3.3 Illustrative Case Studies45Denote by p the transition probability from i to j. The transition rates on theijspace shown in Fig. 3.5 are for instance p =p1, p =p , p =p and so on. Suppose01 13 2 35 3that: p1 > p2 > p3 that is transitions across more levels are hardly. Observe that: p = min (p , p )ij k ik kj(3.10)This is valid for any states i, j, k of the system. Consequently the distances d ,ijdefined by: d =1-p , satisfy the NA axiom.ij ijd = max (d , d )ij k ik kj(3.11)Fig. 3.6 shows the energy barriers between different states. The NA tree shown in Fig. 3.5 is equivalent to the one dimensional hierarchy of energy barriers as shown in Fig. 3.6.p3 p1 p2 p1 p1 p2 p101234567Fig. 3.6 Energy barriersThe studied case corresponds to jumps of arbitrary distance with transition rates depending only on the highest energy barrier between initial and final state. For instance p =p =p2, p =p and so on. The fact that the transition depends only on12 56 25 3the highest barrier is a scale effect. In order to illustrate the evolution of a particle in such a space as shown in Fig. 3.5 or in Fig. 3.6 consider the processes diagram presented in Fig. 3.7. Fig. 3.7 shows the PSM frame associated to one level of states for Cayley tree.
