Kubo formulas

The dlffuslvltles parallel to the pore walls at equilibrium were determined form the mean square particle displacements and the Green-Kubo formula as described In the previous subsection. The Green-Kubo Formula cannot be applied, at least In principle, for the calculation of the dlffuslvlty under flow. The dlffuslvlty can be still calculated from the mean square particle displacements provided that the part of the displacement that Is due to the macroscopic flow Is excluded. The presence of flow In the y direction destroys the symmetry on the yz plane. Hence the dlffuslvltles In the y direction (parallel to the flow) and the z direction (normal to the flow) can In principle be different. In order to calculate the dlffuslvltles the part of the displacement that Is due to the flow must of course be excluded. Therefore,... 

It is well known that each transport coefficients is given by a Green-Kubo formula or, equivalently, by an Einstein formula ... 

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 ... 

The starting point of the calculation is a Mori-type rephrasing of the Green-Kubo formula for the viscosity, which is given by Eq. (155). Thus... 

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... 

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... 

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... 

The Modified Kubo Formula for the Generalized Friction Coefficient... 

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... 

The Einstein relation (159) or the expression (157) of the dissipative part ffiep(m) of the mobility constitute another formulation of the first FDT. Indeed they contain the same information as the Kubo formula (156) for the mobility, since p(co) can be deduced from 9ftep(oo) with the help of the usual Kramers-Kronig relations valid for real co [29,30]. Equation (156) on the one hand, and Eq. (157) or Eq. (159) on the other hand, are thus fully equivalent, and they all involve the thermodynamic bath temperature T. Note, however, that while p(oo) as given by Eq. (156) can be extended into an analytic function in the upper complex half-plane, the same property does not hold for D(co). 

Correspondingly, one can write a Kubo formula relating the generalized friction coefficient y(co) to the random force correlation function ... 

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... 

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. 

Equation (189) allows to derive the function T (go) from a given effective temperature reff(oo). Conversely, it provides the link allowing one to deduce 7eff (oo) from the modified Kubo formula (187) for p(oo). 

Similarly, out-of-equilibrium, it is not possible to rewrite the Kubo formula for y(co) in a form similar to Eq. (162) with reff(o)) in place of T ... 

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]. 

Let us emphasize that the function (co) is not the effective temperature involved in both the modified Einstein relation (184) and the modified Nyquist formula (185). Instead it is the quantity which appears in the modified Kubo formula for the mobility [Eq. (187)]. Let us add that one could also have... 

Equation [116] is the central result of this section and is one of the most important equations in nonequilibrium statistical mechanics. It is from this point that one makes contact with the more familiar relations relating transport coefficients to time correlation functions under the guise of the so-called Green-Kubo formulas.i2.39,40... 

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