Langevin equation critical

Another critical result, which provided a more microseopic view of the Langevin equation, was the proof by Zwanzig that when the dynamics of a system obeying the classical Hamiltonian... 

The characteristic time scale for the motion of the particle in the parabolic top barrier is the inverse barrier frequency, the sharper is the barrier, the faster is the motion. Typically, atom transfer barrier are quite sharp therefore the key time scale is very short, and the short-time solvent response becomes relevant instead of the long-time overall response given by the ( used in Kramers theory (see eq.(20)). To account for this critical feature of reaction problems, Grote and Hynes (1980) introduce the generalized Langevin equation (GLE) ... 

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen [58] (Chapters 8 and 14). In our calculations, we shall encounter stochastic differential equations of the form... 

Applying these methods to systems in the vicinity of the non-equilibrium critical points, the conclusion was drawn [72] that the mesoscopic approach contains excess information about spatial particle distribution the details of how the whole system s volume is divided into cells become unimportant as — oo. The possibility to employ expansion in inverse powers of vo -similarly to a complete mixing case - was also discussed. Asymptotically it leads to the Focker-Planck equation equivalent to the Langevin-like equation. 

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