Suppose we have a steady state universe with a gas chamber resembling that of Maxwell's demon that is used to power this hypothetical heat engine as molecules transfer to their respectable sides based on temperature. Now, suppose we ran this machine for infinity where eventually it reaches a thermodynamic equilibrium. However, every so often, Schrodinger's wave equation could allow one of these molecules to switch sides, hence powering the engine as it again transfers to its respectable side. What flaw in this experiment prevents this machine from becoming a perpetual motion machine and breaking the 2nd law of thermodynamics as the molecules transfer back and forth every so often for infinity?
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4The tunneling of the molecules would not select for faster-than-average and slower-than-average molecules, and hence not be a variant of Maxwell's demon. Additionally, you should not mix classical thermodynamics and quantum mechanics in this way - either stay classical, or do full quantum statistical mechanics. – ACuriousMind Mar 30 '15 at 21:09
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The tunneling of the molecules would not necessarily choose sides, but their still remains a probability that they would tunnel to the other side. – user1939991 Mar 30 '15 at 21:35
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1@user1939991 The Daemon would then need to measure which side the hotter molecule is on, or wait until it knows the hotter molecule is on a given side. Either way, it must make a measurement. This is not what in principle costs work, though. Are you familiar with the "Cost of forgetting" arguments that shows that the Daemon must do work to erase its memory of the foregoing state measurements, and that this is why the system ultimately can't violate the second law? We've actually built and tested a working MD in the laboratory- the scheme itself is not a problem till it needs to erase memory. – Selene Routley Mar 30 '15 at 22:15
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2nd law is a law of macroscopic thermodynamics, inferred from experience with macroscopic systems with limited set of macroscopic variables and understood to be valid for such systems. 2nd law says nothing about special mechanical systems, so their behaviour cannot contradict this law. The systems involved in the Maxwell demon setups, as opposed to thermodynamic systems, are mechanical systems that are fully described in terms of mechanics, which is not restricted by laws of thermodynamics. – Ján Lalinský Mar 30 '15 at 23:23
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Mechanical systems can be described less accurately with thermodynamic variables and with some additional assumptions, it can be shown that these descriptions obey 2nd law of thermodynamics (in a probabilistic sense). – Ján Lalinský Mar 30 '15 at 23:25
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@JánLalinský "2nd law says nothing about ..." is wrong. Every system has entropy; 2nd law says the entropy won't decrease in isolated system. If system is finite the entropy may be hard to fix precisely, that's all. – Andrew Steane Jun 22 '19 at 08:23
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@AndrewSteane I suppose you can find $a$ definition of such non-decreasing quantity in terms of coordinates and momenta . However, making sure it has other required properties of thermodynamic entropy (which 2nd law says something about) is not trivial. What definition of entropy would you use for a two-body system with gravitational interaction? – Ján Lalinský Jun 22 '19 at 12:23
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@JánLalinský If the temperature of the degrees of freedom you have in mind (position, momentum) were zero then the entropy would be zero. In practice neither will be exactly zero. – Andrew Steane Jun 22 '19 at 15:07
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@AndrewSteane I don't see how that answers my question. What definition of entropy are you talking about? – Ján Lalinský Jun 22 '19 at 15:21
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@JánLalinský It can be defined either the thermodynamic way, or the statistical way. Thermodynamic way: $dS = dQ_{\rm rev}/T$ which can be integrated. Statistical way: $S = -k_B \sum p \ln p$. – Andrew Steane Jun 22 '19 at 15:39
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@AndrewSteane those are the usual definitions that I already know and both have limited applicability - the first to macroscopic thermodynamic systems, the second to probabilistic models. What I was asking is what is your definition of entropy of a simple mechanical system, such as two body system with gravitational interaction. In other words, how do you define entropy in mechanics, where there is no heat and no probabilities, just coordinates and momenta. – Ján Lalinský Jun 22 '19 at 23:00
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@JánLalinský The way we usually do mechanics is to imagine the state has been observed sufficiently to reduce the probability distribution to one outcome so $S=0$. My point was that as soon as the state is less well specified then entropy is non-zero. The relation involving $dQ_{rev}$ only works in thermal equilibrium but even simple mechanical degrees of freedom can in principle be brought to dynamic equilibrium with a reservoir. They then fluctuate over time. Obviously that is not normally the situation in mechanics, but this discussion was about basic concepts and their applicability. – Andrew Steane Jun 23 '19 at 09:20
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@AndrewSteane Maxwell's daemon system is an externally controlled parameter mechanical system. One can assign information entropy to this system, but that is not going to change its behaviour. Probabilistic description often works, but for this special case we have more detailed description which says thermodynamic entropy will decrease without loss of heat. This system does not obey 2nd law of thermodynamics. It realizes the exception that statistical physics discovered. It does not matter that information entropy cannot decrease. – Ján Lalinský Jun 23 '19 at 10:13
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@JánLalinský There is no exception to 2nd law (if there were then it would be exploitable for conversion of heat to work). Maxwell daemon has to clear its memory; that's the step that requires heat output. Overall the thermal efficiency is same as for other reversible heat engines. The best ref I know for this is my own thermodynamics text book, but ultimately I learned it from work of Landauer, Bennett, Szilard and others. – Andrew Steane Jun 23 '19 at 11:46
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There is no exception in practical sense, because it is not practically possible to measure state of such vast number of molecules. But in mechanics motions temporarily violating 2nd law are possible, due to reversibility of EOM. Re your argument on memory, even if Maxwell's daemon had to exchange heat or work with some other body, it won't need to do it with the purely mechanical system it is controlling. Proper control of the door is all that is needed to separate the molecules and break 2nd law for such system. What other processes need to take place in the controller is immaterial. – Ján Lalinský Jun 23 '19 at 12:27
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Let us continue this discussion in chat. – Ján Lalinský Jun 23 '19 at 12:28
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please note that tunnelling does not change energy levels http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html – anna v Jun 23 '19 at 13:30
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Firstly, the assumption that you could build a working Maxwell Daemon (a three state machine with Szilard-engine actuation hardware) to extract work from the system already gainsays the assumption of a steady state universe.
The problems of mixing classical and quantum statistical mechanics aside, the tunnelling here is in principle no different from classical translational motion through the door operated by the classical Maxwell Daemon. Here, the Daemon would work by waiting until it knew there was a random fluctuation making one side of the wall hotter. Then it could do its stuff - through a Szilard engine.
But then the standard objections to the Maxwell Daemon would apply. See my answer here for more details. The Daemon itself can and does work - we've actually built one in the laboratory - the problem is that it must do measurements to work and if it is a microscopically reversible system (look up Loschmidt's paradox), it thus transfers the entropy of the gas states it is watching to its computer memory. When this memory fills up, it must erase it, a process which is also microscopically reversible, then this entropy must be transferred to the states of the external system - now we're back to square one.
The entropy of all theromodynamic systems fluctuates up and down, with the size of the fluctuations inversely proportional to the size of the number of particles so you would indeed see local, short lived fluctuations. Indeed, over longer and longer times, you would eventually see bigger and bigger imbalances purely by chance. This line of reasoning leads to the Boltzmann Brain problem - look this up.
Selene Routley
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tunneling does not change energy levels http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html – anna v Jun 23 '19 at 13:34
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To answer this I need first to clarify what the scenario is. I think it is the following. A gas is in two parts of a vessel, and the two parts have reached thermal equilibrium with one another so no heat engine can operate between them. However, every now and then a molecule will change sides, whether by tunneling or another process. The two sides are then slightly out of equilibrium (says the argument) so now a heat engine can run and extract a little energy, returning them to equilibrium, and now the whole process can repeat ad infinitum. The gas slowly cools overall, and useful work is extracted. This would break the 2nd law if it happened.
The flaw in the reasoning here is not essentially different from the flaw in applying this same reasoning to two parts of an ordinary gas in a single chamber. The transfer of molecules between sides is happening all the time by thermal fluctuations, so people have suggested that one could simply wait for a suitable thermal fluctuation, and then run an engine to exploit it. There's a nice analysis of a ratchet working on this principle in the Feynman lectures on physics. One finds that in fact it acts just like any other heat engine. In order to exploit a thermal fluctuation it needs to have access to another region at a lower temperature, and it exports heat to that other region in just such a way that entropy overall is conserved (if we assume the whole apparatus is reversible; if it is not then the efficiency is worse just as Carnot's theorem says).
The case of a Maxwell daemon is one where the heat engine works through an intermediate stage in which information is stored internally. However, eventually the internal memory has to be cleared and this is the step which unavoidably involves in increase of entropy in the environment (as a previous answer correctly asserted). As I understand it, Landauer proposed this resolution and Bennett worked it out more thoroughly.
To conclude, then, you can indeed extract mechanical work by exploiting a thermal (or other type of) fluctuation in some system A, where A is in internal equilibrium, but only by having access to system B whose temperature is lower than that of A. If the process acts reversibly then one finds that the heats exchanged at the two places A and B are in proportion to $T_A/T_B$ so it achieves the same efficiency that the Carnot cycle achieves.
Andrew Steane
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"whether by tunneling or another process." please note that tunnelling does not change energy levels http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html . what oter process you envisage? – anna v Jun 23 '19 at 13:32
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@annav "other process" could include percolation and things like that; or exploiting a random opening of the trap door owing to thermal fluctuation. Tunneling would move a density fluctuation from one place to another. – Andrew Steane Jun 23 '19 at 13:55