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Physica A Statistical Mechanics and its Applications

General nonequilibrium statistical ensemble formalism and its application to quantum computing

Bi Qiao, Harry E. Ruda

Published Jun 3, 2003• 3 citations
Publisher
DOI
10.1016/s0378-4371(03)00365-0

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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.
What kind of ensemble has to be used for describing general nonequilibrium behavior is one of the unsolved problems of nonequilibrium statistical mechanics. We would like to propose a formalism for nonequilibrium many-body systems which follows the spirit of Gibbs' ensemble theory. We shall set forth a new perspective of Kirkwood's time-smoothing operation to define the initial density matrix. We will show that such an initial density matrix may be employed to prove the positivity of entropy production.
In this work, we propose a transition wave equation from quantum to classical regime in the framework of the von Neumann formalism for ensembles and then obtain an equivalent scaled equation. This leads us to develop a scaled statistical theory following the well-known Wigner-Moyal approach of quantum mechanics. This scaled nonequilibrium statistical mechanics has in it all the ingredients of the classical and quantum theory described in terms of a continuous parameter displaying all the dynamical regimes in-between the two extreme cases. Finally, a simple application of our scaled formalism consisting of reflection from a mirror by computing various quantities including probability density plots, scaled trajectories and arrival times are analyzed.