Kubo formula susceptibilities

The ionic susceptibility/conductivity is a function of the trajectories of the charges at equilibrium that is, y (m (o>) is proportional to the ACF spectrum of the E-projection of the steady-state velocity. One may regard Eq. (394) as a convenient (for numerical calculations) form of the Kubo formula [69] for the diagonal component of the conductivity tensor... 

Transforming the correlator (4.230) by the Kubo formula, one arrives at the longitudinal dynamic susceptibility... 

D. Effective Temperature in an Out-of-Equilibrium Medium The Link with the Kubo Formulas for the Generalized Susceptibilities... 

Interestingly, each one of the two FDTs can be formulated in two equivalent ways, depending on whether one is primarily interested in writing a Kubo formula for a generalized susceptibility %(co) [namely, in the present case, p(m) or y(o))], or an expression for its dissipative part [namely, the Einstein relation... 

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... 

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... 

The effective temperature can, in principle, be deduced from independent measurements, for instance, of [Rep) ) and D(co) [or of Ofey(m) and C/r/r(oo)]. However, experimentally it may be preferable to make use of the modified Kubo formulas for the corresponding generalized susceptibilities. The Kubo formula for p(o>) [and also the one for y(co)] cannot be extended to an out-of-equilibrium situation by simply replacing T by Teff (co) in Eq. (156) [and in Eq. (162)]. In the following, we will show in details how the Kubo formulas have then to be rewritten. 

There are a number of different ways to determine the quantum mechanical formulas for the susceptibilities XabQ., co). Perhaps the simplest and most elegant procedure is due to Kubo.11 We follow this procedure here. 

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See also in sourсe #XX -- [ , , ]

See also in sourсe #XX -- [ , , ]

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