Glass transition in systems without static correlations: a microscopic theory
We present a first step toward a microscopic theory for the glass transition in systems with trivial static correlations. As an example we have chosen N infinitely thin hard rods with length L, fixed with their centers on a periodic lattice with lattice constant a. Starting from the N-rod Smoluchowski equation we derive a coupled set of equations for fluctuations of reduced k-rod densities. We approximate the influence of the surrounding rods onto the dynamics of a pair of rods by introduction of an effective rotational diffusion tensor D and in this way we obtain a self-consistent equation for D. This equation exhibits a feedback mechanism leading to a slowing down of the relaxation. It involves as an input the Laplace transform v_0(l/r) at z=0, l=L/a, of a torque-torque correlator of an isolated pair of rods with distance R=ar. Our equation predicts the existence of a continuous ergodicity-breaking transition at a critical length l_c=L_c/a. To estimate the critical length we perform an approximate analytical calculation of v_0(l/r) based on a variational approach and obtain l_c^{var}=5.68, 4.84 and 3.96 for an sc, bcc and fcc lattice. We also evaluate v_0(l/r) numerically exactly from a two-rod simulation. The latter calculation leads to l_c^{num}=3.45, 2.78 and 2.20 for the corresponding lattices. Close to l_c the rotational diffusion constant decreases as D(l) ~ (l_c - l)^γwith γ=1 and a diverging time scale t_ε~ |l_c - l|^{-δ}, δ=2, appears. On this time scale the t- and l-dependence of the 1-rod density is determined by a master function depending only on t/t_ε. In contrast to present microscopic theories our approach predicts a glass transition despite the absence of any static correlations.