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Lecture notes in physics

Non-Equilibrium Formalism

Jonas Fransson

Published Jan 1, 2010• 0 citations
Publisher
DOI
10.1007/978-90-481-9210-6_4

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The reduced dynamics formalism has recently emerged as a powerful tool to study the dynamics of non-equilibrium quantum impurity models in strongly correlated regimes. Examples include the non-equilibrium Anderson impurity model near the Kondo crossover temperature and the non-equilibrium Holstein model, for which the formalism provides an accurate description of the reduced density matrix of the system for a wide range of timescales. In this work, we generalize the formalism to allow for non-system observables such as the current between the impurity and leads. We show that the equation of motion for the reduced observable of interest can be closed with the equation of motion for the reduced density matrix and demonstrate the new formalism for a generic resonant level model.
Since Gibbs synthesized a general equilibrium statistical ensemble theory, many theorists have attempted to generalized the theory to non-equilibrium domain, however the status of the theory of non-equilibrium phenomena can not be said so well established as the Gibbsian ensemble theory. In this work, we present a formalism for the non-equilibrium statistical ensemble based on a subdynamic kinetic equation (SKE) rooted from the Brussels-Austin school and followed by some up-to-date works. The constructed key is to use a similarity transformation between Gibbsian ensembles formalism based on Liouville equation and the subdynamic ensemble formalism based on the SKE. Using this formalism, we study the spin-Boson system, as cases of weak coupling or strongly coupling, and obtain the reduced density operators for the Canonical ensembles easily.