Considerations on the relaxation time in shear-driven jamming

We study the jamming transition in a model of elastic particles under shear at zero temperature, with a focus on the relaxation time $\tau_1$. This relaxation time is from two-step simulations where the first step is the ordinary shearing simulation and the second step is the relaxation of the energy after stopping the shearing. $\tau_1$ is determined from the final exponential decay of the energy. Such relaxations are done with many different starting configuration generated by a long shearing simulation in which the shear varible $\gamma$ slowly increases. We study the correlations of both $\tau_1$, determined from the decay, and the pressure, $p_1$, from the starting configurations as a function of the difference in $\gamma$. We find that the correlations of $p_1$ are more long lived than the ones of $\tau_1$ and find that the reason for this is that the individual $\tau_1$ is controlled both by $p_1$ of the starting configuration and a random contribution which depends on the relaxation path length -- the average distance moved by the particles during the relaxation. We further conclude that it is $\gammatau$, determined from the correlations of $\tau_1$, which is the relevant one when the aim is to generate data that may be used for determining the critical exponent that characterizes the jamming transition.