Kubo identity

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

The combinatorial point of view is reminiscent of the classical cumulant formalism developed by Kubo [39], and indeed the structure of Eqs. (25) and (28) is essentially the same as the equations that define the classical cumulants, up to the use of an antisymmetrized product in the present context. In further analogy to the classical cumulants, the p-RDMC is identically zero if simultaneous p-electron correlations are negligible. In that case, the p-RDM is precisely an antisymmetrized product of lower-order RDMs. 

As A x was supposed stationary the integral is independent of time. The effect of the fluctuations is therefore to renormalize A0 by adding a constant term of order a2 to it. The added term is the integrated autocorrelation function of At. In particular, if one has a non-dissipative system described by A0, this additional term due to the fluctuations is usually dissipative. This relation between dissipation and the autocorrelation function of fluctuations is analogous to the Green-Kubo relation in many-body systems 510 but not identical to it, because there the fluctuations are internal, rather than added as a separate term as in (2.1). 

We observe that the level density is recovered if the observable is the identity, A = I, since i dt = Tp. The above formula may be generalized to situations where the observable itself depends on time, for example, A = R exp(+iHt/h)S exp(- iHt/h), where R and S are other observables, as is the case in the Green-Kubo-Yamamoto-Zwanzig formulas for the transport and reaction rate coefficients. 

W. H. Miller The expression for the reaction rate (in terms of a flux-flux autocorrelation function) obtained by myself, Schwartz, and Tromp in 1983 is very similar (though not identical) to the one given earlier by Yamamoto. It is also an example of Green-Kubo relations. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

These two identities are called Kubo-Martin-Schwinger boundary conditions [47]. From the commutation relation, it follows that... 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

Silicon thin film thermal conductivities are predicted using equilibrium molecular dynamics and the Grccn-Kubo relation. Periodic boundary conditions are applied in the direetions parallel to the thin film surfaees (Fig. 5). Atoms near the surfaces of the thin film are subjeeted to the above-described repulsive potential in addition to the Stillinger-Weber potential [75]. Simulations were also performed adding to each surface four layers of atoms kept frozen at their crystallographic positions, in order to eompare the dependence of the predieted thermal eonduetivities on the surface boundary eonditions. We found that the thermal eonduetivities obtained using frozen atoms or the repulsive potential are identical within the statistical deviations, exeept for the in-plane thermal eonduetivity of films with thickness less than 10 nm [79]. Therefore, in the present study, we present only the predietions obtained with the repulsive potential. 

As noted above, the SCF theory of the conductivity deserves further attention. Nevertheless in the case of the simple mean field theory of Section 3.8.5, Edwards (1970b) notes that the conductivity is identically zero for the localized states of E < 0. The details proceed rather simply as before. In the Kubo-Greenwood-Peierls (Kubo (1956) (1957)) expression for theton-ductivity, it is necessary to obtain the average of the product of two Green s functions... 

Here we prove the Kubo s identity for any operator and Hamiltonian H. It states... 

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