I want to calculate elastic constants at finite temperatures. I read the LAMMPS document, and also the Elastic_T/DEFORMATION example on the LAMMPS github repository, and checked this literature: Clavier et al.
I found that in the literature and LAMMPS document, the NVT ensemble is used for ‘deformation method’. Based on my understanding, the steps to calculate elastic constants at a finite temperature are, first using NPT to equilibrate the system, and then do a series of NVT simulations with deformation.
It appears you may be trying to do a bit too much pattern recognition from preexisting scripts, which may be causing confusion. Before you worry about the MD, make sure you are confident in your understanding of the thermodynamics of mechanical properties and how they are measured in experiments. The MD is really just replicating those conditions in a simulation.
For instance, whether or not you do an initial NPT simulation is not as important to understand as why you do one. The purpose of the initial NPT is to make suren you’re starting with a system near its mechanical equilibrium at a specific P and T. So, you may or may not see such a step in someones script depending on if they assume the input is an equilibrated sample.
For the mathematical theory, it is especially helpful to read this resource: 5: The Isothermal-Isobaric Ensemble - Chemistry LibreTexts it strikes the right level of mathematical detail, giving you enough knowledge to at least figure out what you’re asking (never easy!) but not drowning you in calculation steps.
In addition to the (nice) resources provided by @srtee, you can also look at this other paper that was published during my PhD work. You will find more details on the different methods you can use in molecular modeling to compute the elastic constants of materials there. We also discussed the different constants you get from using NVT or NVE ensemble (i.e. adiabatic or non-adiabatic).
That said, note that the scripts you used are the previous finite deformation simulations in the NVT ensemble and not the second derivative calculation detailed in the paper you cite.
Another helpful source of information to me at that time were the 5th and 7th tomes of the Landau-Lifschitz series which make the connection between statistical physics, thermodynamics and material mechanics. The full lecture is not required, as only some (obvious) chapters are self-sufficient.