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

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... 

To relate Z t) with the friction coefficient we use the Green-Kubo formula and the Einstein relation [18, 19, 20] ... 

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