Kubo equation

This equation, called the Kubo equation, is equivalent to the Einstein equation. However, it is easier to estimate self-diffusion coefficients from the slope of the mean-square displacements. 

The starting point is the Kubo equation with a Kawasaki-Ferrell decoupling approximation, as discussed earlier for dilute solutions " ... 

INVESTIGATION OF GALVANO-MAGNETIC PROPERTIES OF TRANSITION METAL ALLOY SYSTEMS USING THE KUBO-GREENWOOD EQUATION... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

The Green-Kubo result demands that this be equated to the negative of the natural rate of change of the first energy moment, Eq. (260), which means that... 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Such equation is termed the KMS (Kubo, Martin and Schwinger) relation and describes the conditions of periodicity to be obeyed by a correlation, in particular the Green functions. 

Following FerrelK, the second term in Equation 2 can be expressed as a Green-Kubo integral over a flux-flux correlation function. The transport is due to a velocity perturbation caused by two driving forces, the Brownian force and frictional force. The transport coefficient due to the segment-segment interaction can be calculated from the Kubo formula(9 ... 

Much of the recent literature on RDM reconstruction functionals is couched in terms of cumulant decompositions [13, 27-38]. Insofar as the p-RDM represents a quantum mechanical probability distribution for p-electron subsystems of an M-electron supersystem, the RDM cumulant formalism bears much similarity to the cumulant formalism of classical statistical mechanics, as formalized long ago by by Kubo [39]. (Quantum mechanics introduces important differences, however, as we shall discuss.) Within the cumulant formalism, the p-RDM is decomposed into connected and unconnected contributions, with the latter obtained in a known way from the lower-order -RDMs, q < p. The connected part defines the pth-order RDM cumulant (p-RDMC). In contrast to the p-RDM, the p-RDMC is an extensive quantity, meaning that it is additively separable in the case of a composite system composed of noninteracting subsystems. (The p-RDM is multiphcatively separable in such cases [28, 32]). The implication is that the RDMCs, and the connected equations that they satisfy, behave correctly in the limit of noninteracting subsystems by construction, whereas a 2-RDM obtained by approximate solution of the CSE may fail to preserve extensivity, or in other words may not be size-consistent [40, 42]. 

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. 

In the work of Brown [5] and Kubo and Hashitsume [45] the starting equation is the Gilbert equation (3.43), in which the effective field is increased by a fluctuating field yielding the stochastic Gilbert equation. This equation can, as in the deterministic case, be cast into the Landau-Lifshitz form as... 

The relation between the osmotic pressure II and the polymer concentration, referred to as the equation of state for the solution, is often used for a critical comparison between theory and experiment (or simulation). Kubo and Ogino... 

The above equations are limited to treatment of exchange between two environments the general analysis for n environments has been described in detail by Anderson (2), Kubo (64), and Sack (121). 

The equation proposed by Kubo et al. (1983) can be used for 10 < ReJ < 2000 (experiments took place in upflow mode) ... 

We confine ourselves here to situations where the mean free path is due to elastic collisions with impurities or, in alloys, liquids and amorphous materials, with the non-periodic field in such materials. According to the Kubo-Greenwood formula, the current at temperature T is given by (31), where, instead of (32), o(E) is defined by the equation... 

The cancellation of g when l > a in the equation for the conductivity was first discussed by Edwards (1962). He showed that if /> a, although (42) remains valid in principle (as follows from the Kubo-Greenwood equation (34)), one can write... 

For we take the value given by the Kubo-Greenwood equation, namely equation (34) of Chapter 1. Mott (1970) showed that localized regime, and Mott and Kaveh (1985b) give a corrected version... 

But approximations a and b above were also based on this, and therefore this is the overall condition for the validity of (4.7) as a master equation. The parameter / rc is a measure for the effect of the external influence during one correlation time, and has been called the Kubo number. 

Introduced by R. Kubo, J. Mathem. Phys. 4, 174 (1963) under the title Stochastic Liouville Equation . The adjective stochastic is here used in the sense of pertaining to stochastic phenomena in contrast to our use as a synonym of random - as in the title of this chapter. 

Example. Kubo constructed the following model to illustrate line broadening and narrowing due to random perturbations.510 In equation (1.2) suppose that co = (o0 + a (t), where a>0 and a are constants and (r) is the dichotomic Markov process (IV.2.3). As only takes two values we may abbreviate... 

Kubo calls this relationship the Second Fluctuation Dissipation Theorem. For its proof it should be noted that the modified Langevin equation can be written as... 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

To overcome this difficulty Kubo, Mori, Zwanzig, and others introduced a time-dependent friction coefficient, writing, instead of the simple Langevin equation, the... 

The external potential is included in these equations. The self-energy is defined as a function or operator in the Keldysh space according to Eq. (20). obeys the Kubo-Martin-Schwinger boundary conditions (16). The ( )-projections of the Kadanoff-Baym equations can written as... 

If we restrict ourselves to the Kubo-Anderson model, the problem does not seem entirely intractable. The shape of the decay distinguishes the limits the fast limit has an exponential decay the slow limit has a Gaussian decay. In the intermediate regime [Equation (7)], the details of the decay shape can be used to find both Aw and r0l. 

Figure 1 Two-dimensional contour plots of the log of the Raman-echo correlation function, In Cre(ti, T3), showing the effect of changing the rate of solvent-induced perturbations. All cases give a Raman line with the same FWHM (5 cm-1) and FIDs with similar decay times but give very different Raman echo results, (a) Fast modulation (b) intermediate modulation (A, = 3.32 cm-1, r( = 1.60 ps) (c) slow modulation. Calculations are based on a single Kubo-Anderson process [Equations (7)-(9)].

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