sethoflagos

FYI How long is "enormously long compared to the age of the universe"? A trillion years for a 1m cube initially at stp? Well let's rescale. For a 1mm cube this reduces the timeline by a factor of 10^9 ie once in a millenium. So within a km^3 of the earth's atmosphere (a 10^18 scaling factor), the frequency of events is over 30 MHz. Back-of-an-envelope tells me that expansion of 0.5 mm^3 @ 2 bar into 0.5 mm^3 of vacuum releases oto 60 microJoules. 2 kW of acoustic energy is rather a lot. If it came from a point source half a kilometer away, that's still ~0.3 microWatts / m^2 which would interfere with normal conversation. And the same sound intensity level is coming from all directions, no matter where you are. And of course, we're only considering the most extreme events. Lesser events presumably occur far more frequently yet with still comparable energy outputs. We would all be deafened by the noise!

My textbooks may be in Lagos, but I'm not : we've been stranded in Holland since February due to the COVID19 restrictions. Hence I've now the time to research some topics I've left sitting on the shelf for a rainy day. I fail to see the relevance of concrete pouring here. You've been adding bits and pieces to your postings! Yes, the Mandl example is representative of the OP. And I have much the same issues with it, at least in the set up. 1) Between state a) and state b), the centre of mass of the gas has moved L/4 from its original position where L is the width of the box. This requires external work i) to set it in motion ii) to stop it again. We'll leave aside the compression element (insufficient info to evaluate) but that issue remains open. 2) In state b) the gas is acted on by a reaction force P*area from the left hand wall that is only opposed by the acceleration forces of the gas towards the right hand wall (at sonic velocity!). It's internal energy being converted pro rata to bulk kinetic energy. At the right hand wall, the reverse happens (KE back to U) until the gas CoM is L/4 to the right of origin and the gas is stationary again. And now, in the absence of frictional forces, we have a simple undamped harmonic oscillator. The gas will NOT "very rapidly ... return to the uniform state of Fig. 2.1(a)" as Mandl states, rather it will remain in an eternal ping-pong state cycling between expansion and compression. Just as it is impossible to leave this cycle without introducing new external inputs, how can it be less impossible to enter it? I really don't know how Mandl proceeds from these initial premises, but it doesn't appear to be a great start. There's more. A lot more. But until we're straight on 1st Law constraints, these examples have nothing to say on the 2nd. We seem to be going backwards.

Yup and Synthesis, respectively.

As in the Hegelian sense of 'thesis - antithesis - synthesis'? Then that's amusing for reasons I won't go into here. Undisputed. You asked me earlier which formulation of the 2nd Law I used. Perhaps, in hindsight it was a fair question and deserved a more complete answer. This is my own personal overview of the heirarchy: Classical Entropy (chemists' version?) : dS = dQ/T and all that follows from that, is the underlying principle behind internationally recognised methodologies of performing thermodynamic engineering design (obviously with appropriate allowances for real process behaviour). Gibbs' Entropy : - kB * Summation of Pi ln(Pi) I don't use, but is I understand synonymous with the above per E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics, 391 (1965) Boltzmann H theorem : Flawed (same Jaynes) due primarily to the molecular chaos assumption - ie that collisions are uncorrelated. This is where my dialogue with swantont foundered, and it continues to sow confusion. Boltzmann Entropy : k log W. Boltzmann's improvement on H-theory, and from which Gibb's Entropy follows (same Jaynes). von Neumann Entropy : the - Summation nj ln(nj) form shows it's affinity with Gibbs' Entropy for which it is an extension (not a refutation) into the quantum realm. I understand that the collision correlation issue is addressed through terms covering the quantum entanglement between interacting regions under the umbrella 'entanglement entropy'. Zachos, C. K. (2007). "A classical bound on quantum entropy". Journal of Physics A: Mathematical and Theoretical. I'm pretty clueless on quantum mechanics, but if von Neumann's entanglement entropy yields quantitatively the same collision correlation as classical conservation of linear and angular momenta (the 'applied' world) and I'd be very surprised if it didn't, then all the above, with the exception of H-theory seem to be mutually consistent. It's obvious which flavour I use as an everyday tool, but in principle, they all seem equivalent. The 2nd Law is the 2nd Law (as someone once said). Therefore I see no problem with statistical mechanics as such. But I take issue with simplistic statistical methods that ignore standard conservation laws.

Nice picture! Let's run with it. The red hot poker is initially one region in thermal equilibrium at T1, the bucket of water another at T2. When brought into contact, the poker starts losing heat at T1 the bucket absorbs heat at T2. The heat transferred across the boundary is identical but the bucket is gaining entropy much more quickly than poker is losing it due to the inequality of their absolute dQ/T values. Therefore, when everything settles down, to thermal equilibrium at a new temperature, the system entropy will have increased quite considerably. In real cases, numbers can be put to these nett changes with a high degree of accuracy. So far, so good. But this is the bit I'm having difficulty getting my head around. I can take a snapshot at any instant of this new equilibrium state and draw an irregular partition carefully excluding any 'fast' particles close to the boundary, but capturing 'slow' ones. Effectively, I've now got a system in 'perfect' thermal equilibrium and a precisely known entropy, comprising two regions at arbitrarily different temperatures whose entropies sum to significantly less. The picture must be wrong somewhere, and I can think of a couple of glib dismissals, but these are giving me no useful insight. Without partition functions the number of microstates in any ensemble is uncountable. It answers what I see as statistical mechanics' version of the UV catastrophe, though the analogy is probably a poor one.

No worries. Majcek & Meijer Statistical Thermodynamics covers the relationship between state variables and the canonical partition function quite comprehensively. However, I'm sure you're aware that quantification of the latter seems to head off into QFM land even in the simplest cases.

Please try not to be offended by anything I post below. The issues you raise are deep ones and I fully recognise that a spectrum of opinion across intelligent parties is not only to be expected, but is very much a positive thing. If we refuse to look at an object from another viewpoint, our understanding of the object can never improve beyond speculation. We need the other's viewpoint. I'll try to be less robust in my presentation, but please remember that I'm schooled in producing an end product that nails a small target within limited budgets and tight deadlines. Not much opportunity to sit around philosophising. Does the stress here introduce a value judgment of a methodology based in its ontological classification (ie. what name you give to it)? I'll attempt to paraphrase Feynmann "If it disagrees with experiment, it's wrong". Conversely, all methodologies that agree like-for-like with measurements of the real world must be equivalently valid. In my experience ontology carries with it many hidden traps which makes me very wary. It's a branch of metaphysics and maybe best left to the theologians to play with. I'm faced here with trying to guess what real world differences result from such an ontological separation of what I currently view as different aspects of the same thing. To me, the distinction is one between experimental investigation of particle-particle interaction at the microscopic level (eg kinetic theory) compared to the empirically developed relationships between state variables in the bulk. Neither is necessarily complete in either their formulation or interpretation, but they are measurements taken on the same 'thing', only at different scales. They should therefore reduce to expressions of the same. If some particular interpretation suggests differences, then we must fall back on the weight of experimental evidence. Can this count as the additional material you refer to? Consider this: An isolated system comprises a number of contiguous but measurably different regions which when plotted on a thermodynamic diagram yield a scatter plot centred around a central point P. This represents a system that is not (yet) in thermal equilibrium. Each region has a mass m and plots to a local equilibrium centre a distance r from P, and can be viewed as carrying a 'moment of disequilibrium' (my terminology) of magnitude mr(t) since obviously we have interest in the time evolution of the system(s). Clearly, the total moment of disequilibrium about P is dominated by those regions furthest away from P. We'll come back to this. From a kinetic point of view, regions interact by the passage of a particle carrying a particular packet of mass and energy from one region into another resulting in the plotted points for each region moving in equal and opposite steps. I personally picture such an event as equivalent to equal and opposite transfers of Q, W and mass between regions, but that's simply a preferred style - they appear exactly synonymous to me. Now let's focus on those more distant regions. We've not yet established that these regions must converge toward P (that comes later) but let's say that there is some hidden mechanism that preserves r. So now we have each of these regions moving stepwise clockwise or anticlockwise around an annular band centred on P. ie under the influence of a sequential transfer of particles between regions (regions performing work on each other in my language) we have an isolated system traversing a reasonably clearly defined closed thermodynamic cycle, regions at opposing sides of the cycle exchanging Q, W, and m as necessary to maintain the status quo. Now remove the constraint on r, and let each region gradually make stepwise moves statistically approximating to a stroll in the direction of P as I hope we all agree is what it must do. The system as a whole remains just as dynamic as ever, particles are exchanged with the same kind of frequency, the number of regions remains unchanged, each region remains in motion with the same size steps, and the system continues to trace out a closed cycle , only with reduced diameter and reduced overall moment of disequilibrium. Ultimately, all regions meet at P, so by definition, they are all at the same thermodynamic temperature and thermal equilibrium has been established. But there is still a closed cycle whirring away behind that point. There is still a huge number of microstates (Boltzmann's W value) consistent with the energy content defined by point P on the chart. And each region remains in a state of constant fluctuation as they move stepwise through the W possible permutations defined by pointlike, stationary P which they visit in a well- orderly sequence particle exchange by particle exchange, tracing out that closed cycle. Are we okay so far? Many thanks for that. I think my Smith and van Ness ended up in the library of one of my trainee engineers.

24 hours has passed which I guess is enough. For what it's worth, T2 = T1 + W/2nCv which defines the maximum value of Q2*(T2-T1) for the system, but that's now by the by. Certainly there was no ad hominem intended (and many sincere apologies if it appeared otherwise). I was simply the stating that the heat pump example you asked me to consider seemed to shed no light at all on the OP paradox. I present three main grounds in support of this assertion. Firstly, in the OP case, the Youtube presentations claim that the 2nd Law has been broken by 'statistics'. ie that some quantity of system entropy has somehow vanished. I think I've demonstrated clearly enough that the heat pump at least preserves initial total system entropy. So here, it seems uninformative. Secondly, the presentations claim that the gas has somehow contracted from some initial equilibrium state to occupy half of its original volume purely through its own internal mechanics. ie that the contraction happens without any external nett energy exchange with the environment. Again, for the heat pump case I've demonstated that all volumetric changes have exactly matching Q and W terms, so again, it seems uninformative. Lastly, as stated earlier, it dawned on me on Saturday that there seemed to be a strong conservation of occupied volume arising from the conservation of angular momentum. I don't see how the heat pump example was leading us toward such a conclusion. In hindsight, perhaps a centrifuge would have been a more effective guiding light - the volume restoring forces here are quite explicit and macroscopically large. In conclusion, it's become clear that the Youtube proposal of a broken 2nd Law is nothing but a red herring. Their real stumbling block arises from overlooking fundamental 1st Law constraints. Their loss of control of the 1st Law simply results in entropy being undefined, even in qualitative terms. Lies, damned lies and statistics again. Anyway many thanks to you all for your assistance in clarifying and solving the OP paradox so completely. I'm most grateful for your time and patience.

You're joking, surely! The nearly-fully conical brass instruments: flugle horn, euphonium, and tuba are hardly rare, and they present a by far greater challenge to in-tune harmonics compared to those instruments with a conventional Webster horn profile. Hence the prevalence of more-than-three valves on these instruments with additional compensating loops, especially on the tubas. Really beautiful sounds in the hands of professional performers, but in the hands of amateurs? Sheesh! Worse that trying to keep in tune with herd of great Highland bagpipes! A friend of mine, Rowuk on Trumpetboards.com is also a professional zink player when he's not tootling his baroque trumpets. I think he'd have a word or two to say about your assertion that 'the playing technique is lost'. Sat through a concert by a zink trio myself in York minster not so many years ago. They seemed to know which end to blow down.

I've provided an analysis of your heat pump system and have thus answered your question. However, I need you to explain to me why this provides a solution to the OP. You could perhaps help by providing me with a simple expression for T2 in my worked example. After all, you do understand your pet a lot better than I do (plus you have the textbook it came from).

The congratulations really aren't necessary, sincere or otherwise. I provided you with a standard 1st year BSc thermodynamic calculation and in return you provide some ontological classification of my methodology? What's the point? This is a non sequitur. How many versions of the truth can there be? Neither of you need any help from me in that. IMHO you both ensnare yourselves in petty formalistic rituals that simply obscure. You're no nearer to answering the OP paradox now than you were when you first posted at 9.51 pm on Friday. If you don't know then just say so. Frankly, with so many of you stumbling over the 1st Law constraints, there seems little point in discussing the 2nd.

The gas has increased in entropy by W(T2-T1/2)/T1T2 (= W/T1 - W/2T2) Reservoir 1 has decreased in entropy by W/T1 Reservoir 2 has increased in entropy by W/2T2 W/T1 - W/2T2 - W/T1 + W/2T2 = 0 Hence no nett change in entropy. Your examples don't come into it. Your assertion (paraphrasing) "it is reversible therefore it is isentropic" is a clearly flawed assumption. My version of the 2nd Law is that there are no versions. On more than one occasion I've answered your question quite fully with the statement that "The 2nd Law is the 2nd Law". If you're unhappy with that answer then I'm afraid that's your problem. If there is an alternative version to consider, then either it yields precisely the same results (and is therefore identical in all but name and is therefore superfluous), or it is wrong. It's not rude to evade semantic entrapment. Simply prudent.

Both in the text and on the associated sketch you refer to Stage 2-3 as 'adiabatic compression'. Adiabatic processes may be isentropic but not necessarily so. Because that's how you defined those two isothermal stages. So you do acknowledge that the entropy changes are precisely as I defined them in my last post. You had not mentioned them previously. This appears to be in direct conflict with your earlier statement: ... which I think you need to gracefully withdraw. No shame. Just own the error. Not in dispute.

Some unnecessary complications in here aren't there? The 2nd Law is the 2nd Law. Your process extracts a certain amount of work from a pressurised gas in a W=Q process so W = Q1 and dS1 = W/T1 ( - W/T1 to reservoir 1) Let us say half this shaftwork is used to isentropically compress the gas in stage 2 where dS = 0 by definition The remaining W/2 is spent in a further W=Q process at T2 so Q2 = W/2 and dS2 = - W/2T2 ( +W/2T2 to reservoir 2) So what's happened to the gas overall? We've heated it increasing it's internal energy by W/2 and increasing its entropy by W(T2-T1/2)/T1T2 So okay, you've moved W/2 worth of heat from T1 to T2 but where is the nett change in entropy? There isn't one. 2nd Law is good. dS = dQrev/T by definition. Your process may be ideally reversible, but it absolutely is not isentropic other than the stage you call 'adiabatic compression'. If you also called this stage 'isentropic compression' it may help alert you to the fact that isothermal compression processes are very far from isentropic. So if you've drawn inferences from this line of thinking that you believe will help me with my box problem (I no longer have one), I fear that you have managed to confuse yourself. In passing, I would strongly recommend sketching out thermodynamic processes on HS diagrams rather than PV. On an HS diagram ideal isothermal processes are horizontal lines, ideal isentropic processes are vertical. With practice, you can see at a glance whether there's a 2nd Law infringement. Invariably, such infringements arise from human error.

My understanding is that the difficulties of applying the microcanonical ensemble to the real world are pretty well documented. Following this thread has highlighted to me some of the more blatant pitfalls that people can fall into. I didn't name and shame the Youtube channels concerned in the OP since actually, some of them are pretty well-informed - their message doesn't crumble under research, but in this instance ... I'm 61. I'm not blind to my own limitations. Those I forget, my better half reminds me of quickly enough.

It's late and a combination of your obscurantism and swansont's bloody-minded negativity has exhausted my patience (which to be frank, I've never had in great excess). It must be obvious to you by now that I've been clear in my own mind since way back on page 1 of this thread where the Youtube presentations break down. The key lies in conservation of angular momentum which is a topic I usually shy away from. Turns out, it can be quite useful on occasion. Do we have anything more to discuss? If your earlier posting was a joke, then I'm sorry I didn't get it and took its meaning at face value.

I'm sorry that the standard engineering techniques for analysing thermodynamic systems cause you such consternation. Better engineers than me have approved my approach and given me repeat contracts to lead the detailing out their power design projects since I turned thirty, and that was a very long time ago. Your pride blinds you. I'm out of here. Just stick a heat engine between T2 and T1 heat reservoirs and you've got free energy. Well done! You've solved the worlds energy problems! Have a ball!

Do you think this is the first time I've ever seen a sketch for a heat pump that claims to break the 2nd Law? Really? I've just worked through the expression for the shaft work of your blasted adiabatic compression stage wondering why the hell am I spending my time on this hair-brained stuff when I'm not being paid for it, and you have the gall to say I'm messing around wasting everyone's time. HOW DARE YOU! Let's be clear on this - you are the one who has come up with an invention that any patent office would chuck in the waste-paper basket unread! I had about 12 hours work left to properly research and analyse your scheme so that I could help you through your misunderstanding. I've been helping my junior engineers overcome such hurdles for many years. But now shall I bother? What the hell do you think. Grow up!

I've not changed the conditions one iota. Even ideal systems have to observe basic symmetries. Better. Now sum all those microstates where the container remains unchanged from its original state, and we have a workable thermodynamic ensemble. The mean free path in air is what? 100 nm? There may be a lag between arrival and collision, but it's trivial in a macroscopic system. We don't need a 1-1 match up. The entire reaction vector to the momentum of a single particle can be split anyway you like between the remaining N-1 particles in the system. In a sense it is. So long as motion of one particle is exactly matched by some ensemble in contrary motion to fill the void, then sanity is preserved and you have the restoring force you've been asking for. Not when you've transferred nett momentum into your box. But now you have a non-canonical microstate. Do you really think you can both have your cake and eat it? They all require work in a sense. However small fluctuations are reversed almost immediately by the inertial reaction mechanisms discussed above, and are inherently reversible in nature. Larger fluctuations imply a long term resistance to those reaction forces that increase in direct proportion to the scale of the disturbance and I see no internal mechanism that could account for such a resistance.

Looks to me like you've just heated up some high pressure gas from T1 to T2 at roughly constant pressure. Close the cycle, put some preliminary numbers to it and we can discuss.

However all the particles are in one side of the box. So lets drop in a partition isolating a system of N particles on one side from an absolute vacuum on the other. Please don't reply 'there is no partition'. Yes it is. Ball and box is a two-body system with regular interchange of (at the very least) momentum between them. If you think that this is awkward, just wait until we start considering the regular interchange of torque. Every action has an equal and opposite reaction. Every particle arrival or departure within a space brings with it changes in mass, linear momentum, orbital angular momentum, axial spin etc and each must be balanced precisely by an equal and opposite external action. Sit it stationary at the CoM and forget about it. It isn't a problem until something bumps into it. I'll ask the mice. The box is a mathematical artefact introduced solely to define the system geometrical constraints. A condition of it being considered non-interacting is that we must be able to delete it from the snapshot and see no nett change to the system. Therefore the only states it is admissible to consider are those where every single particle linear momentum vector is exactly balanced by an equal and opposite vector. Just as the box could transfer momentum to and from the system, so can the octants transfer momentum between each other. While it must be possible for one octant to momentarily grab an extra particle or two (I'll agree with you that far), there are one or two particles on the imminent verge of leaving it probability one ensuring that the system never departs by a measurable amount from equilibrium. Serious departures demand serious external work.

Every fibre of my being feels this principle to be true. Though the name is unfamiliar to me. Thank you. I'll try to remember the spelling! Fire away, I'm all ears.

The 2nd Law is the 2nd Law. Let us assume dS/dt >= 0 for an isolated system. You're in the same trap as swansont. At each bounce, there is an interchange of momentum with the box. The box is actively participating in the thermodynamic process, so you now have to consider it's inertia, thermal capacity, temperature and entropy. Your system is an almost perfect vacuum. You want to discuss vacuums? They're pretty well defined, I think. It's rapidly becoming irrelevant in your idealised case. I'm more interested at this point in what the temperature of your box is, since this is now setting the conditions necessary for thermal equilibrium. Is it? Never heard of it. Your box is an active part of the thermodynamic system, which is why I ask what its temperature is, because that will define the equilibrium state. I suspect with only one single particle the equilibrium temperature is to all intents and purposes absolute zero, so your particle will shed its last little shred of momentum to the box and become stationary wrt the system CoM. Equilibrium has been reached. Real world processes tend to follow non-analytic PVT paths, which is why we characterise them by a sequence of analytic ideal steps, often an infinite number. There may well be no physical piston or heat exchanger involved but is entirely appropriate to fabricate a few for calculation purposes. See https://en.wikipedia.org/wiki/Thermodynamic_process_path#:~:text=A thermodynamic process path is,-entropy (T-s) diagrams. "A thermodynamic process path is the path or series of states through which a system passes from an initial equilibrium state to a final equilibrium state and can be viewed graphically on a pressure-volume (P-V), pressure-temperature (P-T), and temperature-entropy (T-s) diagrams. There are an infinite number of possible paths from an initial point to an end point in a process. In many cases the path matters, however, changes in the thermodynamic properties depend only on the initial and final states and not upon the path." At last! Conservation of linear momentum. It's not permissible in this simple NVE case to conveniently transfer some to the box because that is the equivalent of introducing external Q and W terms. Precisely! A low one. You objected to my earlier suggestion of 42, but it wasn't entirely flippant. Enough to constitute a system that isn't too swamped by quantum effects. Having said that, if you take the st. dev. of momentum from the Maxwell-Boltzmann distribution and plug it into the Heisenberg Inequality, some very interesting expressions arise that look very 2nd Law-ish to me. I suspect a very deep link in there somewhere, which actually strongly enforces my trust in the 2nd Law.

No, the CoM of the ball (wrt the system CoM) is not fixed. It was you who introduced the single ball case Needlessly offensive phrasing. My 'agenda' is simply to respond to points you raise in support of arbitrary random changes of state. There's a principle we use in Chem Eng that the transition between any two thermodynamic states is path independent i.e. when a body of gas transforms from one PVT state to a new PVT state, it is perfectly admissible to break it down into a sequence of smaller idealised steps, knowing that the overall changes remain fixed for purposes of calculating the enthalpy changes for instance. Hence, in the case in question, I analyse the change as an ideal isentropic compression (external work performed with no change in entropy) followed by a possible reversible cooling stage (shedding heat and entropy into the environment). The Youtube presenters haven't specified the final temperature of the system, but they have stated that the entropy has fallen, therefore the cooling stage is a valid means of estimating the minimum transfer of entropy. My 40 years work experience screams out 'THE ENTROPY OF THE ENVIRONMENT HAS RISEN BY AT LEAST AN EQUAL AMOUNT'. Must have. Otherwise all my design calculations over the last 40 years have been wrong and numerous production facilities around the world are fundamentally unsafe. A little over-dramatic maybe, but my hackles have been raised.

I do not believe it is possible for all particles to be on one side of the box in the absence of shaftwork or equivalent energy input. We're not able to compute system evolution collision by collision, but what we do know for sure is that the outcomes of each collision are very far from random - the entire discipline of physics is based on the symmetries of conserved quantities through such events. That deterministic principle does not suddenly vanish simply because we have insufficient computing power to handle the long term dynamics of macroscopic systems: those single event symmetries remain intact, instant by instant, eon by eon. In my experience, truly random behaviour is rarely observed and then only in a very few, very special scenarios. The rest is merely 'complicated'.