We performed extensive molecular dynamics (MD) simulations, supplemented by Mode Coupling Theory (MCT) calculations, for the square shoulder model, a purely repulsive potential where the hardcore is complemented by a finite shoulder. For the one-component version of this model, MCT predicted [Sperl et al., Phys. Rev. Lett. 104, 145701 (2010)] the presence of diffusion anomalies both upon cooling and upon compression and the occurrence of glass-glass transitions. In the simulations, we focus on a non-crystallising binary mixture, which, at the investigated shoulder width, shows a non-monotonic behaviour of the diffusion upon cooling but not upon isothermal compression. In addition, we find the presence of a disconnected glass-glass line in the phase diagram, ending in two higher order singularities. These points generate a logarithmic dependence of the density correlators as well as a subdiffusive behaviour of the mean squared displacement, although with the interference of the nearby liquid-glass transition. We also perform novel MCT calculations using as input the partial structure factors obtained within MD, confirming the simulation results. The presence of two hard sphere glasses, differing only in their hardcore length, is revealed, showing that the simple competition between the two is sufficient for creating a rather complex dynamical behaviour.

We investigate the slow dynamics of a simple glass former whose interaction potential is the sum of a hard core and a square shoulder repulsion. According to mode coupling theory, the competition between the two repulsive length scales gives rise to a complex dynamic scenario: besides the fluid-glass line, the theory predicts a glass-glass line in the temperature-packing fraction plane with two end points. Interestingly, for critical values of the square-shoulder parameters, such end points can be accessed from the liquid phase. We verify, via extensive numerical simulations, the existence of both points through the observation of an unconventional subdiffusive/logarithmic dynamical behavior. Unexpectedly, we also discover that the simultaneous presence of two end points generates special loci in the state diagram along which the dynamics is identical at all length and time scales.

We study the distribution of the maximum of a set of random fitnesses with fixed number of mutations in a model of biological evolution. The fitness variables are not independent and the correlations can be varied via a parameter ℓ = 1, ..., L. We present analytical calculations for the following three solvable cases: (i) one-step mutants with arbitrary ℓ (ii) weakly correlated fitnesses with ℓ = L/2 (iii) strongly correlated fitnesses with ℓ = 2. In all these cases, we find that the limit distribution for the maximum fitness is not of the standard Gumbel form.