A first-principles derivation of the master equation is systematically given based on Kikuchi's ansatz, which is then applied to non-interacting (ideal) and interacting lattice-gas systems. The former application to an anomalous diffusion, observed by MID simulation of beta-AgI, makes its mechanism clear in terms of relaxation modes such that the anomalous diffusion is due to non-diffusive (collective) modes. It is also shown in random systems that anomalous frequency-dependent conductivities, made up of Jonscher and nearly constant loss regimes, are reduced to a single master curve. The case of interacting lattice-gas system is discussed on the ab-plane of Rb3H(SeO4)(2) by a pair approximation of the path probability method, where a spontaneous strain involved in the ferroelastic phase turns out to be a proton-trapped state originated in an attractive strain energy mediated by a proton-displacement interaction, and the transition to superprotonic phase is due to an off-trapping of protons. This mechanism is confirmed by no phase transition without the attractive strain energy. (C) 2008 Elsevier B.V. All rights reserved.