We are simulating a percolation dynamic system where we obtain first or second order phase transitions depending on certain parameters. For certain values it is clear that we are in the first-order vs the second-order regime. However, for other values the finite size effects make it very difficult to distinguish between a first and second order phase transition. The need to run and analyze larger systems and more realizations creates a large burden on our computation resources.
Another well-known feature of phase transitions is the critical slowing down. From preliminary tests that we've done, it seems that the graph of iterations (time) as a function of the critical parameter has a markedly different signature for first and second order phase transitions. This is interesting but without some sort of theoretical basis, we can't rely on it too strongly.
Is there any published research (or good argument) about why a first vs second order phase transition would have different critical slowing down curves? Obviously, the more quantitative detail the better.