# Entropy and Conservation of Momemtum

I have read a post that interests me greatly Heat engines and "Angular momentum" engines?

I have often wondered what the relationship is for example between Conservation of Angular Momentum and Entropy

For example

I have a simple system with two possible (but not equally probable) microstates consisting of 8 particles contained in a circular boundary

Entropy tells me I will evolve from the ordered to the disordered system

However Conservation of Angular Momentum tells me that the Angular Momentum of the system (Positive L ) must be conserved so I will not evolve to the disordered system (L= Zero) without an external force (torque)

This to me makes sense as without an external force the sum of the counter and counterclockwise vectors can not change.

However if the system can exchange heat with its surroundings then the vectors can change (thru Maxwell Diffuse Reflection) and evolution to the disordered state is possible.

The problem is that the interaction of heat at the boundary is normal to the surface so is not a torque but provides a deceleration that changes the amount of Angular Momentum ?

How can I bring these two laws together. Are they one and the same thing as suggested by @Nathaniel ?

i am seeking an intuitive answer as my calc is greatly deficient

• Angular momentum can be exchanged between particles, it is total angular momentum that is conserved, not particle angular momentum. Oct 8, 2016 at 4:18

In principle, with a perfectly circular boundary that is incapable of exerting any torques the angular momentum of the gas will be conserved. This, like the volume of the system, is one of the constraints that limit the available phase space. In practice, all atoms are sticky because of Van de Waals forces, so any surface used to confine the system will exert torques.

There are some exceptions to the above statements in the realms of cryogenic physics. Superfluid helium, for example, can flow without resistance, so a circular container of superfluid helium could exhibit a conserved total angular momentum that is independent of temperature. Likewise, the electrons (Cooper pairs of them, at least) in a conductor flow without resistance.

If you want to think about how this can be consistent with the laws of thermodynamics, imagine a box full of gas at a given temperature hurtling through space. The total momentum of the box+gas is conserved, and non-zero, and it will stay that way. Similarly the energy of box+gas is conserved, if we pretend it can be perfectly insulated.

I am not sure how you arrived at the conclusion that those two states are not equally probable. I suppose you are simply assuming it.

Order and disorder are qualitative concepts, and entropy does not correspond to "amount of disorder" (how do you even quantify this?). Entropy is a mathematical entity whose maxima characterizes an equilibrium state for a thermodynamic system under constraints.

Even if one state is more probable than another, this doesn't mean a system will spontaneously move over to the more probable state. One must check whether a transition is possible. A fuel may increase entropy of universe by burning, but it does not spontaneously combust if you keep it in the open. Similarly in your example conservation of momentum perhaps forms a barrier that locks the system into that particular state, irrespective of probabilities.

• sorry my wording was misleading I meant the macrostate in not equally probable I mean that they are more potential microstates that reflect the chaotic macrostate than the ordered one. Oct 8, 2016 at 6:25
• Looking again at my diagram this equally isn't clear as they are still ordered pairs but if we were to expand the potential momentums across all vectors then it would be clearer. I think @Sean Lake grasped what I was asking and his answer identified how a torque could be provided. Oct 8, 2016 at 6:31