statistical mechanics - Why is the canonical ($NVT$) ensemble often used for (classical) molecular dynamics (MD) simulations?

Molecular dynamics (MD) simulation is a common approach to the (classical) many-body problem. It relies on integration of Newton's equations of motion to simulate the trajectories of many (e.g., ~1,000-100,000) particles.

In my limited exposure to the MD literature, a recurring theme that I think I see (at least on the chemical physics end of things) is that MD simulations are often performed in the canonical ($NVT$) ensemble. Why does this seem to be the case?

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Here are the common thermodynamic ensembles:

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  Microcanonical ensemble ($NVE$)

  
  
  
  
  • The system is isolated.
  
  
  • The total energy $E$ is fixed.
  
  
  
  • Every accessible microstate has equal probability. That is, if $\Omega$ is the number of accessible microstates, the probability that a system is in a particular microstate is $\frac{1}{\Omega}$.
  
  
  • Please correct me if I am wrong, but I think that the microcanonical ensemble satisfies ergodicity -- time averages can be replaced with ensemble averages.
  
  
  
  


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  Canonical ensemble ($NVT$)

  
  
  
  
  • The system is not isolated. The system can exchange energy with a heat bath. The total energy of the system + bath is fixed. The average or equilibrium energy of the system is constant.
  
  
  • The absolute temperature $T$ is well-defined. (Is $T$ fixed? I think so.) $T$ is given by the temperature of the heat bath.
  
  
  • The probability of finding the system in some microstate $i$ with energy $E_i$ is given by the Boltzmann distribution: $$p_i = \frac{e^{-\frac{E_i}{k_B T}}}{\sum_i e^{-\frac{E_i}{k_B T}}}$$
  
  
  
  
  


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  Isothermal-isobaric ensemble ($NPT$)

  
  
  
  
  • The absolute temperature $T$ and the pressure $P$ are fixed.
  
  
  
  

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Looking at this non-exhaustive list of choices, it seems that we can eliminate the $NVE$ ensemble from consideration because "real world" chemistry involves energy exchange with the environment.

MD simulations typically do not model chemical reactions, but still, I would say that most chemistry in the "real world" occurs at nearly constant pressure (e.g., atmospheric pressure). So the $NPT$ ensemble seems like a reasonable candidate.

What about the $NVT$ ensemble? Constant temperature perhaps seems reasonable for equilibrium, "real world" chemistry, but I am not so sure about constant volume.

Now let's jump back to my very rudimentary of MD simulations in the literature. In MD simulations, molecules sit in a simulation box to which periodic boundary conditions are applied. From reading some literature articles it seems that the $NPT$ ensemble is used for equilibration -- to obtain the simulation box size that gives an average pressure of, for example, 1 atm. Then, the system is simulated in the $NVT$ ensemble -- that is, the simulation box's dimensions are held fixed, hence fixing the system volume. It is from this simulation in the $NVT$ ensemble that ensemble averages are computed and the system's chemistry is analyzed.

Why is the $NVT$ ensemble used for MD simulation production runs?

Answer

To amplify on something in the Ron's answer: Fixed energy is hard to maintain numerically; the slight computational errors accumulate over time. The "thermalization" effects serve to fix this, and keep the overall system with a (relatively) stable average energy (large N).