# Why do people care about reversibility in molecular dynamics simulations?

When carrying out a molecular dynamics simulation, one has to choose an ensemble. Depending on the problem, one usually works in the microcanonical (NVE), canonical (NVT), or grand canonical ensemble (NPT).

In the NVE ensemble, maintaining reversibility of the simulation is very easy, as the total energy is conserved, so simply starting at the endpoint of the simulation with reversed momenta should reproduce the initial trajectory but in reverse.

In the NVT and NPT ensembles, one has to maintain a constant temperature and/or pressure. Taking NVT as an example, there are many different ways to keep a constant temperature using different thermostats. Thermostats are all methods of keeping temperature constant by exchanging heat with a hypothetical heat bath. There is a range of complexities of thermostats and there's no need to discuss the subtleties of these here. The important point is only that some commonly-used thermostats, such as Nose-Hoover chains, are deterministic and hence result in reversible dynamics. On the other hand, there are Langevin thermostats which are very effective and efficient but are stochastic. So, short of playing a pseudo random generator backwards, the dynamics are not reversible. (I've never seen that reversing a random number generator thing done, but I think it's at least possible in principle?)

I've seen it mentioned many times in the literature that Langevin thermostats result in dynamics which are not reversible. Because I've seen this mentioned so many times, I've always believed it to be important, but I have no idea why one would care about being able to use the end of a simulation as new initial conditions and then propagating the simulation in reverse.

So, why does the reversibility of a molecular dynamics simulation matter? In what situations does using a deterministic thermostat/barostat provide a distinct advantage over a stochastic one?

• I'm only aware of one case where reversible dynamics are required and that is non adiabatic simulations. In these cases sometimes the dynamics jumps over a crossing point so it has to back propagate and then adjust the time step to get it right. So perhaps the other use cases is that the results can be refined in reversible dynamics but not really with stochastic. – Cody Aldaz Sep 3 '20 at 0:57
• @CodyAldaz I think you can do the back-tracking even without time-reversible dynamics because you can simply store the previous geometries and forces and just reset all of those values without actually propagating the dynamics backwards. I might be misunderstanding your point though. – jheindel Sep 3 '20 at 3:22
• Tangentially, yes, a time-reversible RNG is possible. In fact most common pseudo-RNGs are technically reversible, i.e. their state transition map is invertible. Most RNG implementations don't have an API for that, though, and even if they did, you'd have to be careful to call the RNG in the exact reverse order. It's also possible to use a "non-sequential" RNG of the form $r_{t,i}=f(t,i)$ where the output $r_{t,i}$ is calculated by hashing the time step $t$ and a unique ID $i$ used to distinguish random numbers used during the same time step. However, this tends to come at a computational cost. – Ilmari Karonen Sep 3 '20 at 15:39

This is an excellent question! Reversibility in MD is useful because:

1. Time-reversibility in a numerical integrator leads to a doubling of the accuracy order (see Propositions 5.2 and Theorem 6.2 here).
2. Reversible maps can be readily Metropolized, for example in a hybrid Monte-Carlo scheme. This gives an easy way to enhance the sampling efficiency and eliminate the finite time-step error for sampling.
3. Reversibility is aesthetically pleasing, since the exact equations of motion (both classical and quantum) are reversible.

I am sure there are other reasons I have missed. These are the ones that come to mind.

I would argue the main reason this is important is philosophical, linked to the history of science and determinism (as proposed by Laplace). Newtonian mechanics is mathematically reversable while any observation in a "real world" system is one of irreversibility and increasing entropy, which is why we end up with Loschmidt paradox. From a theoretical point of view, keeping a link to a Hamiltonian in the NVT ensemble (as in the Nosé-Hoover thermostat) preserves a number of nice mathematical properties, including time-reversible equations.

In practice, the finite precision of a molecular simulation means than even an NVT run isn't reversible. Small errors grow until the final state has lost the precision required to get back to the initial state when we reverse the time integrator. The fact this point is often ignored show how infrequently this is a concern in most MD studies. In getting thermodynamics properties from MD, we deliberately average the system for long enough that all dynamic information is lost (all of phase space covered satisfying the ergodic hypothesis) so the reversibility is irrelevant. Even for non-equilibrium thermodynamics we work with an ensemble of phase spaces trajectories where the individual trajectories are unimportant.

A little known work by Verlet using integers appears to provide exact reversibility using MD. Some interesting recent work by Bill Hoover explores some applications of exactly reversable integrators.

• +1. Welcome to our site Ed !!!! We hope to see much more of you here. And thank you for your contribution! – Nike Dattani Sep 4 '20 at 18:17
• Thank you @NikeDattani :) I've been on regular old programming stackoverflow for years but only just found "matter modelling", great to have a forum for the MD community – Ed Smith Sep 4 '20 at 18:26