Microscopic theory for the glass transition in a system without static correlations

– We study the orientational dynamics of infinitely thin hard rods of length L, with the centers-of-mass fixed on a simple cubic lattice with lattice constant a. We approximate the influence of the surrounding rods onto dynamics of a pair of rods by introducing an effective rotational diffusion constant D(l), l = L/a. We get D(l) ∝ [1 − υ(l)], where υ(l) is given through an integral of a time-dependent torque-torque correlator of an isolated pair of rods. A glass transition occurs at lc, if υ(lc) = 1. We present a variational and a numerically exact evaluation of υ(l). Close to lc the diffusion constant decreases as D(l) ∝ (lc − l) γ , with γ = 1. Our approach predicts a glass transition in the absence of any static correlations, in contrast to present form of mode coupling theory. When a system of interacting species is cooled sufficiently fast it will typically undergo a glass transition. In the last two decades there has been a significant progress in microscopic understanding of this transition. The first approach is the so-called mode coupling theory (MCT) [1, 2], first suggested and mainly worked out by Götze and coworkers. For reviews see Refs. [3, 4, 5]. MCT yields a closed set of equations for time-dependent correlators. This set involves as an input static correlators which depend only on thermodynamic variables like temperature T , density n, etc. With decreasing T (or increasing n) local ordering described by peaks of the static correlators grows. When the peaks’ magnitude becomes large enough MCT predicts an ergodicity breaking transition at a critical temperature Tc (or density nc). This dynamic transition is interpreted as an ideal glass transition. A different approach has recently been proposed by Mézard and Parisi [6, 7]. Inspired by the replica theory for spin glasses they developed a microscopic replica theory for structural glasses. This theory predicts a static glass transition at a temperature Tf which is lower than Tc. One of its characteristic features of this transition is the vanishing of the configurational entropy at Tf . The physical picture behind that replica theory was suggested earlier by Singh et al. [8]. Despite providing new valuable insights, these microscopic theories can not be complete, because there exists a purely dynamical mechanism which leads to a glass transition which is not based on the existence of any static correlations. As an example, let us consider a simple