Hi and welcome.
So, first, I would like to share the information above...
It may help if you can provide some references or links. Sadly the forum software won't allow you to provide a working hyperlink until you've made a few posts. That's to reduce the amount of advertising and spam etc. This means your first post just may not be as productive as you hoped - sorry.
A request of mutual benefit:
You could start to increase the number of posts you've made by replying to this forum thread:
https://www.thenakedscientists.com/forum/index.php?topic=86280.msg704124#msg704124
The basic question in the OP (Original Post) was about how or why this site might attract and retain new members. The OP (Original Poster) @Zer0 hasn't had many replies and even just a short sentence from a new member would be appreciated.
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And, second, to ask for any other examples of systems with known uneven laws of energy partitioning.
I can at least offer one example of such a system. The behaviour can sometimes depend on the substance rather than on the container in which it is put.
A common example (of ergodicity breaking) is that of spontaneous magnetisation in ferromagnetic systems, whereby below the Curie temperature the system preferentially adopts a non-zero magnetisation even though the ergodic hypothesis would imply that no net magnetisation should exist by virtue of the system exploring all states whose time-averaged magnetisation should be zero.
https://en.wikipedia.org/wiki/Ergodic_hypothesis#Phenomenology
Best Wishes.
Hi.
I've re-read your posts but without glancing at the articles or paper your talking about, it's not really all that clear (to me) what is being done.
Consider the usual ideal gas,...
You could be using a purely Newtonian model for that. However this later statement...
Hence, not all the equal energy states are populated,...
suggests that you could be using a Quantum Mechanical model. The later post with the diagram seems to emphasize angular momentum as a Newtonian concept instead of as an Angular momentum operator acting on a QM system. Your model may be a mixture or hybrid (which is also common).
This statement may be incorrect:
Hence, not all the equal energy states are populated, which makes the system nonergodic.
I'm not sure if you mean "energy states" as the term is used in QM, or some reference to paths through position and momentum space.
I think we need to establish what ergodicity is and/or how you are considering it.
Ergodicity can be considered as a property of an observable where an average over time is equal to an average over multiple systems (an ensemble). You might be considering ergodicity as the property of a system suggesting that it will visit (and re-visit) all parts of position-momentum space over large times. The two definitions or approaches are compatible but it's easier to consider the first definition and then the second will follow for Newtonian systems. For Quantum Mechanics the first definition or approach is nearly always easier to use (the second version or definition is just complicated for QM and some discussion appears later).
You ( @napdmitry ) might already know this but if anyone else is reading then a diagram like this one will help:
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Diagram snipped from a video - original creator was the London Mathematical Laboratory, I think.
The diagram shows values for an observable X observed in 5 independent but identical experiments (well, identical as much as possible, but the variable X is inherently random. For a deterministic system, the initial value of X was random). The random variable X(t) is ergodic if the average you obtain over the 5 different systems at one instance of time (the red shaded average of values) is equal to the average you would obtain by just following any one system over time (the green shaded average of values). Formally, you consider the limit as the number of identical systems and the time over which an average is taken → ∞.
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1. Now, if you're using a Quantum mechanical model, then your observables are given by operators (... usual QM..). So, you might be tempted to consider the position and momentum of a gas particle as an observable BUT they are certainly not clearly defined straight line paths like that shown on one of your later diagrams. Particles can be found in all sorts of places with some probability and uncertainty relationships will limit simultaneous position and momentum determinations. Overall it would be a difficult task to demonstrate that these sorts of observables won't be ergodic.
2. If you're using Newtonian mechanics, then particles should have the straight line paths as illustrated in your diagram and you can know the position and momentum simultaneously etc. However, the entire model was idealised. Any perturbation will put the particle onto a different path. For example, the usual assumption in an ideal gas is that collisions between particles are negligible but some assumption about randomness of trajectories is also made. Exactly how you reconcile all of the assumptions is open to interpretation and it doesn't matter, we just want those assumptions and see what consequences follow. However one reasonable and practical interpretation is that there could be collisions or close approaches where some inter-atomic forces deflect the particles. They are rare enough to be ignored in most calculations but sufficient to perturb the trajectories and ensure that over long durations of time, the particles do travel on all paths and visit all parts of the position - momentum space.
But what is the law of partitioning, instead of the even one? That is the problem.
You might be talking about partitioning between different ways of accounting for energy macroscopically. For example, an ideal gas will tend to have an equal amount of motion (kinetic energy) in the x-, y- and z- directions.
On a more microscopic level, I have to agree with @alancalverd , the Boltzmann distribution describes the situation quite well in most situations.
Provided a system can be considered to be in contact with a heat bath that maintains temperature T and a given microstate of your system has energy Ei then the probability of the system being in that microstate is given by:
Pi =
with Kb = Boltzmann constant; Z = partition function =
Various developments are possible from there.
---This post is already too long. I'm ending. It's the best I can do for a discussion at the moment, sorry.
Best Wishes.