Exactly solvable toy model that mimics the mode coupling theory of supercooled liquid and glass transition

A toy model is proposed which incorporates the reversible mode coupling mechanism responsible for ergodic-nonergodic transition of the mode coupling theory (MCT) of structural glass transition with trivial Hamiltonian. The model can be analyzed without relying on uncontrolled approximations inevitable in the current MCT. The strength of hopping processes can be easily tuned and the ideal glass transition is reproduced only in a certain range of the strength. On the basis of the analyses of our model, we discuss about a sharp ergodic-nonergodic transition and its smearing out by "hopping."