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Fuza xitong yu fuzaxing kexue

A Framework for Non-Equilibrium and Equilibrium Statistical Ensembles

Fang Jin-Qing

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

In this paper, we establish a non-equilibrium statistical ensemble formalism through a subdynamic equation.A framework constructed to study the weak or strong interaction in the case of the spin-boson model, so that it is easy to get a reduced density operator for the quantum canonical ensembles. A strongly correlated quantum network based on the Anderson model has also been discussed.

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In this paper, we establish a non-equilibrium statistical ensemble formalism through a subdynamic equation.A framework constructed to study the weak or strong interaction in the case of the spin-boson model, so that it is easy to get a reduced density operator for the quantum canonical ensembles. A strongly correlated quantum network based on the Anderson model has also been discussed.
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.
Since Gibbs synthesized a general equilibrium statistical ensemble theory, generalizing the Gibbsian theory to non-equilibrium phenomena has been a dream for some theorists. However, the status of the theory for the non-equilibrium ensemble formalism is not as well established as the Gibbsian ensemble theory. In this work, we explore the formalism behind the non-equilibrium statistical ensemble theory based on the sub-dynamic kinetic equation originating from the Brussels-Austin school and developed further by others. The key is establishing a similarity transformation between Gibbsian ensembles and the sub-dynamic ensemble. Using this way, we study a spin-Boson model with strong coupling. A reduced density operator for the canonical ensemble can be easily obtained and the relevant decoherence-free property is revealed.
In this work,we present a nonequilibrium ensemble formalism based on the subdynamic kine-tic equation.The constructing procedure is to take a similarity transformation to Gibbsian ensemble formalism.This allows one to obtain subdynamic ensemble formalism for projected density operator.This formalism can be considered as a generalization of the equilibrium quantum canonical ensemble formula in the sense as 1) if the similarity operator is unitary,then the new formula is just an effective representation of the old canonical ensemble formula,2) if the similarity operator is non-unitary,then the new formula is an extension of the old formula,which represents a kind of non-equilibrium quantum canonical ensembles formula and reflects irreversibility of the system,and 3) if the similarity operator can be deduced by some approximations,such as Markovian/non-Markovian approximations,then the new formula can expose some irreversibility characteristics of the system which can not be gained from the equilibrium quantum ensemble formulas.As an application,we study a quantum strongly correlated network described by an Anderson model.A Green function based on this formalism is also introduced.We find the complex entropy of the system to be related to appearance of the local magnetic vectors in the nodes of the network,where the image part of the entropy represents an evolution rate of the entropy.
In this work, we present a general non-equilibrium ensemble formalism based on the subdynamic equation (SKE). The constructing procedure is to use a similarity transformation between Gibbsian ensemble formalism and the non-equilibrium ensemble formalism. The obtained density distribution is a projected one that can represent essence part of (irreversible) evolution of the density distribution, by which a generalized reduced density distribution for the quantum canonical ensembles is studied and applications in Cayley tree and spin network are discussed.
A framework for statistical-mechanical analysis of quantum Hamiltonians is introduced. The approach is based upon a gradient flow equation in the space of Hamiltonians such that the eigenvectors of the initial Hamiltonian evolve towards those of the reference Hamiltonian. The nonlinear double-bracket equation governing the flow is such that the eigenvalues of the initial Hamiltonian remain unperturbed. The space of Hamiltonians is foliated by compact invariant subspaces, which permits the construction of statistical distributions over the Hamiltonians. In two dimensions, an explicit dynamical model is introduced, wherein the density function on the space of Hamiltonians approaches an equilibrium state characterized by the canonical ensemble. This is used to compute quenched and annealed averages of quantum observables.