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Given the large difference between simulation timescales and the timescales on which we normally interact with ensembles of molecules, the time average of a molecular dynamics simulation does not always match what we would expect to observe experimentally. Effectively, there is an ergodicity problem as the molecular dynamics simulation is not exploring the entirety of the relevant phase space.

What techniques exist to improve the handling of non-ergodic systems for molecular dynamics simulations?

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    $\begingroup$ Can you give an example of what you mean? Because I wouldn't say that the MD simulation taking too long in practice is a problem of non-ergodicity. Like if some protein takes one second to fold, you probably can't use MD but this is not because the system is non-ergodic. Usually to circumvent this type of issue, people use something like replica-exchange MD, some type of MC, umbrella sampling, or metadynamics. All of these are methods that destroy the meaning of the sequence of frames in exchange for enhanced sampling of the system. $\endgroup$ – jheindel Oct 26 at 17:41
  • $\begingroup$ I am thinking of situations where there are energetic barriers that prevent the exploration of the entire relevant phase space on the timescale of the MD simulations, though where the physical system does explore these regions on experimental timescales (as an example; other similar situations are interesting too). While using the phrasing "non-ergodic" to describe such a situation may be using a slightly loose definition of ergodicty, I believe I have heard it used by some people in the field. $\endgroup$ – 2ndQuantized Oct 26 at 21:21
  • $\begingroup$ @jheindel You could write up an answer including the different techniques you mentioned (replica-exchange, umbrella sampling, etc.) as those are the kinds of techniques I am thinking of. If you or others know of more approaches, they could be included in answers too. $\endgroup$ – 2ndQuantized Oct 29 at 21:35
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    $\begingroup$ @jheindel it would be great if you could write up an answer! $\endgroup$ – Nike Dattani Nov 20 at 22:46
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Metadynamics

One way to make your dynamics explore more of phase space is to force it out of regions it has already explored. Metadynamics does this dynamically by intermittently adding bias potentials (typically Gaussians) to the system Hamiltonian. These bias potentials are functions of user defined collective variables, which should be chosen such that states of interest can be clearly distinguished from each other. The figure below gives a good example of how metadynamics proceeds; a common analogy is that you are exploring a potential energy surface that is slowly with sand/water around the current state. In the long time limit, the bias potentials should sum up to the free energy, but with the opposite sign.

Example of Metadynamics

The main difficultly of this method is determining what collective variables need to be considered in order to fully describe your system. Its an area of active of research to develop fairly general collective variables (or at least guidelines for their selection) for a variety of systems. As an example of what these collective variables can look like, consider [1], a paper by Michele Parrinello, who originally proposed the concept of metadynamics. In this paper, they define an enthalpic and an entropic collective variables in order model crystal formation of aluminum and sodium.

  1. Pablo M. Piaggi, Omar Valsson, and Michele Parrinello PRL 119, 015701 (2017)
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