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 ∑_{b^{'}}T_{b^{'}}≫1 we find 〈P_{ab}〉=P_{a}T_{b}/∑_{b^{'}}T_{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 b^{'} coupled to the second Hamiltonian. That result confirms transition-state theory in its general form. For tunneling through a very thick barrier the condition ∑_{b^{'}}T_{b^{'}}≫1 is relaxed and independence of formation and decay of the tunneling process hold more generally.