sethoflagos

I meant the necessity of the early universe itself being spatially infinite in this case as opposed to a (near) pointlike spatial singularity.

My immediate thought was that cooking may convert some indigestible starch to a more digestible form. But then I started wondering how many calories you would expend in thawing out frozen chips in your stomach.

Yusuf Automotive LPG is an undeveloped market in the US. In Europe it is developed a little more but nothing like as much as in the Middle East, Asia and Australia. Therefore there is little experience on this site regarding your question. Also, the proportion of propane to butane specified for each country varies with climate. In Britain and Ireland, automotive LPG is almost pure propane while some Mediterranean countries specify up to 80% butane. I don't know the standard for Iran (I had to ring a friend to check you had moved onto unleaded gasoline!) but I would expect your LPG to be butane rich and therefore a little more 'gasoline-like' in its combustion characteristics than that used in northern Europe. LPG octane numbers are on the low side (~90) and therefore I personally would be very reluctant to conduct this experiment particularly on an unmodified high compression engine. (Unless I wanted it to be converted to a low compression engine!). I'm sure you must know a good local car mechanic. He is likely to have a much better idea how to proceed given your particular vehicle and the locally available fuels.

Engineering Databook SI Version (aka GPSA Databook), Gas Processors Suppliers Association, 11th Edition 1998 has probably helped resolve more day-to-day issues for me over recent years than any other. But most enjoyable .... probably Pump Handbook, Kurassik & al, 3rd Edition 2001. Comprehensive, very readable, more Aha! moments per chapter than you can shake a stick at. Honourable mention to 'Understanding Atmospheric Dispersion of Accidental Releases', Devaull et al.,AIChE 1995 just for the unfortunate title really.

I've seen a number of references to space flowing across a black hole event horizon at the speed of light. If space can be lost in this way, what happens to the vacuum energy associated with that space? i.e. Does it increase the mass of the black hole?

Many apologies for this OP as I'm sure it has been raised many times before. IFF the universe were truly infinite in spatial extent (and I appreciate the 'unknowability' aspect of this), would this imply perforce that whatever the energy density of the earliest moments of the universe, it too would have been spatially infinite? In such circumstances, would GR predict gravitational effects to be compressive, tensile or ... just undefined.?

By applying an external magnetic field to the system, haven't you introduced new potential energy terms boosting the system total energy? As the magnetic particles descend this new energy gradient to a new equilibrium position how is this energy going to be dissipated other than in the form of heat? As a practical example, you might compare and contrast the removal of particulates from eg coal-fired power station flue gas by electrostatic precipitators. The pretty efficient separation of gas and solids suggests an entropy decrease, but on the other hand, electricity is consumed, and that's going to end up releasing heat somewhere or other.

Wow! A third of a million views! Best come clean since we're under such scrutiny: After much deliberation, I've come to the conclusion that at least one of these statements is correct. A little more thought on my part at the time, and I should have realised this for which I apologise. If the premise of the OP were correct, the kinetic energy necessary for convective flow would be sourced by a reduction in internal energy of the flowing fluid (as is normal in fluid processes). No external shaft work necessary. From this it clearly follows that the OP case is a far-from-equilibrium peturbation for which, I understand, entropy is usually regarded as undefined until equilibrium is reestablished. So even if there was some mechanism by which this process could occur, the 2nd Law would, contrary to the OP, remain intact. Happy new year!

Could one choose to define 'pressure exchange' as an 'exchange of gravitational potential energy'? Not clear on whether pressure is something that can be 'exchanged'. 'Propagated into another medium' for sure, but 'exchanged' usually tends to imply a conserved quantity, doesn't it?

It's effectively a time reversed Coe and Clevenger test - the progressive fluidisation of a packed bed of buoyant particles rather than the more industrially significant progressive compaction of denser particulates from a suspension. I agree with joigus that it's not saying anything significant about pressure here. No measurements seem to be taken so I question whether it's actually saying anything about anything.

Internal energy is primarily associated with fixed volume, closed processes where dU = CvdT holds. Enthalpy is associated with open, fixed pressure processes where dH = CpdT holds. In open processes the product stream is running into space already occupied by something else and that something else needs to be moved somewhere. The additional work required is contained in the PV term in your H = U + PV formula. Hence either in chemistry lab experiments or industrial processes, we generally work with enthalpy changes rather than changes in internal energy. We could use U throughout, but then we'd have to keep track of all the extraneous PV terms and it's easy to miss the odd one or two. There are many 'wrinkles' to the above but I believe you're more interested in the broad overview, and at that level the above works fine for me. Further elaboration can be found in the appropriate textbooks.

Don't worry. I found the answer I was looking for at http://www.astronomy.ohio-state.edu/~dhw/A825/notes2.pdf and http://library.oapen.org/download/?type=document&docid=459733 Le Chatelier's Principle is embedded in Navier-Stokes.

How is Le Chatelier's Principle Accommodated in Statistical Thermodynamics?

It became quite clear after the first dozen or so posts to this thread that the insight I was looking for to answer the OP was not going to emerge. However I am actually quite grateful for this relentless exercise in mudslinging at my methodologies that you and your partner in crime have been engaged in over the last few days. They're not often questioned when employed in my professional activities, but there's always the chance that they might be and it's as well to be forewarned what arguments might be presented, whether these arguments have any significant validity, and what the most effective counter-arguments are. So thanks for that. As things stand, I'm learning more from those who are not posting and could, than from those who are, so as I've made all the points I felt like making and you're just arguing for the sake of it, perhaps it's time to draw this process to a close? Rhetorical question. You're not obliged to respond.

"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." - Philip E. Bloomfield, William A. Steele, "Thermodynamic processes," in AccessScience, ©McGraw-Hill Companies, 2008 Do you understand and accept this principle? If it were so, you would see that the path I choose to take from state A to state B is irrelevant. Obviously, I pick a sequence of paths that are analytic. "Of course, one can especially prepare the system to be in this state; for example compressing the gas into half the space and inserting a partition" - Franz Mandl, "Statistical Physics, 2nd Edition", ISBN: 978-1-118-72343-2, June 2013 What compressed the gas? The partition? You are prevaricating with ad hominem trivia. Seems to be an established habit on this site among some.

On the contrary, it is you who is attempting to extend some unspecified microscopic action to an unspecified macroscopic effect. I know that my formulae work for macroscopic systems. You are the one making the extraordinary claim that they are somehow inappropriate so please demonstrate (with numbers!) why my formulae don't work or withdraw your objections to them. THESIS: ANTITHESIS: SYNTHESIS: Replace the box with a sphere of the same volume, and have the particles converge radially inward to occupy a concentric sphere of half the volume. The principle of the OP is perfectly preserved and the CoM issue goes away. The compression issue remains. Even Mandl invoked a piston to prepare Fig 2.1(b)

So that long walk off a short pier shouldn't exhaust you There is a difference between 'arbitrarily small' and 'zero'. But compression is by far the larger work issue, and this most definitely is NOT arbitrarily small. Even isobaric compression requires 0.5 PV Joules of external work. That's oto 20% of its initial internal energy for heaven's sake. This rises to W = ln(2) PV (=0.7 PV) for isothermal compression; and a minimum of W = ((2^0.4 -1)/ 0.4) PV (= 0.8 PV) for adiabatic compression of a diatomic gas; W = ((2^0.667 -1)/ 0.667) PV (= 0.88 PV) for the monatomic case. This is the source for the 'explosive' energy release when you relax your constraint and allow the gas to undergo free expansion back to its original state. Only it can never find its way back to its original state because it has no mechanism to get rid of its bulk kinetic energy other than by recompression. This puts the lie to Mandl's case that "it would only last a very small fraction of a second". No, we are now engaged in fast, real-time energetic dynamics that are well-understood. Your steady accumulation of 'zero work' over countless eons has led to the paradoxical situation that you cannot now return to the original equilibrium state other than by precise reversal of the process that got you there. A process that you have already stated takes far longer than the age of the universe. Mandl's Fig. 2.1 (a) is a really good starting point for testing wacky ideas, because it is by definition a perpetual motion machine of the third kind. The slightest (and here I do mean arbitrarily small) nudge in the wrong direction turns it into either a perpetual motion machine of the second kind breaking the 2nd Law, or a perpetual motion machine of the first kind breaking the 1st Law. Which kind of wacky are you wishing to defend?

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.