3
$\begingroup$

In my rough understanding Molecular Dynamics using Classical Newtonian mechanics is a 6N dimensional non linear system. 6N dimension because we have 3 position vectors and 3 momentum vectors for each N particles. Nonlinearity because of the terms in force fields as far as I understand. In principle this system can exhibit chaos. So my question is the following: Is there any scenario where by the system enters a stable/ semi stable limit cycle or periodic trajectory? I have read that the system cannot go into a point attractor since the volume of probability density function must be conserved and cannot become zero. If the system can enter periodic oscillations or limit cycle what does that imply?


UPDATE : Can someone give me reference suggestions like books, papers, video lectures or any other document links related to this topic? I also wanted to know whether the assumption of ergodicity in the system prohibits limit cycles ?

( @Wrzlprmft answered my question about ergodicity in chats. I am copying it here :

In case of a system with attractors, you can consider ergodicity as follows: On the one hand you take one arbitrary condition, evolve it for some time (to discard transients) and then start pooling the ensuing trajectory for the time average. On the other hand you take a bunch of arbitrary initial conditions, evolve them for some time (to discard transients) and pool the resulting states (one per initial condition) to obtain the ensemble average. In case of a single attractor, the time and ensemble average are the same, namely the average over all states on the attractor. In case of two or more attractors, the time average only reflects the states of one attractor, while the ensemble average reflects the states of all attractors. Hence such systems are not ergodic. )

$\endgroup$
  • $\begingroup$ One slightly snarky answer would be that the system is simple enough (or has a strong enough attractor I guess) that it can overcome the accumulation of numerical errors in your simulation. $\endgroup$ – Jon Custer Oct 21 '16 at 13:28
  • $\begingroup$ @JonCuster : Sorry I did not understand. $\endgroup$ – dexterdev Oct 23 '16 at 17:59
  • $\begingroup$ @JonCuster Update: Now I understand your comment. :) $\endgroup$ – dexterdev Nov 16 '17 at 12:26
1
$\begingroup$

Is there any scenario where by the system enters a stable/ semi stable limit cycle or periodic trajectory?

A limit cycle is a periodic trajectory.

That being said, despite all non-linearity, we are still looking at a Hamiltonian system here. As you already mentioned, phase-space volume is conserved in such systems (Liouville’s theorem). Now, a defining quality of an attractor is that it has a basin of attraction that becomes the attractor through time evolution. As the basin of attraction has to be a true and open superset of the attractor, it’s volume inevitably shrinks upon time evolution. This contradicts Liouville’s theorem and hence there cannot be any attractors in Hamiltonian systems – they only exist in dissipative systems. (Also see this answer of mine.)

For periodic or fixed-point solutions, there is another simple argument via time symmetry, which applies to these systems. If the system converging to a periodic or fixed-point attractor is a solution, then the time-reversal, i.e., the solution becoming aperiodic or starting motion, must also be a solution. But then the respective solutions cannot be attractors anymore. Every attractor would have to be a repellor when time is reversed.

There are periodic solutions to such systems, though. Consider, e.g., a solid state where all molecules are on a lattice and perform oscillations where one half of all oscillators oscillate in an identical fashion and anti-phase to the other half.

$\endgroup$
  • $\begingroup$ By semi-stable limit cycle, I meant a case where by the trajectory can converge from one direction and diverge in another direction (when thinking in 2D) like here : art.tze.cn/ApaDownLoadRef/m.20071207-m300-w001-008/images/… $\endgroup$ – dexterdev Oct 21 '16 at 16:48
  • $\begingroup$ I am convinced by the fact that system cannot enter point attractors (sinks). But why not periodic attractors? Why cannot the volume of pdf be conserved in that case? $\endgroup$ – dexterdev Oct 21 '16 at 17:42
  • 1
    $\begingroup$ @dexterdev: See my edit. $\endgroup$ – Wrzlprmft Oct 21 '16 at 19:02
  • $\begingroup$ So is this equivalent to saying that Energy is not conserved in the limit cycle case ? (say for example in simple harmonic motion case the energy is conserved in the periodic orbitals, but in the limit cycle case energy is not conserved) $\endgroup$ – dexterdev Oct 22 '16 at 16:53
  • 1
    $\begingroup$ @dexterdev: So is this equivalent to saying that Energy is not conserved in the limit cycle case ? – Yes, at least in what is covered by your model or simulation. To have an attractor, you need to lose energy at some point. Usually energy is also fed into the system at another point. A typical example would be the damped and driven pendulum. Energy is lost through friction and fed into the system through the driver. $\endgroup$ – Wrzlprmft Oct 22 '16 at 17:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.