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arXiv:1506.04577 · Physical Review E

Linear response and modified fluctuation-dissipation relation in random potential

Fattah Sakuldee, Sujin Suwanna

Published Nov 16, 2015• 3 citations

Abstract

In this work, a physical system described by the Hamiltonian H(ω)=H(0)+V(ω)(t) consisting of a solvable model H(0) and external random and time-dependent potential V(ω)(t) is investigated. Under the conditions in which, for each realization, the potential changes smoothly so that the evolution of the system follows the Schrödinger dynamics, and that the average external potential with respect to all realizations is constant in time, an adjusted equilibrium state can be defined as a reference state and the mean dynamics can be derived from taking the average of the equation with respect to the configuration parameter ω. It provides extra contributions from the deviations of the Hamiltonian and evolves the state along the time by the Heisenberg and Liouville-von Neumann equations. Consequently, the Kubo formula and the fluctuation-dissipation relation (FDR) are modified in the sense that the contribution from the information of randomness and memory effects from the time dependence is also present. The modified Kubo formula now has a contribution from two terms. The first term is an antisymmetric cross correlation between two observables measured by a probe as expected, and the latter term is an accumulation of the propagation of the effects from the randomness. When the considered system is in the adjusted equilibrium state at the time the measurement probe interacts, the latter contribution vanishes, and the standard FDR is recovered.

DOI: 10.1103/physreve.92.052118

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In this work, a physical system described by the Hamiltonian H(ω)=H(0)+V(ω)(t) consisting of a solvable model H(0) and external random and time-dependent potential V(ω)(t) is investigated. Under the conditions in which, for each realization, the potential changes smoothly so that the evolution of the system follows the Schrödinger dynamics, and that the average external potential with respect to all realizations is constant in time, an adjusted equilibrium state can be defined as a reference state and the mean dynamics can be derived from taking the average of the equation with respect to the configuration parameter ω. It provides extra contributions from the deviations of the Hamiltonian and evolves the state along the time by the Heisenberg and Liouville-von Neumann equations. Consequently, the Kubo formula and the fluctuation-dissipation relation (FDR) are modified in the sense that the contribution from the information of randomness and memory effects from the time dependence is also present. The modified Kubo formula now has a contribution from two terms. The first term is an antisymmetric cross correlation between two observables measured by a probe as expected, and the latter term is an accumulation of the propagation of the effects from the randomness. When the considered system is in the adjusted equilibrium state at the time the measurement probe interacts, the latter contribution vanishes, and the standard FDR is recovered.

We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.

Abstract We study the response of a quantum system induced by a collision with a quantum particle, using the time-independent framework of scattering theory. After deriving the dynamical map for the quantum system, we show that it encodes a non-perturbative response function obeying a general fluctuation-dissipation relation. We show that Kubo's formula emerges autonomously in the Born approximation, where the time-dependent perturbation is determined by particle's evolution through the potential region.

We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.

We derive a general set of fluctuation relations for a nonequilibrium open quantum system described by a Lindblad master equation. In the special case of conservative Hamiltonian dynamics, these identities allow us to retrieve quantum versions of Jarzynski and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula.

The Kubo formula is an equation that expresses the linear response of an observable due to a time-dependent perturbation. It has been extended from closed systems to open systems in recent years under the Markovian approximation, but is barely explored for open systems in non-Markovian regimes. In this paper, we derive a formula for the linear response of an open system to a time-independent external field. This response formula is available for both Markovian and non-Markovian dynamics depending on parameters in the spectral density of the environment. As an illustration of the theory, the Hall conductance of a two-band system subjected to environments is derived and discussed. With the tight-binding model, we point out the Hall conductance changes from Markovian to non-Markovian dynamics by modulating the spectral density of the environment. Our results suggest a way to the controlling of the system response, which has potential applications for quantum statistical mechanics and condensed matter physics.

The Kubo fluctuation-dissipation theorem relates the current fluctuations of a system in an equilibrium state with the linear AC-conductance. This theorem holds also out of equilibrium provided that the system is in a stationary state and that the linear conductance is replaced by the (dynamic) conductance with respect to the nonequilibrium state. We provide a simple proof for that statement and then apply it in two cases. We first show that in an excess noise measurement at zero temperature, in which the impedance matching is maintained while driving a mesoscopic sample out of equilibrium, it is the nonsymmetrized noise power spectrum which is measured, even if the bare measurement, i.e. without extracting the excess part of the noise, obtains the symmetrized noise. As a second application we derive a commutation relation for the two components of fermionic or bosonic currents which holds in every stationary state and which is a generalization of the one valid only for bosonic currents. As is usually the case, such a commutation relation can be used e.g. to derive Heisenberg uncertainty relationships among these current components.