# Questions about symplectic conservation of MD

MD method is widely employed in different areas. However, as we all known that the limitation in timescales and length scales and the stiffness problem due to high frequency molecular vibrations are still important and difficult issues to be solved. While, characteristics of symplectic conservation is important for numerical methods. I found that only a few leteratures discussed this issue, and seldom new symplectic methods were widely adopted expect for the classical leap-frog Verlet  algorithm whose characteristics of symplectic conservation was proofed later.

I have several questios as fellows:

Which are the key issues of the development of MD?

Whether symplectic conservation is still not paid enough attention for the reason that the time scales limitation is so short that the dissipative effect is still not obvious?

Can the symplectic algorithm paly an important role in the development of MD.

I am not familiar with this area , if anything wrong, please point it out.

Best regards

teng zhang

### Symplecticity & Reversibility for MD equations ...

Since the Hamiltonian flow is reversible and symplectic, one should aim to devise algorithms that are reversible and symplectic. Reversible or symplectic algorithms exhibit superior long-time conservation properties (the error in Hamiltonian and other adiabatic invariants are bounded over exponentially longer time intervals). Consequently, these are preferred and are used commonly (in almost all MD codes). The classical leap frog (or Verlet) algorithm is second order, symplectic, reversible, and SIMPLE!. This algorithm is explicit, and can be formulated as an operator splitting type algorithm (see also under the name Geometric integrators and Suzuki-Trotter expansions), which is one of the best ways to devise symplectic and reversible schemes. However, using more operator splits than the traditional Verlet scheme does not fetch you much. In fact, it has the same computational complexity as operating with Verlet although the order of the method is increased. But, now you make your algorithm more complex than simple Verlet. Besides, accuracy ("as determined by order") is not that important once you have at least second order accurate (reversible & symplectic such as the Verlet) algorithms since MD equations are inherently chaotic and we are interested only in thermodynamical or kinetic properties in MD as determined by ensemble averages. On the other hand, stability of the algorithms is more important since it dictates your step size.
Having said that, multiple time stepping schemes that are both reversible and symplectic are in common use as well. These algorithms split the force fields into slow and fast fields in order to speedup the MD computations with slightly larger step sizes. However, symplecticity of these algorithms introduces certain parametric resonance effects into these algorithms, which restricts the maximum step size that can be taken even with these algorithms.  Anyway, I hope that I have given reasonable amount of information for you to dig up much more. But, the bottom line is symplectic and reversible schemes are used most commonly for integrating MD equations. There is a vast literature on this topic.

### Dear Phani: Thank you for

Dear Phani:

Thank you for presenting this summary about the symplecticity & reversibility for MD. I learned a lot from this. I am a junior graduate student of mechanics and have no background on this, when I saw many literature on this, however, the main algorithms are still the classical ones and I do not konw whether the currently popular algorithms are symplectic conservation, I was puzzled. I am suddenly enlightened thanks to the information from Phani.