First of all sorry for my English - it is not my native language.

During my engineering studies at the university the thermodynamics professor told us that the "second law of thermodynamics is not true for a low number of molecules".

At that time scientists were already talking about micro technologies where single molecules do some work. (Today such technologies already exist).

Now I'm wondering if a perpetual motion machine of the second kind (a machine gaining energy by cooling down the environment) would be possible in nanotechnology. As a result such microchips would lower the entropy of the universe.

I was thinking about the following theoretical experiment:

Brown's molecular movement was discovered by observing objects in fluids in amber stones. Due to Brown's molecular movement these objects are moving. The energy comes from the environmental heat. If the object was a small, very strong magnet and metallic parts are placed near the amber then the moving magnet should induce eddy currents in the metallic parts. This would mean that there is a heat energy flow from the amber to the metallic parts even if the metallic parts are warmer then the amber.

Now the question: Would such a device be possible in nanosystem technology or even microsystem technology?

(I already asked a physics professor who only said: "maybe".)

  • $\begingroup$ Dear MArtin I thought of your question today when this one was asked today it might be worth following - maybe nanotechnology thermodynamics is on the ascendant $\endgroup$ Sep 4 '13 at 9:12
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    $\begingroup$ Related, with respect to Maxwell's demon: Exorcism of Maxwell's Demon. Also of interest are Brownian ratchets, particularly Feynman's analysis in his Messenger lecture The distinction between past and future (link), 22:05 onwards. $\endgroup$ Sep 8 '15 at 11:00

The second law holds on average for systems of any size, large or small. If you have an isolated contraption containing just a few atoms, and you run it through some procedure (maybe as simple as waiting 5 seconds, or maybe more complicated), there is some probability that the atoms will wind up in a lower-entropy configuration at the end of the procedure than the start. That's what your professor was referring to.

However, the probability of random entropy reduction is not high, and certainly not 100%. The key point is that the average change in entropy, upon many repetitions of the procedure, cannot be negative for any procedure.

When people hear this, they get an idea: "I'll run the procedure 100 times, and check each time whether or not the entropy got randomly lowered, and somehow I'll only use that 1 successful run to power the perpetual motion machine while throwing away the 99 unsuccessful runs." Unfortunately, it doesn't work that way! One would refer to this whole process (involving 100 runs of the original procedure) as being just one run of a different (more complicated) procedure, which also now involves extra effort / energy to check whether or not the entropy was lowered each time. (Otherwise you wouldn't know which of the 100 runs to use). That checking process creates enough entropy to undo the benefit of repeated runs.

This kind of stuff is commonly discussed under the heading of Maxwell's Demon.

If you have a small magnet diffusing around by brownian motion, it will indeed transfer energy to a stationary metal nearby via eddy currents. Unfortunately, the reverse process happens at exactly the same rate: The electrons in that metal randomly jiggle, creating currents that create magnetic fields that push on the diffusing magnet, thus transferring energy from the metal to the magnet. The total energy transfer rate is equal in both directions, or more specifically, as long as both parts start at the same temperature, they stay at the same temperature.


A better way to define entropy is to say "We don't know exactly what the microstate (microscopic configuration) of a system is, instead our best information is that there a probability distribution of possible microstates. Then the entropy is $S = k_B \sum_n p_n \log p_n$ where $p_n$ is the probability of microstate $n$.

Note that entropy is observer-dependent, in the sense that one observer may have more information about the probability distribution than another. In more concrete terms, a system might be disordered to one observer, but a different observer knows a "magic recipe" for undoing that disorder. For example, I could create a seemingly-unpolarized beam of light by switching the polarization of a laser every nanosecond according to a pseudo-random sequence. I know the sequence, and therefore I can use a waveplate to get back a perfectly polarized beam with no intensity losses. But for somebody who doesn't know my pseudo-random sequence, the beam really needs to be treated as unpolarized, and they cannot polarize it without losses, according to the 2nd law.

A configuration of a small number of molecules might randomly become "more ordered" in some sense, but that doesn't mean it has a lower entropy. Until you measure it, you don't know that it became more ordered, and therefore you cannot make use of that order. All you have is a probability distribution for what the microstate is. As time passes the probability distribution changes and the entropy that you calculate from it either stays the same or goes up. Well, in a certain sense, it does not go up by Liouville's theorem in classical mechanics or unitarity in quantum mechanics ... but it often turns out that you wind up with useless information about the microstate, i.e. information that cannot be translated into a "magic recipe" for undoing the apparent disorder as in the polarization example above. In that case, you might as well just forget that information and accept a higher entropy. See this question.

When you make a measurement, there is a probability distribution for the possible measurement results and the possible microstates following the measurement. Some of the measurement results may leave you with a low-entropy configuration (you pretty much know what the microstate is from the measurement). But if you appropriately average over all possible measurement results, the overall entropy increases on avearge as a result of the full measurement process.


Now I'm wondering if a perpetual motion machine of the second kind (a machine gaining energy by cooling down the environment) would be possible in nanotechnology. As a result such microchips would lower the entropy of the universe.

A thousand times no. If a concept lowers the entropy of the universe in a non-superficial sense it is always false.

As for your "device", it violates the 2nd law. For an alternative device that still obeys thermodynamics, I would present the information-to-energy converter. In short, it uses electronics (which we might as well call micro or nano electronics) to control the state of atoms within an array, and the state of these atoms are affected by thermal fluctuations. By reading their state, storing it to a computer, and then making an intervention that makes them "hold" or not hold their state, the machine can get energy from thermal fluctuations.


In a greater cosmic sense, this machine can do work by making its environment colder. The thermal energy from the environment is what goes to doing the work. The caveat is that it needs a computer to control it.

Pro tip: there will always be a caveat to 2nd law violations.

No problem you say, we'll just make computers nearly perfectly efficient, and then free energy forever, right? That is the deep realization from this - that computers can't be made perfectly efficient. There is a minimum energy to change a bit.

$$ E_{min} = k T \ln{(2)}$$

From all this, the problem with your "device" is glaringly obvious. If your device has an information connection to a computer than uses energy to do computations, then it can work. This seems almost identical to sending electricity to a wire, but it's not. Electricity moves energy from place A to B. Here, I'm talking about a computer server located at A, then communicates with a device in place B, and the temperature of B's environment is converted to work. This work could actually be electricity. Imagine the device in your room making electricity to run a lamp. That energy then becomes thermal in your room makes a circle around your room. This is impossible as a stand-alone system, but the server located at A is host to the non-ideal entropy increase because of the computation it does.

  • $\begingroup$ This doesn't address the OP's question about small particle numbers, since it's true that the second law can be violated with high probability by systems with small numbers of particles. $\endgroup$
    – user4552
    Sep 2 '13 at 15:58
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    $\begingroup$ @BenCrowell My answer says exactly that the 2nd law can be violated locally with that system with a controlled information input to the system. The novelty of this question compared to all the other Maxwell's Demon questions is the focus on nanotechnology. The reference describes something that meets the requirements of the nanomachine in the OP. Unless I'm wrong, of course. $\endgroup$ Sep 2 '13 at 16:12
  • $\begingroup$ You don't have to have any controlled information input. The second law simply isn't valid for small particle numbers. $\endgroup$
    – user4552
    Sep 2 '13 at 16:19
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    $\begingroup$ @BenCrowell I don't see what that has to do with turning heat into work. Perhaps your definition of "work" includes these unusable anomalies? That still wouldn't fit the OP's definition of a machine "of the second kind" unless you thought that an isolated system with nanobots can turn uniform thermal energy into work. In that case you would just be wrong. I see what definitions you use to reply "yes" to the OP, and I just don't think that fits the question's intention. $\endgroup$ Sep 2 '13 at 17:11
  • $\begingroup$ Alan I think Ben is focusing on explaining the observed fluctuations in the OP's ingenious example, which is a necessary concept for the OP to understand. But I'm with you insofar that I believe the OP is asking about whether fluctuations driving nanomachines can beget large scale, continuous (i.e. "perpetual", not being given in a short term fluctuation to be taken back thereafter) conversion of heat to work, to which question I believe experimental evidence must be brought to answer in the negative. Also, neat demonstration that the second law implies Landauer's principle BTW +1. $\endgroup$ Sep 3 '13 at 1:33

Interesting question. I'm not completely convinced by Cristi Stoica's argument that this is equivalent to Maxwell's demon, since the traditional analysis of Maxwell's daemon assumes that the second law applies to the daemon, whereas we know that the second law has a significant probability of being violated for systems with small numbers of particles.

But it's not necessarily true that you can take many small systems, each of which violates the second law with significant probability, and put them together to form a larger system that also violates the second law with significant probability.

For example, say I put three helium atoms in a box, and keep track of how many are in the left half and how many are in the right. The entropy of the state with all three on the left is lower than that of the state in which two are on the left and one is on the right. If we observe the system's entropy over time, it will often go down. But if I try to scale up the system by putting it together with some large number of such systems, i.e., by putting not just 3 but $3n$ molecules in the box, where $n$ is large, then the probability of spontaneous decreases in entropy becomes negligible.

I also don't think this is hypothetical or dependent on future technological advances like the ability to build nanomachines. Nuclei, atoms, and molecules are all $n$-body systems with relatively small $n$, and people who work in these fields deal every day with the fact that the application of thermodynamics to them is only approximate. For example, nuclear physicists refer to the temperature of a nucleus all the time, but they're conscious that they're making an approximation by doing so.

  • $\begingroup$ I think that drawing the OP's attention to essentially the Fluctuation theorem (see the Wiki page and also Sevick, E. M.; Prabhakar, R.; Williams, Stephen R.; Bernhardt, Debra Joy, "Fluctuation Theorems", Annual Rev. of Phys. Chem., 59, pp. 603-633) I think you do help the OP's understanding and in particular explain the short term energy conversion he cites, so +1. But I also think one has to substantiate the claim "I also don't think this is hypothetical or dependent on future technological advances like the ability to build nanomachines", which is where I feel that experimental ... $\endgroup$ Sep 3 '13 at 1:09
  • $\begingroup$ .. evidence is essential to show that such fluctuations don't scale up. I'm not sure whether the works of E. T. Jaynes are wonted to you but these for me really shake the simple statistical-type reasoning that is often applied to fluctuations. In particular the Boltzmann "stosszahlansatz" (assumption of molecular chaos) can often only be applied once, as later changes to the system leave the states of molecules of a gas correlated, thus begetting the difference between the Gibbs (informational) and Boltzmann ("experimental", i.e. defined only when you have big systems) entropies, with ... $\endgroup$ Sep 3 '13 at 1:15
  • $\begingroup$ ..the former unchanged in things like irreversible volume changes, the latter always increasing. See E. T. Jaynes, "Gibbs vs Boltzmann Entropies", Am. J. Phys. 33, number 5, pp 391-398, 1965, also many of his other works at bayes.wustl.edu/etj/articles . I think one has to look to experimental results rather than simply laws of large numbers, in particular, experimentally observed boundary conditions of the universe to explain Loschmidt's paradox. $\endgroup$ Sep 3 '13 at 1:21
  • $\begingroup$ @WetSavannaAnimalakaRodVance: Interesting comments, thanks. What I'm claiming is a commonplace in physics, well tested empirically, and not hypothetical is that small systems can violate the second law. I don't claim the same is true for the inability to scale up such violations. $\endgroup$
    – user4552
    Sep 3 '13 at 1:50
  • $\begingroup$ Ben, totally agreed that not hypothetical for small systems. After reading E. T. Jaynes and also many articles on Stanford Dictionary of Philosophy to do with foundations of probability, my trust in theoretical statistical mechanics is utterly shaken, so experiment is the only thing I can get my well fried brain around these days. $\endgroup$ Sep 3 '13 at 2:02

That's a great idea. Maxwell also had an idea to break the second law of thermodynamics. He imagined a demon who would sort the fast from slow molecules. As you can see on Wikipedia, at Maxwell's demon > Criticism and development, Leó Szilárd and Léon Brillouin, and later others, proved that the demon, or equivalently a nanobot, when doing this sorting, would actually increase the total entropy. So the answer seems to be no. But I don't know how this applies to your interesting experiment.


I agree with your professor that "the second law of thermodynamics is not true for a low number of molecules". But precisely because of this, we cannot use systems composed by small number of particles to systematically break this law. But, all these arguments are in principle, so I think your idea deserves closer analysis, based on its particularities. For instance, the magnet should be small enough, so that Brownian motion would be significant enough to induce current in the metal plates. The currents will generate magnetic fields, which will oppose the movement of the magnet, and will also affect the molecules in the amber stone. These molecules will also be affected by the heating due to the currents in the plates, so this may seem on the one hand to lead to thermal equilibrium and cancel the effect, but on the other hand, to increase the Brownian motion and amplify the effect. In these places I would check the balance to see if it can be positive.


I do think that we are in uncharted territory here and I agree with Cristi Stoica that your argument deserves deeper examination, which I don't have the time nor likely the expertise to look at closely. I have wondered about nanomachines and the second law for some time, but I do believe we can appeal for insight to the null result of the grandest experiment of all time together with the result of a not so grand experiment (one done by us mere humans). First let me quote from the Wikipedia Page for "Entropy in Thermodynamics and Information Theory, where the following experiment is cited:

Shoichi Toyabe; Takahiro Sagawa; Masahito Ueda; Eiro Muneyuki; Masaki Sano (2010-09-29). "Information heat engine: converting information to energy by feedback control". Nature Physics 6 (12): 988–992. arXiv:1009.5287. Bibcode:2011NatPh...6..988T. doi:10.1038/nphys1821. "We demonstrated that free energy is obtained by a feedback control using the information about the system; information is converted to free energy, as the first realization of Szilard-type Maxwell’s demon."

Now, let's think about this in-the-laboratory Szilard engine as a thought experiment and note particularly that it definitely scales the possibility of realising a Szilard engine up to the nano-technology, or more importantly, the molecular biology and microbiology scale.

Now for the grandest experiment of all time: the evolution of life on Earth. After 3 billion (and likely more like 4 billion) years of evolution, wherever biologists look, the basic powerhouses driving life (photosynthesis, mitochondrial ATP production from chemical energy ingested by cells and so forth) are well understood and almost certainly universal. It seems to me, that if there were a way around the second law through biological nanomachines (particularly with a variation on the Szilard engine, since the experiment above shows how it can scale to biological systems), evolution would almost certainly have found it by now and this second-law-violating process would be meeting all the needs of the creatures that evolved it. One could imagine that such creatures would swiftly become utterly dominant in the biosphere, sweeping all else aside. But this hasn't happened yet, although I believe there is an animal who delusionally believes that all its needs can be gotten for free from the environment, and that is just what that animal seems to be doing to the biosphere around it, so we have some anecdotal experiment evidence about how a second law violating animal might behave and how it might affect the biosphere. And yet, after all this time of life on Earth, all the creatures are using energy deriving processes altogether in keeping with the second law. Indeed the weird life (tube worms and the like) discovered in the last few decades around "volcanic smokers" deep in the ocean, almost poetically seem to be paying homage to Carnot's thought experiment, drawing work from heat and chemical potential blasting out of their central, hot life-giving "smoker" and casting the excess entropy out into the icy darkness surrounding their tiny thriving neighbourhood.

There are no biological perpetual motion machines of the second kind, and this is an extremely strong experimental null result. In particular, I believe it likely, given the experimental Szilard engine, that the Earth as an evolutionary computer (in the spirit of Douglas Adams) has likely "thought about" variations on the Szilard engine realised in molecular machines many times, so the null result is particularly pointed as a confirmation of Landauer's principle, a particularly pithy equivalent of the second law. An excellent paper showing exactly how the Szilard engine / Maxwell Daemon must heed the second law is Charles Bennett, "The Thermodynamics of Computing: A Review", Int. J. Theoretical Physics, 21, 12, 1982. Bennett used reversible mechanical gates ("billiard ball computers") to thought-experimentally study the Szilard Engine and to show that Landauer's Limit (the minimum amount of work needed for computation) arises not from the cost of finding out a system's state (as Szilard had assumed) but from the need to continually "forget" former states of the engine by casting the information-theoretic Shannon entropy of the sequence of gas particle states recorded by the Daemon out into the surrounding world encoded as increased complexity there.

Footnote: The following is not part of the answer so please don't read it as such but simply some results that come from the above line of thinking and which I would like to publish some time, so the following has not been peer reviewed (I'm just recording it here to establish priority since it is nowhere else on the internet). One of the ideas about nanotechnology that often bubbles to the surface in my mind is that there seems to be a lack of "big picture" thinking about how we might design nanotechnology. Thinking of the Earth as a computer in the spirit of Douglas Adams quickly shows, even by crude calculation, that evolution is way smarter than any group of designers is likely to be so that sythetic evolution by computer presents itself as a possibility. So what energy resources do we need to "Out-Doug the Earth?" (i.e. design complex nanosystems through synthetic evolution). One can use Landauer's principle to get some lower bounds that will apply even with reversible, e.g. quantum, computing as follows: the initialisation of each bit of storage needed even for reversible algorithms at the beginning calls for the expenditure of energy $k\,T\,\log 2$. Likewise, if a reversible algorithm's storage requirements grow throughout its running, its Landauer principle energy requirements are $k\,T\,\log 2$ for each bit of growth. We could imagine brute force search by some faroff in the future quantum computer if one could abstract the configuration space of all possible evolving biospheres to think of this gathering of possibilities as a colossal database. The memory storage initialisation Landauer limit energy needs of this databaseis likely to be greater than the total output of the Sun for a considerable fraction of if not all its lifetime so brute force would seem to be out. So let's think about a reversible synthetic evolution algorithm (indeed such reversible algorithms seem to exist in the literature) equivalent to the early biosphere. Estimates on the number of prokaryotes in the biosphere are of the order of $10^{31}$ organisms. With reasonable rough estimates of the combinatoricsarising in the asexual horizontal genetic transfer of these organisms and estimates of the frequency of this asexual transfer, we find we would need a "truly random" sequence input to our genetic algorithm to simulate genetic mixing at the rate $R$ of roughly $10^{29}$ to $10^{30}$ bits per second merely to keep abreast of the Earth's evolution. Sexuality in eukaryotes means that the bits per organism are much higher, but there are many fewer of them in the biosphere so that they too would require a random sequence of about the same rate as the set of all prokaryotes on the Earth: the basic sexual and asexual lifeforms are thus searching the configuration space at roughly the same rate, which is reasonable: if two organism classes in the same ecological niche (e.g. producer eukaryotes and prohayotes) searched the configuration space at greatly different rates, one would evolve much faster than the other and out-compete it. Now, if the input random sequence is truly random, a reversible algorithm taking this sequence as an input must grow in storage at at least this rate, so our basic, reversible genetic algorithm Landauer limit is $R\,k\,T\,\log 2$. So suppose we want to simulate the Cambrian explosion (10 million years) in 100 years for the good of our great grandchildren. We get $10^5\,R\,k\,T\,\log 2$ watts as our Landauer limit power requirements, or about $3\times10^{15}$ watts or about 30 grams per second: about a thirtieth of the total solar power lighting the Earth and roughly three orders of magnitude greater than the current human rate of energy consumption. And this is the Landauer limit. So can we use a lower $T$? Not with any benefit on Earth: the second law shows that refigeration will only worsen this result. So the natural place to do this computation is on a dwarf planet sized computer in the Oort cloud and possibly gather and solar energy by a near-Sun collector and beam energy to it. Even so, with reasonable estimates of how big the dissipation surface area would need to be to keep it cool. I calculate a few tens of kelvin as its steady state temperature when in steady state radiant heat exchange with the cosmic background microwave radiation is reached.

  • $\begingroup$ @Nathaniel I believe you should take a good look at this question, given your considerable expertise in this field. $\endgroup$ Sep 2 '13 at 2:57
  • $\begingroup$ This is a good point "evolution would almost certainly have found it by now and this second-law-violating process would be meeting all the needs of the creatures that evolved it". Anyone who has studied photosynthesis or cellular mechanics sufficiently will agree that cells are highly optimized nanotechnology. Their energy conversion processes have a large number of complicated steps. If a molecular gate could have turned into the demon in a conventional sense, one of these parts would have. $\endgroup$ Sep 2 '13 at 14:14
  • $\begingroup$ The idea of arguing from biology is cute, but I don't think it works here. As I argued in more detail in my answer, the second law can be violated at the microscopic scale (small particle numbers) without providing any possible way of scaling up the violation to larger scales. So lack of violation at larger scales does not prove lack of violation at smaller ones. Even a virus is a macroscopic system in terms of particle number. $\endgroup$
    – user4552
    Sep 2 '13 at 15:54
  • $\begingroup$ @BenCrowell I don't think I'm arguing anything - I'm merely bringing an experimental result to the question "Is a perpetual motion machine of the second kind possible using nanotechnology?" which was the OPs question, NOT whether or not there were second law violations at nanoscales. As you point out, indeed there are such violations: even the Maxwell's daemon will win over short time scales until its memory overflows and has to be erased (see the Bennett paper). The second law becomes less and less ,,, $\endgroup$ Sep 2 '13 at 18:01
  • $\begingroup$ ... meaningful as you shrink the scale and the law of large numbers gets weaker (more often violated). The biology result shows that it is unlikely that such small scale violations will realize "perpetual motion of the second kind", a phrase coined by Carnot (IIRC) and clearly implying everyday scale machines. I cited the actual Maxwell Daemon experiment was cited to show that, through Brownian motion, the Daemon idea becomes applicable to "nano" scales and not just molecule scale - therefore it makes it plausible to be a "tool" that biological systems could "use". $\endgroup$ Sep 2 '13 at 18:06

This sounds very similar to the ideas proposed at olsonb.com back in 2012 which are currently being updated in 2015 and discussed more.

Daniel Sheehan has studied Johnson Noise and determined it would not affect his devices significantly, so it is still open to debate as to whether eddy currents would immediately transfer back an equal amount of brownian motion. What if some of the radiation randomized in such a way that the heat never made it back into moving the magnet? How are we 100 percent certain, without actually doing the experiment? we can't just make assumptions, using circular reasoning, that the system will remain in equilibrium because the second law is always true. It could be that eddy currents get randomized, more than the magnet's motion itself, and not all the eddy current johnson noise pushes back on the magnet perfectly (what if the eddies are scattered and some of the magnetic field points elsewhere?)

Likely this idea of extracting energy from brownian motion will be falsified rather than verified, but it's worth experimenting and trying. The idea that a number of microscopic violations cannot become a macroscopic violation is on shaky grounds and is hand waving. How do we know? Where are the empirical studies? In order to remain falsifiable the second law has to be empirically tested. There may be limits to the second law (many have stated the second law is more of a general principle rather than an outright law: olsonb.com has a quote from Maxwell himself, at the top of the page)

See the following interesting proposals:

PDF: http://olsonb.com/articles/olson-brownian-interference-designs3-2015.pdf

Eddy current demon:


For a gravity exploitation (geometry causes gravity to be switched off! virtually):

...see the milk-demon-2015.pdf on the site

But there are many more, that's just a few. This is uncharted territory, that is being charted via the olsonb project (plan b).

There are youtube videos listed on olsonb site which show nano magnets vibrating in distilled water. A strange paradox is that a bunch of nano magnets in high entropy disorder, will tend to clump together over time after random brownian motion - why are nano magnets becoming lower entropy over time, and not higher entropy? Entropy is a confusing subject. It still takes energy to break apart clumps of nano magnets and put them into a high entropy state (usually we consider it taking energy to put something into a low entropy state, but in the case of nano magnets, it takes energy to put them into a high entropy scattered state?)

Entropy and information theory is still not fully understood with many interesting paradoxes.


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