Random-matrix approach to transition-state theory

To model a complex system intrinsically separated by a barrier, we use two random Hamiltonians, coupled to each other either by a tunneling matrix element or by an intermediate transition state. We study that model in the universal limit of large matrix dimension. We calculate the average probability < P-ab > for transition from scattering channel a coupled to the first Hamiltonian to scattering channel b coupled to the second Hamiltonian. Using only the assumption Sigma T-b'(b') >> 1 we find < P-ab > = PaTb/Sigma T-b'(b'). Here P-a is the probability of formation of the tunneling channel or the transition state, and the T-b' are the transmission coefficients for channels coupled to the second Hamiltonian. That result confirms transition-state theory in its general form. For tunneling through a very thick barrier the condition Sigma T-b'(b' )>> 1 is relaxed and independence of formation and decay of the tunneling process hold more generally.