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Physical Review Letters

Reformulation of steady state nonequilibrium quantum statistical mechanics

Selman Hershfield

Published Apr 5, 1993• 139 citations
Publisher

Abstract

Starting from the usual formulation of nonequilibrium quantum statistical mechanics, the expectation value of an operator A in a steady state nonequilibrium quantum system is shown to have the form 〈A〉 =Tr{${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}$A} /Tr{${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}$}, where H is the Hamiltonian, \ensuremath{\beta} is the inverse of the temperature, and Y is an operator which depends on how the system is driven out of equilibrium. Because 〈A〉 is not expressed as a sum of correlation functions integrated over real time, one can now consider performing nonperturbative calculations in interacting nonequilibrium quantum problems.

DOI
10.1103/physrevlett.70.2134

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Starting from the usual formulation of nonequilibrium quantum statistical mechanics, the expectation value of an operator A in a steady state nonequilibrium quantum system is shown to have the form 〈A〉 =Tr{e\ensuremath−\ensuremathβ(H\ensuremath−Y){\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}e\ensuremath−\ensuremathβ(H\ensuremath−Y)A} /Tr{e\ensuremath−\ensuremathβ(H\ensuremath−Y){\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}(\mathit{H}\mathrm{\ensuremath{-}}\mathit{Y})}e\ensuremath−\ensuremathβ(H\ensuremath−Y)}, where H is the Hamiltonian, \ensuremath{\beta} is the inverse of the temperature, and Y is an operator which depends on how the system is driven out of equilibrium. Because 〈A〉 is not expressed as a sum of correlation functions integrated over real time, one can now consider performing nonperturbative calculations in interacting nonequilibrium quantum problems.
Entropy in nonequilibrium statistical mechanics is investigated theoretically so as to extend the well-established equilibrium framework to open nonequilibrium systems. We first derive a microscopic expression of nonequilibrium entropy for an assembly of identical bosons/fermions interacting via a two-body potential. This is performed by starting from the Dyson equation on the Keldysh contour and following closely the procedure of Ivanov, Knoll and Voskresensky [Nucl. Phys. A {\bf 672} (2000) 313]. The obtained expression is identical in form with an exact expression of equilibrium entropy and obeys an equation of motion which satisfies the HHH-theorem in a limiting case. Thus, entropy can be defined unambiguously in nonequilibrium systems so as to embrace equilibrium statistical mechanics. This expression, however, differs from the one obtained by Ivanov {\em et al}., and we show explicitly that their ``memory corrections'' are not necessary. Based on our expression of nonequilibrium entropy, we then propose the following principle of maximum entropy for nonequilibrium steady states: ``The state which is realized most probably among possible steady states without time evolution is the one that makes entropy maximum as a function of mechanical variables, such as the total particle number, energy, momentum, energy flux, etc.'' During the course of the study, we also develop a compact real-time perturbation expansion in terms of the matrix Keldysh Green's function.
We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I\ensuremath{-}V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
By mapping steady-state nonequilibrium to an effective equilibrium, we formulate nonequilibrium problems within an equilibrium picture. The Hamiltonian in the open system is rewritten in terms of scattering states with appropriate boundary condition. We first study the analytic properties of many-body scattering states, impose the boundary-condition operator in a statistical operator and prove that this mapping is equivalent to the linear-response theory in the low-bias limit. In an example of infinite-UUU Anderson impurity model, we approximately solve the scattering state creation operators, based on which we derive the bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ to construct the nonequilibrium ensemble in the form of the Boltzmann factor ${e}^{\ensuremath{-}\ensuremath{\beta}(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y})}$. The resulting effective Hamiltonian is solved by noncrossing approximation. We obtain the $I\text{\ensuremath{-}}V$ features of Kondo anomaly conductance at zero bias, inelastic transport via the charge excitation on the quantum dot, and significant inelastic current background over a wide range of bias.
Quantum response theory is a cornerstone of statistical physics.However, the standard Kubo formalism is restricted to isolated equilibrium systems.We here generalize Kubo's results to open quantum systems in nonequilibrium steady states.We derive three different, but equivalent, forms of the quantum response function.We discuss for each of them the role of the noncommutativity of quantum operators and introduce a steady-state extension of the Kubo transformation.We show in particular that the equilibrium response vanishes for perturbations that commute with the unperturbed Hamiltonian, while the steady-state response does not, highlighting the profound difference between the two linear response theories.
In this paper, we formally demonstrate that the nonequilibrium density matrix developed by Hershfield for the steady state has the form of a McLennan-Zubarev nonequilibrium ensemble. The correction term in this pseudoequilibrium Gibbs-like ensemble is directly related to the entropy production in the quantum open system. The fact that both methods state that a nonequilibrium steady state can be mapped onto a pseudoequilibrium, permits us to develop nonequilibrium quantities from formal expressions equivalent to the equilibrium case. We provide an example: the derivation of a nonequilibrium distribution function for the electron population in a scattering region in the context of quantum transport.