Kubo response theory

Lindner presented an important paper on this type of systems already in mid-1960s (102). She considered the whole system as an ensemble of microcrystallites with different orientations of the principal axis of the ZFS tensor with respect to the magnetic field and applied the linear response theory of Kubo and Tomita (103). The expression for the nuclear Ti for the case of S = 1 could be written as ... 

The important step of identifying the explicit dynamical motivation for employing centroid variables has thus been accomplished. It has proven possible to formally define their time evolution ( trajectories ) and to establish that the time correlations ofthese trajectories are exactly related to the Kubo-transformed time correlation function in the case that the operator 6 is a linear function of position and momentum. (Note that A may be a general operator.) The generalization of this concept to the case of nonlinear operators B has also recently been accomplished, but this topic is more complicated so the reader is left to study that work if so desired. Furthermore, by a generalization of linear response theory it is also possible to extract certain observables such as rate constants even if the operator 6 is linear. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

Linear response theory, applied to the particle velocity, considered as a dynamic variable of the isolated particle-plus-bath system, allows to express the mobility in terms of the equilibrium velocity correlation function. Since the mobility p(co) is simply the generalized susceptibility %vx(o ), one has the Kubo formula... 

Equations (161) and (162) are two equivalent formulations of the second FDT [30,31]. The Kubo formula (162) for the generalized friction coefficient can also be established directly by applying linear response theory to the force exerted by the bath on the particle, this force being considered as a dynamical variable of the isolated particle-plus-bath system. We will come back to this point in Section VI.B. 

Let us now come back to the specific problem of the diffusion of a particle in an out-of-equilibrium environment. In a quasi-stationary regime, the particle velocity obeys the generalized Langevin equation (22). The generalized susceptibilities of interest are the particle mobility p(co) = Xvxi03) and the generalized friction coefficient y(co) = — (l/mm)x ( ) [the latter formula deriving from the relation (170) between y(f) and Xj> (f))- The results of linear response theory as applied to the particle velocity, namely the Kubo formula (156) and the Einstein relation (159), are not valid out-of-equilibrium. The same... 

A published derivation of the Green-Kubo or fluctuation-dissipation expressions from the combination of the FR and the central limit theorem (CLT) was finally presented in 2005. This issue had been addressed previously and the main arguments presented, but subtleties in taking limits in time and field that lead to breakdown of linear response theory at large fields, despite the fact that both the FR and CLT apply, " were not fully resolved. ... 

Transport coefficients of molecular model systems can be calculated by two methods [8] Equilibrium Green-Kubo (GK) methods where one evaluates the GK-relation for the transport coefficient in question by performing an equilibrium molecular dynamics (EMD) simulation and Nonequilibrium molecular dynamics (NEMD) methods. In the latter case one couples the system to a fictitious mechanical field. The algebraical expression for the field is chosen in such a way that the currents driven by the field are the same as the currents driven by real Navier-Stokes forces such as temperature gradients, chemical potential gradients or velocity gradients. By applying linear response theory one can prove that the zero field limit of the ratio of the current and the field is equal to the transport coefficient in question. 

As discussed in the Introduction, considerable effort has been devoted to the study of transport properties. Specifically, viscosity estimates are of potential importance in the rational design of lubricants. In these simulations, two main approaches have been used. The earlier of the two focused on a linear response theory based Green-Kubo formula ... 

Proton conductivity of the systems studied has been considered in the framework of the Kubo linear response theory [154], i.e., the conductivity has been written as... 

The familiar shear modulus of linear response theory describes thermodynamic stress fluctuations in equilibrium, and is obtained from (5b, lid) by setting y = 0 [1, 3, 57], While (5b) then gives the exact Green-Kubo relation, the approximation (lid) turns into the well-studied MCT formula (see (17)). For finite shear rates, (lid) describes how afflne particle motion causes stress fluctuations to explore shorter and shorter length scales. There the effective forces, as measured by the gradient of the direct correlation function, = nc = ndck/dk, become smaller, and vanish asympotically, 0 the direct correlation function is connected... 

Abstract. Kubo s paper on linear-response theory provided a unified language to describe a wide variety of transport phenomena, both quantum and classical, in a suitable microscopic language. The paper has been crucial for subsequent developments in numerical simulation. 

It is sometimes difficult to indicate precisely which paper heralds an important change in a particular field. In retrospect, one can often observe precursors and parallel developments in other publications however, Kubo s paper on linear-response theory, although it certainly did not appear in a scientific vacuum, clearly marks a watershed. 

Kubo s linear-response theory provides the full, quantum-mechanical relation between the response of a system to external perturbations and the spontaneous decay of fluctuations in the unperturbed system. Of course, the paper had important predecessors Nyquist s [1] paper on thermal noise in resistors and Onsager s [2] seminal paper on the relation between decay of macroscopic and microscopic fluctuations, to name but the earliest. 

In fact, Kubo s paper also provided much of the language for the subsequent development of nonequilibrium molecular dynamics simulations however, nonequilibrium molecular dynamics adresses problems that could not be handled in the context of the original linear-response theory (see, e.g. Ref. [4]). 

As is clear from the above, linear-response theory has stimulated many important developments yet Kubo s approach has not been uncontroversial. In particular, in a paper that cannot be found in most libraries, van Kampen [7] has criticized the assumption made in linear-response theory that, for sufficiently weak fields, the change in the phase-space density is linear in the applied perturbation. In fact, due to the exponential divergence of phase-space trajectories, even the weakest perturbation will, on a macroscopic time-scale, result in large changes in the phase-space density however, as was shown in the subsequent numerical work by Ciccotti et al. [8], on the microscopic time-scales that are usually... 

Kubo R (1957) Linear response theory of irreversible processes. J Phys Soe Jpn 12 570... 

Let us first consider nonequilibrium properties of dense fluids. Linear response theory relates transport coefficients to the decay of position and velocity correlations among the particles in an equilibrium fluid. For example, the shear viscosity ti can be expressed in Green-Kubo formalism as a time integral of a particular correlation function ... 

ESR linewidth is expected to show a significant frequency dependence when the external field Hz and the exchange field Hgx are of comparable size. In the framework of a linear response theory Kubo and Tomita (1954) calculated this factor c. The result is shown in fig. 34 and compared to experimental results of the Gd-resonance experiments in YP, ScP and YAs. Figure (34) clearly demonstrates that almost all data fall onto an universal curve which is discussed in detail by Sperlieh and Jansen (1974). For Hg -C Hz the value of c approaches 1 and for Hg Hz, c amounts to 10/3. The numerical calculations yield He = 36.0 and 40.3 kG for the Gd-Gd exchange in GaP and GdAs, respectively. 

The distinction between electrical and non-electrical processes has another aspect. An external electric field can induce an electric current, and the calculation of this response, as in Kubo s theory, leads to the formula for electric conductivity. In contrast a heat current, for example, is the response to an internal temperature gradient and depends only indirectly on the external forces which set up this temperature gradient. 

Within the framework of linear response theory a phenomenological thermal transport coefficient Lij can be shown to have the form of a Green-Kubo relation... 

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by... 

GK = Green-Kubo LIT = linear irreversible thermodynamics LRT = linear response theory NEMD = nonequilibrium molecular dynamics NESS = nonequilibrium steady state TTC = thermal transport coefficient TTCF = transient time correlation function. 

The thermal conductivity of a material can be calculated directly from equilibrium molecular dynamics (EMD) simulation based on the linear response theory Green-Kubo relationship. " The fluctuation-dissipation theorem provides a connection between the energy dissipation in irreversible processes and the thermal fluctuations in equilibrium. The thermal conductivity tensor. A, can be expressed in terms of heat current autocorrelation correlation functions (HCACFs),/, ... 

The calculation of the thermal conductivity of gas hydrate using EMD and the Green-Kubo linear response theory was repeated recently. In that work, convergences of the relevant quantities were monitored carefully as a function of the model size. Subtleties in the numerical procedures were also carefully considered. The thermal conductivity of methane hydrate was found to converge within numerical accuracy for 3 x 3 x 3 and 4x4x4 supercells. In the calculation of the heat flux vector there is an interactive term that is a pairwise summation over the forces exerted by atomic sites on one another. The species (i.e., water and methane) enthalpy correction term requires that the total enthalpy of the system is decomposed into contributions from each species. Because of the partial transformation from pairwise, real-space treatment to a reciprocal space form in Ewald electrostatics, it is necessary to recast the diffusive and interactive terms in this expression in a form amenable for use with the Ewald method using the formulation of Petravic. ... 

This diffusion constant T> q) can be calculated within the framework of linear response theory by using a Kubo formula ... 

P(co) is an internal field factor and A t) is a time-correlation function which represents the fluctuations of the macroscopic dipole moment of the volume V in time in the absence of an applied electric field. Equations (44) and (45) are a consequence of applying linear-response theory (Kubo-Callen-Green) to the case of dielectric relaxation, as was first described by Glarum in connexion with dipolar liquids. For the special case of flexible polymer chains of high molecular weight having intramolecular correlations between dipoles but no intermolecular correlations between dipoles of different chains we may write... 

Kubo R, Yokota M and Nakajima S 1957 Statistical-mechanical theory of irreversible processes. Response to thermal disturbance J. Phys. Soc. Japan 12 1203... 

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