The problem of response theory in statistical mechanics involves the determination of the density matrix $\ensuremath{\rho}$ from the Liouville equation and the subsequent computation of the response
r from this $\ensuremath{\rho}$. Projection techniques are applied to avoid the entire complicated problem of the full dynamics of $\ensuremath{\rho}$ and to select only that part of $\ensuremath{\rho}$ which is relevant to the response
r. The procedure replaces an inhomogeneous equation by a linear homogeneous integrodifferential equation for response theory. This is a very general equation which can be analyzed in different ways to yield a variety of results. It is shown that the Kubo theory of linear response emerges as the lowest-order approximation. The general equation is solved without approximations for a step-function stimulus, and it is discussed in the context of the steady state.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
\nthe quantum calculation of the d.c. electrical conductivity for an electron system is reconsidered by using an improved form of the long-time solution of the density matrix. The starting time-evolution equation which is written as a linear system in the Liouville space of discrete quantum states, is first solved in terms of Cramer's determinants and next suitably adapted to tranport problems. The calculation of the statistical average of the many-body current density, in the presence of scattering potential, is carried out in the thermodynamic limit, using a characteristic property of collision matrices introduced earlier by van Hove as the so-called
delta-singularity. In contrast to previous methods based on kinetic equations solved by perturbations, Kubo's theory, or Mori type memory functions, the resolution of an integral equation, initially included in Cramer's formulae, is no longer required. Among its major advantages, the method succeeds in overcoming the well known zero frequency divergent behaviour of the conductivity. It also yields an explicit expression of an effective relaxation time which is just constructed out of those diagonal transition sequences responsible for self-energy effects in van Hove's theory. This expression which only assumes the randomization of phases of initial states can be used to get tractable expansions in powers of the collision potential. Its range of validity is limited by the effect of collisions on equilibrium statistics, which is shown to be negligible within the Peierls criterion,
hbar/taullmu,
tau being the relaxation time and
mu the Fermi energy. Finally, the comparison of the present approach with the elementary transport theory reveals that the relaxation time is close to the standard expression, even though the latter would be exactly recovered, in lowest order of collisions, only on disregarding divergent terms arising beyond second order.\n
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
A linear response formalism is developed for evaluation of transport coefficients in quantum many-particle systems. The method is based on a systematic linearization of nonequilibrium Kadanoff-Baym transport equations in an integral form in ac electric fields. Simple and consistent approximations can be performed in the resulting set of transport equations, which replace vertex equations in the Kubo formalism. The method is illustrated by the example of an ac and dc conductivity and conductance for a quasi-one-dimensional electron system with an electron--phonon interaction for three-dimensional acoustical phonons. After approximations standard in metals, the solved transport equations coincide with the Holstein equation. \textcopyright{} 1996 The American Physical Society.