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Physics Letters A

Action principle for nonequilibrium statistical dynamics based on the Lindblad density matrix evolution

A. K. Rajagopal

Published Mar 1, 1997• 15 citations
Publisher
DOI
10.1016/s0375-9601(97)00095-9

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A Lindblad approach is presented to describe quantum dynamics of open systems. It is based on a construction of a Lindblad functional from the microscopic Caldeira-Leggett model for linear dissipation. It leads to a master equation for the reduced density matrix, which preserves positive evolution on short times and asymptotically approaches equilibrium at high and low temperatures. This master equation is applied to study the femtosecond dynamics of vibrational heating, relaxation, and bond breaking at a metal surface. Both a direct solution of the density matrix and an indirect solution with stochastic wave functions are presented. The latter, besides its statistical equivalence to the density matrix, provides a more physical picture about quantum jumps of individual open systems.
In this paper, we outline the main features of the nonlinear quantum evolution equation proposed by the present author. Such an equation may be used as a model of reduced subsystem dynamics to complement various historical and contemporary efforts to extend linear Markovian theories of dissipative phenomena and relaxation based on master equations, Lindblad and Langevin equations, to the nonlinear and far nonequilibrium domain. It may also be used as the fundamental dynamical principle in theories that attempt to unite mechanics and thermodynamics, such as the Hatsopoulos–Gyftopoulos unified theory which motivated the original development of this well-behaved general nonlinear equation for the evolution of the density operator capable of generating irreversible deterministic relaxation to thermodynamic equilibrium from any far nonequilibrium state even for an isolated system.
The theoretical description of materials' properties driven out of equilibrium has important consequences in various fields such as semiconductor spintronics, nonlinear optics, and continuous and discrete quantum information science and technology. The coupling of a quantum many-body system to an external bath can dramatically modify its dynamics compared to that of closed systems; new phenomena like relaxation and decoherence appear as a consequence of the non-unitary evolution of the quantum system. In addition, electron-electron correlations must be properly accounted for in order to go beyond a simple one-electron or mean-field description of the electronic system. Here, we discuss a first-principles methodology based on the evolution of the electronic density matrix capable of treating electron-environment interactions and electron-electron correlations at the same level of description. The effect of the environment is separated into a coherent contribution, like the coupling to applied external electromagnetic fields, and an incoherent contribution, like the interaction with lattice vibrations or the thermal background of radiation. Electron-electron interactions are included using the nonequilibrium Green's function plus the generalized Kadanoff-Baym ansatz. The obtained non-Markovian coupled set of equations reduces to the ordinary Lindblad quantum master equation form in the Markovian limit.