PGH theory

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the... 

PGH theory has its limitations. The derivation depends on three central conditions ... 

PGH theory has been extended. It can be used in conjunction with VTST and optimized planar dividing surfaces,in which case, the energy loss is to be computed along the coordinate perpendicular to the optimal planar dividing surface. In the same vein it has been generalized to include the case of space and time dependent friction. ... 

Kramers derived an expression for the rate in the underdamped and spatial diffusion limits. He did not derive a uniform expression for the rate valid for all values of the damping strength. This is the Kramers turnover problem, which was solved only in the late eighties by Poliak, Grabert, and Hanggi (12) and is known as PGH theory. 

The rate formula Eq. (141) is exact. Approximations enter because the nonequili-brium probability f(E) is not known exactly. Note though that in the equilibrium limit, replacing f E) by /eq( ) in Eq. (141) immediately leads to the KGH estimate for the rate. In the strong coupling limit, energy diffusion is fast and equilibrium is maintained throughout. In this limit, PGH theory reduces to the correct spatial diffusion limited expression. 

This equation is of central importance in PGH theory. Deep in the well, the unstable mode is strongly coupled to the other modes and equilibrium is maintained. This is the boundary condition needed to solve the integral equation. It also allows one to replace the lower limit of the integration by — °° provided that the barrier height is large with respect to k T. 

It is then a matter of some algebra, following the same steps as in PGH theory to find that the energy loss from the particle to the bath, taken over the infinite period motion of the particle at the barrier energy is... 

These equations are forced oscillator equations of motion whose solution in terms of the time dependence of q, z is well known. As in the weak damping version of PGH theory... 

The case of a periodic potential is also of interest. In PGH theory one considers motion of the unstable normal mode. When the potential is periodic, there are an infinite number of unstable modes, each shifted from the other by a constant term. The potential in any one of these coordinates is no longer periodic. The adaptation of PGH theory outside the weak damping regime is still an open problem. 

---