# When is molecular chaos dynamical chaos?

It is very common to have uncorrelated velocities in chaotic dynamical systems. Yet, we should be wary in equating the two quite different meanings of chaos.

Instead of matching dynamical chaos to molecular chaos, it seems much easier to work the other way as the former case does not have a universal definition. Hence, if within some chaotic Hamiltonian system the velocities are uncorrelated, can we say this is molecular chaos? In other words, does molecular chaos exist when particles shrink to an infinitesimal point?

edit: The final question is obviously absurd since molecular chaos concerns uncorrelated velocities occurring as the result of collisions of particles. The point is that particles of finite size are not necessary to define a velocity, so does that mean the molecular chaos 'definition' can be refined further.

Molecular chaos is present in the case of a fully chaotic system. Molecular chaos assures uncorrelated velocities and positions, that lead to a uniformly weighted phase-space distribution on the energy hypersurface. So, then we can apply the ergodic theorem and make the connection between the time average of an observable and ensemble average (calculated from phase-space density) of it.

In another word, in a system with a large number of degrees of freedom, like many particles in a box, and very few constants of motion, phase-space is structureless, and we have molecular chaos. Molecular chaos means uniform distribution of points density on the phase-space.

Notice that molecular chaos is an ansatz or hypothesis, where dynamical chaos is a well-established theory.

• To emphasise further you second paragraph, we should say that dynamical chaos is almost the opposite - it has a great deal of structure in phase space due to UPO families. Feb 24, 2021 at 22:56
• @algae But it's good to remember all that structure is a set of measure zero, i.e., almost all initial conditions in a fully chaotic Hamiltonian system will spread over its phase space over time. Feb 25, 2021 at 8:11

The Molecular Chaos hypothesis is an assumption of perfectly uncorrelated variables, which is conceptually very different from a chaotic dynamics. They have, of course, an interesting interface, as a search for questions with both tags [chaos] and [stistical-mechamics] can show.

As for the title question, molecular chaos is an idealization and, for practical purposes, we could say that: when a chaotic system has evolved for long enough - such that its correlations to have decayed below a measurable threshold - it can be described by molecular chaos.

This could easily be applied to a fully chaotic Hamiltonian system, but you'd need to be more careful with mixed phase spaces or nonconservative systems.