondence between calculatedand observable quantities, it would be desirable to add ingredients like interactions,finite size effects, dimensional dependence, symmetry breaking and symmetryrestoring effects. Real systems are far to belong to the class of ideal ones, and theirnumber of components are not always infinity, meaning that the vast domain of fewparticle systems constitutes a terra ignota from the point of view of statisticalmechanics.In this book, we are presenting different techniques meant to tackle some of thefeatures exceeding the conventional approach. Consequently, we shall introducemethods based on the use of path integration, thermal Green functions,time-temperature propagators, Liouville operators, second quantization, and fieldcorrelators at finite density and temperature. We shall also address the questionof the statistical mechanics of unstable quantum systems.In writing this book, we have benefited from the existing literature. In selectingthe examples about the applications of the various techniques, we have revisitedsome of the most influential books and papers in the field, as indicated in the text.The following is a list of the contents of the book: Chap. 1 contains a revisionof the classical and quantum statistical mechanics formulated for discrete andcontinuous spectra, based on the notion of probabilities and thermal equilibrium.Chapter 2 is devoted to the role of dynamics, with an emphasis on the connectionsvviPrefacebetween Liouville dynamics and statistical mechanics. Chapter 3 contains thenotions of operators and their role in statistical mechanics, particularly in thecombined time-temperature representations. Chapter 4 reviews the Feynmanpath-integral formulation. Chapter 5 explores the principles of statistical mechanicsin terms of geometry. Chapter 6 is a formal continuation of the previous chapter,which specializes in the connections with the statistical ensembles. Chapter 7contains the basic notions which support the use of statistical mechanics forunstable systems.This book is meant to be used for a semester course, following graduate lecturesin quantum mechanics, thermodynamics, electromagnetism and mathematicalmethods in physics. The material of the book is self-contained from the mathe-matical point of view, and the subjects are arranged sequentially.The authors acknowledge the support of the National Research Council ofArgentina (grant PIP-616), and of the Ministry of Economy and Productivity ofSpain (grant MTM 2014-57129-C2-1-P) and of the Junta de Castilla y León (grantBU229P18). The hospitality of the Department of Physics (National University ofLa Plata, Argentina), the Institute of Physics of La Plata (National Research Councilof Argentina), and of the Department of Atomic and Theoretical Physics and Opticsof the University of Valladolid, Spain, where parts of this book have been written,is gratefully acknowledged by the authors.La Plata, ArgentinaJune 2020Osvaldo CivitareseManuel Gadella