sethoflagos

The overall CoM is essentially fixed by your constraint of infinite box mass. The total linear momentum of the system with respect to the total CoM is equal to the linear momentum of the ball wrt the total CoM. What happens at collision with the box wall is observer dependent. An observer travelling with the ball will see the ball suddenly accelerate from rest and deduce that work has been done on the ball providing it with a kinetic energy it did not previously have. An observer stationary wrt the system CoM may (per your assumption) see a specular collision where the kinetic energy of the ball is preserved. A third observer may well see the ball suddenly come to a standstill. All intermediate variations are possible. Single particle thermodynamics really has no meaning due to this observer bias. It isn't until we get into multiple particle systems where particle-particle collisions are possible, that we can shake off this observer bias and establish the concept of thermodynamic equilibrium - one requirement of which will be to set constraints on the motion of the centre of mass of the particles in the system (excluding the box). Maybe I should give you a heads up of where we are heading with this. I'm using a combination of the 1st Law and conservation of linear momentum to establish the necessity for a stationary gas CoM as a priliminary condition of thermodynamic equibrium in, for now, a defined NVE state. The next stage will be to recruit the conservation of angular momentum to establish that there can be no nett flow towards or away from the CoM at equilibrium ie that in the absence of input of external work or torque, the gas remains evenly distributed throughout the available volume. After that, well I trust that any residual ideas of macroscopic systems hopping willy-nilly between unrelated, random microstates will have dissipated. I thank you all in advance for forcing me to think through all this stuff carefully stage by stage. It's over 40 years since I sat through my last thermo lecture!

I specifically stated the centre of mass of the gas because this is precisely the quantity that you are trying to move around relative to the CoM of the box to establish your case.

If the centre of mass of the particles remains stationary at the centre of the box, then it's possible that no nett work has been performed on the gas. Now explain to me how that centre of mass can move towards the sides of the box to any reasonable degree without the box performing work on the gas.

Another interesting rabbit hole. By coincidence, I asked a question the other day about the strength of coupling between a CMB photon emitter and its TV aerial absorber. Mordred informed me that contrary to what I'd inferred from what I'd read of GR, they weren't actually touching since the photon wasn't a valid frame of reference. Obviously, I've no grounds whatsoever for disputing this, and too many Minecraft projects planned to try and get to grips with it. I just accept that for now and the foreseeable future, such fields are beyond my understanding. Not my system. You define your system as you wish. If you can. It's not easy, and perhaps will give some idea of why the microcanonical ensemble has given statistical mechanics some headaches in the past. The canonical and grand canonical ensembles find easier application in the real world and have no aberrant conflicts with classical thermodynamics. There is now an exchange of work between particles and box. It has become your very own 'undisclosed piston'. Do you now see and understand the issue?

An interesting rabbit hole! Firstly, consider what is happening to your system centre of mass, and account for its apparent irregular motion. I look forward to your well considered response (which I'll not preempt here)

My point. We're agreed. Agreed. The V/2 state perforce requires some combination of W and Q which must be reflected in a balancing change in U. 2nd Law implies a minimum increase in U (and hence T) under simple compression. They said it. Obviously. Absolute zero microstates are allowed into their ensemble by their reasoning. How do you propose to explain all the particles appearing in one half of the box? My inference is that there must be something equivalent to an undeclared piston compressing the system which, as you say, violates the conditions of the problem. That in itself is a clarification: the hypothesis is probably BS.

Your point that classical thermodynamics largely time independent, I accepted without seeing the need for further comment. Which was the other point? I'm 100% with Mordred on this issue. Would you be happy to simplify the thread and leave it at that? Then we are in agreement. Obviously. Hope all is clear

You do like to nit-pick! Depends a little on context. Formally, in my day job, it usually infers that the system is in a state of minimum Gibbs Free Energy - e.g there are no bulk convective processes going on within it. For a constant V, T system (I rarely encounter these) it would be a state of minimum Helmholtz Free Energy. In the OP scenario, the presenters start with an equilibrium V, T condition and claim that it can evolve spontaneously to occupy only V/2. This requires a bulk convective flow (eg a piston compressing it) and represents a fundamental change of state. The presenters concentrate on the position distribution of their system and fail to mention any impact on the momentum distribution. Do we infer the temperature has remained constant (breaking the 1st Law and the 2nd)? Has it increased as it would if it had been compressed by a piston (2nd Law preserved but not the 1st)? Or indeed has it decreased. We are left to guess. The only clue we have is that the presenters claim to have 'proven' evolution to a low entropy condition. If we believe them, this eliminates the higher temperature case from consideration. What we are left with is a proposed sudden and significant random change of state breaking both 1st and 2nd Laws. It's perhaps a personal flaw, but I've a habit of ridiculing such proposals by highlighting an extreme case that becomes allowable if their assumptions are correct. Such as a spontaneous jump to absolute zero. Of course the presenters do not state this inference explicitly as it would make them appear very foolish. But I'm quite happy to point out a logical extension of their false reasoning.

I did. A box of volume V with all its contained particles sat in the left half is a non-equilibrium state, isn't it? As were each and everyone of the previous10^(big) intermediate microstates necessary to create this scenario. You're Wikipedia link introduced 'N .... a small number of particles'. I thought it might be useful to firm up the order of magnitude where this concept may have some significance. Something a bit smaller than say N = Avogadro's number. Similarly, you're link stated this small number of particles 'may show significant statistical deviations from that predicted by the second law' without quantifying it by example. So I provided an example. I wasn't changing the parameters. I was merely plugging in representative numbers where they had been left unquantified, woolly, and uninformative. Please read the quote this comment refers to: it paraphrased the Youtube presenters. For me, the relative temperatures of the two states were undefined therefore so were the relative entropies.

Perhaps you didn't read my OP carefully - I dispute that these 'so-called second law violations' exist at all precisely because they ignore the concept of formal states. In particular these examples depict what I presume is a microcanonical ensemble (no heat bath is indicated) which in statistical mechanics (as I understand it at least) has a clearly defined equilibrium NVE state. ie the ensemble consists of all those possible accessible permutations of that number of particles (N) occupying a constant volume (V) within a vanishingly thin band of total energy (E). I trust that you agree that this corresponds to a formal state. The next slide presents (presumably) the same N particles occupying only half the volume, claiming that this an inescapable result of statistical mechanics. Would you agree that this corresponds to an entirely different formal state (with undefined total energy to boot)? Personally, I dispute that such a state could evolve for even the briefest of flickers because in that instant, it 'forgets' its earlier state - the information necessary for restoring it has been irretrievably lost due to the proposed macroscopic drop in entropy. The change would be permanent. This is significant. If we accept the smallest possibility of such an event, we accept higher frequency occurrence of less extreme random deviations and so on until we no longer have meaningful conservation laws - isolated systems would be continuously changing their properties in a continuous random walk with expected deviation propotional to the square root of time elapsed. I am amazed that so many seem to buy into this concept, without apparently the slightest shred of empirical evidence.

Many thanks, swansont. My reading of this passage draws two key inferences: 1) 'Each microstate that the system maybe in' refers specifically and only to the ensemble of microstates whose properties are consistent with those of the initial microstate and for which there is a credible mechanism through which each can be accessed (see ergodic hypothesis). It most definitely does not include any wacky extreme non-equilibrium microstate dreamt up by a Youtube presenter in search of more Patreon support. 2) A snapshot of a small number (like 42) particles doesn't have a precisely defined temperature etc due to the uncertainty principle and the relatively large error bars of a small dataset. However, this measurement problem is just that, isn't it? Hiding away inside the quantum fuzziness is there a possible state of 42 regularly spaced particles all with zero relative motion? I think not. There's no route in and out of such a state. I don't really follow quantum theory but I was under the impression that many of its leading lights were currently touting 'information cannot be destroyed' which pretty much underpins the 2nd Law, doesn't it? Actually there's a third now I think of it. The Wikipedia paragraph you referenced carries no inline references. I was rather hoping to find something on this subject that's been through a proper peer review.

In what way is the 2nd Law 'statistical'? Many notable researchers have used a statistical approach to probe the complexities of thermodynamic systems, but isn't that only because of the computational complexity? Are you claiming that the systems themselves are stochastic in a real sense? If so, then where does the random element creep in?

This has been a common trope running round most of the pop-science channels on Youtube over the last year or so. I won't name names but I guess some of you know some of the channels in question. It starts with a box with N particles randomly dotted around inside it. The presenter then changes the cartoon to one where all the dots are shown on the left hand side of the box and states 'Statistical Mechanics says that all random configurations of particles are possible, therefore sometime eventually this low entropy configuration will occur, therefore the 2nd Law only applies sometimes'. Leaving aside the macroscopic shift in centre of mass that shows dereliction of the 1st Law (an easy fix if they cared about it), they present this extraordinary claim without stating what quantities are preserved in the analysis, what if anything has happened to the momentum distribution in the half box scenario, or whether 'statistical mechanics says' the system will ever find its way back to its original thermal equilibrium. Have I missed something somewhere, or is it all just clickbait BS?