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 with trivial Hamiltonian in the mode coupling theory (MCT) of structural glass transition. 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 Leutheusser-type transition is reproduced only in a certain range of the strength. On the basis of the anayses of our model we discuss about a sharp ergodic-nonergodic transition and its smearing out by " hopping ". The mode coupling theory(MCT) as applied to supercooled liquid and glass transition is the only existing first principle theory for this last stronghold of condensed matter physics with surprising success in describing the initial stages of freezing[1]. However, as one expects, there are lots of formidable issues in applying the theory originally designed for critical phenomena dealing with thousands ofÅngstroms to the short scale problems of at most tens ofÅngstroms. In the original derivation of the idealized MCT self-consistent equation for the density-density correlator, the factorization approximation of replacing the four-body time correlation function by the product of two-body time correlation functions was introduced, which is totally uncontrolled and whose region of validity is unknown. Also nature of the non-ergodic state after the freezing transition is far from clear. The difficulty is compounded if one goes over to the so-called extended MCT where the rapidly varying momentum variable is introduced in the MCT scheme, which was necessary to partially cover the thermally activated processes important after freezing transition has taken place. However, such rapidly varying momentum variable can never be sensibly treated by MCT[2]. Furthermore, physical picture of the hopping processes is still lacking. In the recent years, the ideas and the methodology developed in the spin glass community are brought in to deal with structural glass problems, which was pioneered by Kirkpatrick, Wolynes and Thirumalai[3]. Here mean field type toy models like p-spin models are being analyzed producing deep insights. However, in all these spin glass models the glass transitions are driven by non-trivial Hamiltonians (or free energy functionals)[4]. On the other hand, in the above-mentioned MCT for structural glass, the transition is driven by the reversible mode coupling mechanism, and does not require a nontrivial Hamiltonian. Considering these circumstances we feel it important to develop a mean field type toy model that mimics this reversible mode coupling …