Langevin equation linear response theory

In another paper, R. Kuho (Kcio University, Japan) illustrates in a rather technical and mathematical fashion tire relationship between Brownian motion and non-equilibrium statistical mechanics, in this paper, the author describes the linear response theory, Einstein s theory of Brownian motion, course-graining and stochastization, and the Langevin equations and their generalizations. 

In this equation g(t) represents the retarded effect of the frictional force, and /(f) is an external force including the random force from the solvent molecules. We see, in contrast to the simple Langevin equation with a constant friction coefficient, that the friction force at a given time t depends on all previous velocities along the trajectory. The friction force is no longer local in time and does not depend on the current velocity alone. The time-dependent friction coefficient is therefore also referred to as a memory kernel . A short-time expansion of the velocity correlation function based on the GLE gives (fcfiT/M)( 1 — (g/M)t2/(2r) + ), where r is the decay time of g(t), and it therefore does not have a discontinuous first derivative at t = 0. The discussion of the properties of the GLE is most easily accomplished by using so-called linear response theory, which forms the theoretical basis for the equation and is a powerful method that allows us to determine non-equilibrium transport coefficients from equilibrium properties of the systems. A discussion of this is, however, beyond the scope of this book. 

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... 

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