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Physical Review

Generalized Master Equation for Arbitrary Initial States

Jerome Weinstock

Published Oct 4, 1965• 9 citations
Publisher

Abstract

Exact generalized master equations, for both quantum and classical systems, are derived for completely arbitrary initial states (arbitrary initial "correlations") in the form of a "density expansion." This result is a generalization of a previous equation which was restricted to initially "uncorrelated" states.

DOI
10.1103/physrev.140.a98

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Exact generalized master equations, for both quantum and classical systems, are derived for completely arbitrary initial states (arbitrary initial "correlations") in the form of a "density expansion." This result is a generalization of a previous equation which was restricted to initially "uncorrelated" states.
Generalized master equations determining the time dependence of both diagonal and off-diagonal density matrix elements are derived. The equations are valid for systems of arbitrary size and initial conditions, and have all the generality of the von Neumann equation for the density matrix. They are general enough for use in determining all higher-order contributions to transport coefficients. Their advantage over the von Neumann equation is that they are in forms suited to reduction by time-independent many-body approximation techniques.
An exact generalized master equation is derived for a large quantum-mechanical system in the form of a power series in the density. This derivation is a quantum-mechanical generalization of a previous work by the author for classical systems. The quantum equation can be viewed as a time-dependent analog of the virial expansion of the quantum-mechanical partition function---for both degenerate systems (Bose-Einstein or Fermi-Dirac statistics) and nondegenerate systems. The coefficients of the series, in the quantum master equation, are explicitly given in terms of operators (Green functions) which are determined by the dynamics of isolated groups of particles and are convergent functions of the interaction potential. Equations are obtained for the off-diagonal elements of the density matrix as well as for the diagonal elements. The equation for the diagonal elements is shown to reduce to a Markoffian master equation for asymptotically long times, and an explicit expression is obtained for the "scattering" operator of this asymptotic equation.
For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.
We show that the exact master equation incorporating initial correlations for open quantum systems, within the Nakajima-Zwanzig operator-projection method, is a homegenous master equation for the reduced density matrix. We also derive explicitly the exact master equation for a large class of bosonic and fermionic open quantum systems incorporating initial correlations, the resulting master equation is homogenous. We find that the effects of the initial correlations can be fully embedded into the fluctuation dynamics through the exact homogenous master equation. Also a generalized nonequilibrium fluctuation-dissipation theorem incorporating the initial correlations is obtained.
We derive an exact master equation that captures the dynamics of a quadratic quantum system linearly coupled to a Gaussian environment of the same statistics: the Gaussian master equation (GME). Unlike previous approaches, our formulation applies universally to both bosonic and fermionic setups, and remains valid even in the presence of initial system-environment correlations, allowing for the exact computation of the system's reduced density matrix across all parameter regimes. Remarkably, the GME shares the same operatorial structure as the Redfield equation and depends on a single kernel: a dressed environment correlation function accounting for all virtual interactions between the system and the environment. This simple structure grants a clear physical interpretation and makes the GME easy to simulate numerically, as we show by applying it to an open system based on two fermions coupled via superconductive pairing.
The structure of the initial system-environment state is fundamental to determining the nature and characteristics of the evolution of such an open quantum system. The usual assumption is to consider that the initial system-environment state is separable. Here, we go beyond this simple case and derive the evolution equations, up to second order in a weak-coupling expansion, that describe the evolution of the reduced density matrix of the system for any arbitrary system-environment initial state. The structure of these equations allows us to determine the initial conditions for which the evolution of the reduced density matrix can be written in terms of a set of Lindblad-like equations, once considering the Markov and secular approximations. Moreover, we show that for initial states belonging to a subset of separable states such Lindblad-like equations become actual Lindblad equations and that the evolution of ${\ensuremath{\rho}}_{s}$ thereby obtained is also trace and positive preserving.