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Physical Review

Green's-Function Theory of Nonlinear Transport Coefficients

Tomoyasu Tanaka, K. Moorjani, Tohru Morita

Published Mar 10, 1967• 29 citations
Publisher

Abstract

The nonlinear transport coefficients are expressed in terms of many-time Green's functions. Except for the inhomogeneous term, the equation of motion for the many-time Green's functions is shown to be the same as that for the two-time Green's function. Some simple applications of the method are discussed.
DOI
10.1103/physrev.155.388

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The problem of nonlinear response is considered by employing a general time-evolution equation, and a Green's function which is the transition or conditional probability density for an unperturbed system. Expansion of the Green's function in terms of orthonormal functions enables us to express the distribution function describing the nonlinear behavior by means of matrix products whose elements are composed of correlation functions in the absence of the perturbation. In other words, it is shown how the distribution function induced by a strong perturbation may be calculated by knowing the Green's function without the perturbation. As the special case of the linear response, we have obtained Kubo's relation. The Laplace-transform technique with respect to time is found quite useful in developing the present theory in which the transient effect is also taken into account. As an application of the theory, a new relation valid in the region of the second-order perturbation connecting the transient rise and decay with the stationary alternating perturbations has been obtained.
The present paper is concerned with the retarded many-time thermodynamic Green's function method in classical statistical mechanics. In the formulation of the nonlinear response of a dynamical variable, unaveraged Green's functions are specified and used to discuss the physical meaning of many-time Poisson-bracket nests. Equilibrium ensemble averages of these Green's functions are relevant to the classical Kubo response theory, the latter of which is interpreted from the viewpoints of both the classical ``Heisenberg'' and ``Schrödinger'' pictures. A method for evaluating the many-time thermodynamic Green's functions based on their definition in terms of the resolvent operator is suggested and discussed in relation to the linear dielectric function and the anharmonic oscillator. The many-time Green's function equations of motion are shown to be equivalent to identities based on the resolvent operator. It is shown how microscopic equations of motion can be used to derive the Bogoliubov-Sadovnikov (B-S) hierarchy from the lowest-order many-time Green's function equation of motion.
Thermodynamic perturbation theory is described in terms of temperature-dependent Green functions. The relationship of these functions with time-temperature-dependent Green functions and with correlation functions is discussed and some of their analytic properties are derived. The chain of coupled equations forming the equations of motion of the Green functions is derived and is written in terms of functional derivatives. Two examples of the use of Green functions, discussions of the sharpness of the Fermi surface and of neutron scattering from liquid helium, are given. The use of Green functions in the theory of relaxation phenomena is described.
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
Abstract In the previous chapters, we saw how various physical observables can be expressed in terms of retarded Green’s functions. In many cases we need to calculate the time dependence of these functions. There are several ways of attacking this problem, one of which is the equation of motion technique. The basic idea of this method is to generate a series of coupled differential equations by differentiating the function at hand a number of times. If these equations close, the problem is in principle solvable; and if not, one needs to invoke physical arguments to truncate the set of equations in a reasonable fashion. For example, one can neglect certain correlations. We shall study examples of both situations in this chapter.
Part I: Green's Functions in Mathematical Physics - Time-Independent Green's Functions / Time-dependent Green's Functions / Part II: One-Body Green's Functions - Physical Significance of One-Body Green's Functions / Green's Functions and Perturbation Theory / Green's Functions for Tight-Binding Hamiltonians / Single Impurity Scattering / Extension Theory for Lattice Green's Functions / Part III: Many-Body Green's Functions - Field Operators and Three Pictures / Definition and Properties of Many-Body Green's Functions / The Diagram Technique for Zero-Temperature Green's Functions / Definition and Properties of Matsubara Green's Functions / Diagram Technique for the Matsubara Green's Functions / Three Approximation Schemes of the Diagram Technique / Linear Response Theory / The Equation of Motion Technique for the Green's Functions / Magnetic Systems Described by Heisenberg Model / The Green's Functions for Boson Systems with Condensation / Superconductors with Weak Interaction Between Electrons / Nonequilibrium Green's Functions / Electronic Transport through a Mesoscopic Structure / Appendixes / Index.