In my opinion there is no generally acknowledged meaning associated to convergence as long as molecular dynamics (MD) runs are concerned. You have to specify this meaning depending on the specific goal you set for the simulation which is carried on. For instance you may have convergence towards thermal equilibrium when you run an equilibration step. You may have convergence towards an average conformation or set of conformations depending on the r.m.s. changes in the coordinates of your molecule, and so on...

In contrast, in Molecular Mechanics convergence has a commonly accepted meaning: you have found a set of atomic coordinates corresponding to a local Energy minimum as a function of the chosen force field.

In quantum mechanical ab-initio computations you commonly define convergence according to the values of maximum and average displacements and maximum and average forces on the system you set for your convergence criteria (note that different criteria are possible).

As far as I know, in MD runs such meanings are not accepted as the default for convergence. Thus, if you use such a term, you have to specify clearly the parameters used for its definition.

Also, different quantities converge at different rates. In my own work, this is particularly evident when one attempts to estimate configurational entropy via the mutual information expansion or related approaches. First-order entropy estimates converge relatively fast; going to second order becomes challenging; third order is still almost impossible. Essentially, it is a question of the dimensionality of the probability density function (pdf) that underlies the quantity one aims to compute. Conceivably there could be conformational transitions that occur only when 100 torsion angles all get into the right range at the same time. Computing a converged rate constant for this kind of "combination lock" transition would be really really hard.

As indicated above, it depends what quantity you are looking at. Convergence of Energy (pot, kin, total) Temp, RMSD radius of gyration etc etc. It also depends what your criterion is. For example energy gradient etc.

This is a passage I remembered I read sometime ago. Finally I had time to recover this excerpt from Cramer's book "Essentials of Computational Chemistry". I

"Convergence is defined as the acquisition of a sufficient number of phase points, through either MC or MD methods, to thoroughly sample phase space in a proper, Boltzmann-weighted fashion, i.e., the sampling is ergodic. While simple to define, convergence is impossible to prove, and this is either terribly worrisome or terribly liberating, depending on one’s personal outlook.

To be more clear, we should separate the analysis of convergence into what might be

termed ‘statistical’ and ‘chemical’ components. The former tends to be more tractable than the latter. Statistical convergence can be operatively defined as being likely to have been achieved when the average values for all properties of interest appear to remain roughly constant with increased sampling."

The idea of convergence in MD simulation is based on the "Ergodic Hypothesis", i.e. the relevant phase space should be covered at the long-time limit. In practice, however, this is not always guaranteed, especially, when the system is frustrated.

Thank you very much for this discussion since it enlightens a lot about a possible response we have on a given manuscript where a referee asks us about the convergence of a long MD run. We are not sure though if the numerous analysis and post-processing performed are indicative enough of such "convergence".