We use the equation-of-motion technique to establish a chain of equations for retarded, double-time Green's functions of an ideal gas. Exact termination of this chain allows reproduction of the usual results obtained by other methods.
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.
The effects of exchange and correlations on the diamagnetic susceptibility of an interacting electron gas are investigated using the double-time Green's-function method. The equation of motion of the appropriate Green's function is linearized using a moment-conserving decoupling approximation. The many-body corrections to the diamagnetic susceptibility at metallic densities are found to be small. The present expression reduces to the exact result in the high-density limit. Many of the results of earlier calculations also follow from our expression for the diamagnetic susceptibility.
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.