Hierarchical Treatment of Diffusive Processes

Our plan in this subsection is to examine two ideas that have been advanced to confront such problems. These arguments are put forth not so much with the conviction that they represent the final word on the hierarchical treatment of diffusion, but rather because they serve to illustrate the types of arguments that have been put forth thus far and that will perhaps encourage further efforts. 

Kinetic Monte Carlo approaches. One scheme that has found widespread use in the consideration of diffusive processes is the kinetic Monte Carlo approach in 

The basic idea of the hyperdynamics method in the context of a simple one-dimensional model is illustrated in fig. 12.14. The claim is that by adding a bias potential AVb(q) q is a generalized position) to the bare energy surface, the transition rates can be renormalized in a way that respects the condition provided by eqn (12.64). To see this, we follow Voter s (1997a) analysis. Recalling the definition of the transition rate given in eqn (7.55), we note that 

Our notation (in particular, the S(q)) implies that the saddle point separating the A and B wells is at g = 0. Note that we have separated the Hamiltonian into kinetic and potential energy parts according to the prescription H p, q) = K p) + V q). Our result for the transition rate may be rewritten as 

The subscript on ( )aj, refers to the fact that the the relevant thermal averages are 

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