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Physical Review Letters

Nonlinear Response for Time-dependent External Fields

Janka Petravic, Denis J. Evans

Published Feb 17, 1997• 24 citations
Publisher

Abstract

We generalize nonlinear response theory for autonomous systems so that it describes the response of classical many-body systems to large time-dependent external fields. Our formalism represents the first practical application of response theory to such problems. Our expressions for the nonlinear time-dependent response are tested against nonequilibrium molecular dynamics computer simulation. The relation of our results to known special cases (time-dependent linear response and time-independent nonlinear response) is discussed.

DOI
10.1103/physrevlett.78.1199

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We generalize nonlinear response theory for autonomous systems so that it describes the response of classical many-body systems to large time-dependent external fields. Our formalism represents the first practical application of response theory to such problems. Our expressions for the nonlinear time-dependent response are tested against nonequilibrium molecular dynamics computer simulation. The relation of our results to known special cases (time-dependent linear response and time-independent nonlinear response) is discussed.
Abstract Linear response theory is a general formalism applicable to any physical system slightly departing from equilibrium. It allows us to express the linear response functions and the generalized susceptibilities in terms of the equilibrium correlation functions of the relevant dynamical variables. For instance, in the case of the response to an applied external field, one of the two relevant dynamical variables represents the physical quantity conjugate to the field whereas the other represents the physical quantity whose out-of-equilibrium average is measured. The computation of the linear response functions relies on a first-order perturbation expansion of the density operator (or of the phase space distribution function) of the system coupled to the field. The expressions thus obtained constitute the Kubo formulas of linear response theory.
A time-dependent perturbation solution is derived for a system of first-order nonlinear or linear ordinary differential equations. By means of an ansatz, justified a posteriori, the latter equations can be converted to an operator equation which is solvable by several methods. The solution is subsequently specialized to the case of classical mechanics. For the particular case of autonomous equations the solution reduces to a well-known one in the literature. However, when collision phenomena are treated and described in a classical “interaction representation” the differential equations are typically nonautonomous, and the more general solution is required. The perturbation expression is related to a quantum mechanical one and will be applied subsequently to semiclassical and classical treatments of collisions.
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The exact response theory based on the Dissipation Function applies to general dynamical systems and has yielded excellent results in various applications. In this article, we propose a method to apply it to quantum mechanics. In many quantum systems, it has not yet been possible to overcome the perturbative approach, and the most developed theory is the linear one. Extensions of the exact response theory developed in the field of nonequilibrium molecular dynamics could prove useful in quantum mechanics, as perturbations of small systems or far-from-equilibrium states cannot always be taken as small perturbations. Here, we introduce a quantum analogue of the classical Dissipation Function. We then derive a quantum expression for the exact calculation of time-dependent expectation values of observables, in a form analogous to that of the classical theory. We restrict our analysis to finite-dimensional Hilbert spaces, for the sake of simplicity, and we apply our method to specific examples, like qubit systems, for which exact results can be obtained by standard techniques. This way, we prove the consistency of our approach with the existing methods, where they apply. Although not required for open systems, we propose a self-adjoint version of our Dissipation Operator, obtaining a second equivalent expression of response, where the contribution of an anti-self-adjoint operator appears. We conclude by using new formalism to solve the Lindblad equations, obtaining exact results for a specific case of qubit decoherence, and suggesting possible future developments of this work.
Response theory describes the reaction of observales to perturbations in external fields. We review this formalism for quantum fiels in isolation that have unitary time evolution. An emphasis is put on consequences of causality and the resulting spectral representations for linear and nonlinear response functions, on functional techniques and generating functionals, including the description of the initial state, the evolution, and measurements. We review consequences of time reversal symmetry and relations for the statistics of work, and discuss a large class of quantum correlation functions, and their relation to response functions through fluctuation-dissipation relations. Consequences of conservation laws and gauge symmetries are mentioned briefly.
This is the second in a series of papers which intends to develop a statistical mechanical theory capable of independently treating responses to thermal as well as mechanical disturbances. The essential idea is to develop a method of solving the Liouvill-von Neumann equation for the density matrix of a many-body system in a time-dependent Gibbsian form. The theory is made determinate by defining the relative entropy, which will reflect the character of the entropy production in our model. In this paper, rigorous expressions are obtained for nonlinear responses, which contain all the most general consequences of statistical mechanics regarding the macroscopic variables. In particular, the Onsager reciprocity relations follow from the formula in nonlinear regions. An attempt to treat a stationary state where heat is produced is made in terms of a nonequilibrium density matrix satisfying the Liouville-von Neumann equation.