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Physical Review Letters

Exact Transport Parameters for Driving Forces of Arbitrary Magnitude

V. M. Kenkre, Max Dresden

Published Jul 5, 1971• 20 citations
Publisher

Abstract

A straightforward analysis of the time-correlation function of interest in response theory involves the entire complicated solution of the dynamical problem. Using projection techniques we derive a simpler equation obeyed by the time-correlation function. Laplace transforms are used to solve this equation for the case when the driving force is a step function of arbitrary magnitude. The results are independent of the near-origin approximation of the Kubo formalism.

DOI
10.1103/physrevlett.27.9

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A straightforward analysis of the time-correlation function of interest in response theory involves the entire complicated solution of the dynamical problem. Using projection techniques we derive a simpler equation obeyed by the time-correlation function. Laplace transforms are used to solve this equation for the case when the driving force is a step function of arbitrary magnitude. The results are independent of the near-origin approximation of the Kubo formalism.
Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.
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 rrr 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 rrr. 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 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.
We have analyzed Dyson's UUU-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the UUU matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our UUU matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our UUU matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our U(s){U}^{(s)}U(s) matrix, which is the UUU matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.