[00:00:05]
>> I have to say I like that title crazy distinguished lectures has a really nice ring to it. It's a real pleasure and an honor to be here and I've been having a really interesting and stimulating time all day meeting with meeting with various colleagues and hearing about some of the really terrific work that's going on over here so the title of my talk as you can see is scaling down the laws of thermodynamics and as it suggests I'm going to be discussing on about what thermodynamics looks like so to speak when it is applied to very very small systems so let me start with a kind of introductory slide to motivate why one might want to discuss this issue so as we all know the laws of thermodynamics were originally written down to describe the behavior of macroscopic systems notably the steam engine to pick that up up there on the left these days there's a great deal of interest in trying to understand the precise behavior of microscopic machines particularly biomolecular machines like the A.T.P. Cynthy's depicted up here on the right for those of you who may not be familiar with the A.T.P. Cynthy's it's a truly beautiful biomolecular complex it's made up primarily of 2 pieces the so-called F. 0 domain and the F. one domain the F. 0 domain is imbedded in a membrane and Roughly speaking it works like this there is a proton gradient across the membrane or equivalently a chemical potential difference across the membrane the F. 0 domain allows protons to pass through one at a time and it takes advantage of that chemical potential difference to generate mechanical torque This torque than causes the F. one domain to rotate with respect to the 0 domain and that rotation in turn catalyzes the synthesis of adenosine triphosphate A.T.P. from adenosine Di phosphate and inorganic phosphate so so so this machine is effectively continually recharging the cells battery in an intimate in a matter of speaking.

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Now even though the A.T.P. Cynthy's in the steam engine. Are separated by something like 8 orders of magnitude in length scale ultimately each one of these machines meant to make takes advantage of some source of free energy whether it's a temperature difference in the case of a steam engine or a chemical potential difference in the case of the A.T.P. Cynthy's to generate mechanical or useful work so it becomes natural to ask how do the laws of thermodynamics the laws that were originally written down to describe macroscopic systems like steam engines how and to what extent and in what form do those laws apply to microscopic systems like the A.T.P. Cynthy's or more specifically what do the laws of thermodynamics look like so to speak at the nano scale now in one hour I'm not going to have time to give a comprehensive answer to that question but what I want to do is give an overview of progress that's been made over the past few decades that in my view has helped to clarify what the 2nd Law of Thermodynamics in particular looks like one of the when it applies to very small systems one of the take home messages of this talk is going to be that when fluctuations are properly accounted for then the inequalities that we ordinarily associate with the 2nd Law of Thermodynamics can be reformulated as stronger equal ities and then also discuss how these results what these results have to tell us about thermodynamics of information processing and at the end about the thermodynamic arrow of time so let me begin with a brief kind of refresher here on the 2nd law at least in the context that I'll be discussing it in this talk so let me start with my favorite thermodynamic system that ordinary rubber band imagine I take a rubber band I attach one end to an immovable wall the other end to an ideal spring and this little stick figure at the end is me I'm on the far end of the spring trying to pull with all my might to stretch the rubber band this rubber man is surrounded by let's say a room full of air at ordinary temperature and pressure so if you think about it this has all of the elements in the introductory text book discussion of Therma than.

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Processes there is a system of interest that's the rubber band attached to the spring there's a thermal reservoir with which that system can exchange energy in the form of heat that reservoirs the surrounding air and there is also a work permit or which all denote by lambda which is the distance from the fixed wall to let's say the far end of that spring and off treat this lambda as a variable that I control directly and I use that variable to act on the system of interest now here wondering why I'm not pulling directly on the rubber band why I have this silly little spring over here that will come all answer that question when I look in a moment at a microscopic analogue of the rubber band OK So with these elements in place let me imagine that I subject this process this system to the following thermodynamic process I begin with the rubber band in equilibrium with its surroundings at some initial parameter value let me call a lambda equals A I then stretch the rubber band by changing lambda from A to B. and I imagine that I do this relatively rapidly So typically the rubber bands temperature temperature will go up it will go away from equilibrium with its surroundings and I will also perform a certain amount of work to stretch the rubber band let me let de W. do it that work.

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At the end I'm going to hold the parameter fixed lambda at at its final value B. and I'll let the system come back to equilibrium with its surroundings so we have a situation where the system begins an equilibrium in state A ends in equilibrium in State B. but at intermediate times it's driven away from equilibrium and work is performed on the system and in this context the 2nd law tells us that the work must be greater than or equal to the free energy difference a Delta F. between the initial and final equilibrium states of the system so remember that free energy here this is Helmholtz free energy this is a state function so it has a well defined value and state a a well defined value in State B. and the 2nd law tells us that the amount of work we have to do to stretch the rubber band must be greater than or equal to that to that difference of course the 2nd law is more general than that it applies to situations where you have multiple heat exchangers of particles things like that but for the purpose of this talk this will be the 2nd law of thermodynamics all of the results that I'm going to discuss later on in the talk can be generalized to situations where there are multiple temperatures involved and so forth but for clarity let me stay within within this general context all right now let's think about how these ideas might scale down to a microscopic analogue of stretching a rubber band and that is stretching a single single molecule like the strand of R.N.A. shown over here so this is blue wiggly curve denotes some polymer like molecule namely a piece of R.N.A. I imagine that this piece of R.N.A. is attached to 2 microscopic polystyrene beads using some fancy chemistry and I should I should mention that the scale here is all off the beads are in fact enormous compared to this poor little piece of R.N.A. that stuck between them but this is meant to be a you know theorists artistic rendition I take this whole floppy dumbbell I immerse it in water at say room temperature and pressure I grab one of the R.N.A. strands with a micro pipette a tiny suction device I were add the other R.N.A. strand with.

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Laser to eat with optical tweezers or a laser trap and in this case I again have a work parameter of lambda which is the distance from the micro pipette which is fixed to the center of the laser trap which I which I can manipulate and once again let me imagine that I subject this system to an irreversible process I start in equilibrium at Lambda equals A I stretch the stretch the single molecule performing a certain amount of work on it and I allow it allow the system to come back to a new equilibrium state corresponding to Lambda equals B. And now let me imagine that I go back and I repeat this process over and over again in principle infinitely many times each time evolving from the same initial state of the same final state be at exact using exactly the same schedule to change my work work parameter lambda now even though I'm being very careful to do the same thing to the molecule every single time I repeat the process because we're talking about a very small system it's going to be undergoing different fluctuations every time I stretch the molecules will be flopping around slightly differently each time and as a result the precise amount of work that I perform on the system is going to be different from one realisation to the next so but in this case the 2nd law suggests that the average work is going to be greater than or equal to the free energy Delta after the average work here is denoted by W. and angular brackets and I just literally mean the average over many repetitions of this of this process so what emerges from these kinds of very simple considerations is the following following kind of semi semi quantitative picture for a macroscopic system like a rubber band we get a single value of work as long as we do the process repeat repeat the process identically every time will always observe the same value of work and that work is going to be greater than the free energy difference Delta F. or at best it will be equal to the free energy difference if we stretch it reversibility.

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Whereas for the single molecule we're going to get a distribution of work values Rove W. So think of this as a histogram over many repetitions of the process and in the 2nd largest suggests that the mean of this distribution is greater than the free energy difference Delta F. but now we have substantial fluctuations around the mean in fact we may every once in a while even observe a value of work that's in this tail down here which we would never see for a macroscopic system that would be a shocking event that would be a true true violation of the 2nd law if it happened in a macroscopic system like a rubber band but in a small system like a single molecule we're not going to be so surprised OK So I think this picture was already at least at some level well understood over a century ago by people like Maxwell and bolts Mon and Gibbs who realized that the laws of thermodynamics in the 2nd ball in particular must be understood as statistical statements their statement is statements about averages.

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And this is certainly correct but this this view tends to think tends to make us think of these fluctuations in W. as kind of random boring noise and it turns out that there is a there are actually more interesting than one might 1st expect so these fluctuations in W. fluctuations in the work satisfy some rather strong and unexpected and potentially useful properties and all discussed some of those in in the talk.

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So the 1st of these 1st of these properties that these fluctuations satisfy is shown over here so what this is saying is the following let's imagine that instead of looking at the mean of this distribution which would be the natural quantity look to look at let's instead look at the average of each of the minus beta W. beta here is the inverse temperature in units of Bolton's constant it has units of inverse energy W. again is the work during one realisation of the process angular brackets denotes an average over infinitely many realisations and Delta F. again as before is the free energy difference between the initial and final equilibrium States so the claim that's being made here is that if we take this particular average this is going to be equal to each of the minus beta Delta F. and this is this remains true even if the system is driven substantially far from equilibrium this during the process so this is not just a near equilibrium or linear response kind of result although it certainly remains valid in that regime.

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And if you think about it this places a relatively strong constraint on the work distribution that we might observe so the shape of this work distribution in general is going to depend significantly on on how we carry out the process so for instance if I do the process relatively slowly and gently then I would expect a fairly sharply peak distribution of work values near Delta F. whereas if I do it very fast I would expect the broader diff distribution of of work values with a larger mean but the claim is that whether I do the process slowly or quickly this particular average is going to be independent of that it's only going to depend on the initial and final equilibrium states of the system because those states uniquely determine the free energy difference the free energy difference Delta have.

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OK So so this result over here is stated in the form of an equal of the the 2nd laws already discussed is an inequality Let me now briefly discuss how those 2 are related and it turns out that from equality one can almost immediately the arrive to inequalities to get from here to here or from here to here is literally like a line or 2 of calculation let me discuss these.

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Both in turn so 1st the 1st one that you see up here says that the work is on average greater than the free energy difference I've already discussed that and it turns out that that follows by combining this physical prediction with this mathematical inequality this is the Ensigns inequality that just says that the average of each of the X. has to be greater than or equal to each of the average of X. for any any random variable X. and this just is related to the fact that the function each of the X. always has a positive 2nd 2nd derivative So basically if I apply this Combine this with this result take the log of both sides multiplied by minus K. T. I end up at this inequality assent essentially in the ME immediately so that's nice it shows us that not only is this equality consistent with the 2nd law in some sense it predicts a or implies at least the STIS to distil statement of the of the 2nd law of thermodynamics but it turns out that you can make a stronger statement and to get to that statement let me look at the let me focus on the far left distribution of this distribution of work I'm looking at this tail this far left tail where work is less than or equal to the free energy difference Delta F. and events that are in this tale are often described in the literature as violations of the 2nd Law I like to emphasize that the word violations really needs to be in quotes here nobody is suggesting that the 2nd Law has somehow been overthrown this is just a question of relatively large fluctuations in a very small system so let me zoom in on that far left tail.

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OK so here's that here's that region over here and now let me ask the following question what's the probability that the 2nd law is going to be violated in the sense that I just mentioned by at least an amount Zeta and so data here is some arbitrary positive parameter with units of energy and really what I'm asking is the following if I repeat this if I make this perform this experiment once what is the probability that I'll find a value of work that's no greater than Delta F. minus minus Zeta In other words what's the area underneath this this portion of the tail.

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Of the distribution of the distribution of or so what's the what's the area underneath this tail Well it turns out and that again in about one or 2 lines I'm not going to go through the algebra here but I'll leave that kind of as an exercise you can show that this equality in a plot implies that this probability is no greater than each of the minus Zeta over a K.T. which is saying that this this tail over here the area underneath this tail decays exponentially or faster in the in the in the size of the violation as measured as measured in units of Katie so that's telling us that if we do the experiment many times well we have some chance of seeing a violation of the 2nd law by 2 Katie or maybe by 3 Katie because each of the minus 2 needed the minus 3 are not particularly small in our not terribly small numbers but we have a centrally no problem no chance whatsoever of observing a violation of the 2nd Law by 100 K.T. even though 180 is a very small would be a very small violation on a macroscopic scale each of the minus 100 is such a fantastically small number that we can we can say there is no A For all practical purposes there is no chance that we can observing such a violation.

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OK So so this result is actually consistent with our everyday experience which states that. Shows us that the 2nd law is not just satisfied on average but in fact we never see large violations of the 2nd law but I think it would be different difficult to obtain such a tightly bound directly from directly from macroscopic arguments in fact it can be shown and I won't go through this in detail but it can be shown that this is the tightest universal bound on on the 2nd law within this general context of driving a system from one equilibrium state to another equilibrium state and when I say tightest universal bound what I mean by that is for any choice of love for any choice of Zeta any choice that you want of Zeta I can cook up a I can construct a hypothetical process that in principle could be measured in the lab where this this inequality is saturated in other words this comes as close as possible to being an equality and this is particularly interesting for the choices Ada is equal to 0 so if you think about what this is saying for is equal to 0 this is just saying that the area underneath this tail must be less than or equal to one right and I'm saying that I can get it to be as close to one as possible in other words I can cook up a thermodynamic process where in principle 99 percent of the time I'm going to observe work values that are less than the free energy difference Delta asked for 99.9 percent of the time if you want but then that remaining one percent are point one percent are going to be sufficiently larger than Delta F. that on average the work is going to be greater than or equal to the free energy difference and I actually have teamed up recently with some experimentalists who have who have implemented this in a process where they were able to carry out a process where 65 percent about 65 percent of the realisations were violations of the 2nd Law and the remaining 35 percent were consistent with the 2nd law but of course on average the work was greater than the free energy difference Delta F. So certainly on average the 2nd law remain remains valid.

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OK let me go on to the next prediction that I wanted to discuss and for this prediction and I'm going to talk about thermodynamic cycle so let me explain what I mean by that I'm going to use the word the term forward process to describe to denote the kind of process I've just been talking about up to now where I'm changing some parameter from A to B. and I'll use the term reverse process for the situation where I start with this is the equilibrium start with this system in equilibrium state B. and then I change the parameter from B. back to a using exactly the opposite schedule of the one that I used during the forward process so let's see if I stretched it out 5 centimeters per 2nd during the forward process I'm contract ing at 5 centimeters per 2nd during the the reverse process.

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I'll let the subscripts F. and our denote forward in and reverse each of the is a perfectly valid thermodynamic process in its own right so each of these processes satisfies the 2nd Law of Thermodynamics the minus sign on the on this equation over here just comes from the fact that we've changed the identity is which the identity of the initial and final states.

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And if I add these 2 inequalities together I just get the result that's found down here which says that on average the work that I do to let's say stretch the molecule Plus the work that I get back plus that with the work that I do when I'm contracting the molecule the total work done over the entire cycle must on average be greater than or equal to 0 in other words on average I am doing more work to stretch then I get back by contracting when I allow the molecules to contract and if you think about it this is kind of a No Free Lunch theorem if things worked out the other way I would have a wonderful source of energy I would just stretch and contract this molecule or this rubber band if I could do this in a macroscopic system and if I do this for many many cycles the net result would be to harvest energy from the surrounding thermal environment and and convert that into work and that's certainly the 2nd law would never allow us to get away with something something that that nice so so this is kind of a no no free lunch.

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Statement again is a statement about averages and once again the fluctuations around the average or more important more interesting than one might at 1st guess so if I imagine carrying out this cycle many many times calculating how much work I do during the forward process calculating how much work I get back during the reverse process and if I then construct histograms of these to work distributions then the SR grams will qualitatively look something like this the mean work that I do in stretching the the single molecule is greater than the free energy difference Delta F. that's the position of this vertical line over here and that Delta F. is in turn greater than the mean amount of work that I get back when contracting the single molecule So this is just these inequalities expressed in the form of these these distributions and it turns out however that these distributions satisfy a very nice symmetry property that was a drive by Gavin crooks when he was a graduate student in a chemistry department and.

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Showed that the ratio of these 2 distributions any point along this axis is given by this very simple expression eat of the beta W. minus Delta F. where the symbols beta W. Delta ather mean exactly exactly the same as as before. And again the claim is that this is true even if the system is driven substantially far from equilibrium during during the process kind of one huge consequence of crooks as a result here is is the is the prediction that these 2 histograms are going to cross right at Delta F. because when W. is equal to Delta F. the right side of this equation is equal to one and this has been.

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Verified in experiments and in fact let me let me talk briefly about the 1st experiment that was done to check books as prediction this was carried out by Felix retort and colleagues in Carlos Bustamante his group at U.C. Berkeley and they basically took a single molecule of R.N.A. they stretched it in contracted it stretched it in contracted for I think several 100 cycles like that and then built up these work distributions so let me explain the 2 plots over here the one on the left is the raw data so to speak coming from the experiment so here on the horizontal axis you have the extension of the molecule or equivalently the distance between the 2 beads on the vertical axis is the force that's being exerted by this laser trap on this on this particular bead and that's something that the experimentalists can can measure directly.

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And you have here about. Data from about half a dozen cycles of stretching and contracting the orange curves correspond to the stretching part of the cycle the blue curves correspond to the contraction and there's a couple things that kind of jump out at you one is of course that the orange curves seem to be located at a higher.

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Are located above the blue curves on average except for this area here in the noise or equivalently the work sorry the area under these the orange curves is greater than the area underneath the blue curves and that's just the 2nd law of thermodynamics it's saying you do more work the work is represented by the area under the curve you do more work to stretch the molecule than you do to get to you get back when you're contracting the molecule and you also notice this feature that there is these sudden diagonal lines that are more pronounced in the OR for the stretching portion of the cycle the orange curves but they're also there for the contraction port portion of the cycle for the blue curves and these diagonal lines are easy to understand so the molecule itself our name molecule in its native state it folds up into what's known as a hairpin turn and as you stretch the molecule you know the more you stretch it the extension the length of the molecule increases the force that you're exerting on it also increases and then at some point the molecule pops open this hair pin pops open like that when it does the extension increases suddenly and the tension drops as some of the tension is relieved so that's what you see with the orange a diagonal lines here and on the way back the molecule suddenly really folds and that's that gives you these.

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These features that you see these features that you see down down here OK. OK So again from several 100 such cycles the experimentalists were able to reconstruct distributions both for the wild type molecule this is the molecule that was actually directly harvested from equal why and then for a mutated version of that molecules in the wild type work distributions are represented by the 2 purple histograms and the mutant ones are represented by the gold histograms over here and these histograms are really just the experimental analogs of these the histograms that I had schematic shown schematically up here OK Now in order to test his prediction one needs to have the data in the overlap region between the between those 2 history grams and in the case of the wild type molecule you can see there's a centrally no overlap in other words there just weren't in the statistics weren't good enough to test crooks is to test crooks as prediction and this is why they went back and made a mutated version of the molecule that was slightly easier to unfold and in the in the mutated case there's not a great amount of overlap but there's some overlap right here in the region between W. is equal 21552160 in units of Katie so what's shown in the inset over here is for this overlap region what's plotted is the log of the left side of this equation H. namely the log of the ratio of these 2 distributions as a function of W. in units of Katie and according to crisscross this prediction this should be a straight line whose slope is equal to unity and when they fit this line to.

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The best straight line the best linear fit they've got a slope of 1.06 which is certainly well within the experimental errors where within the statistical errors of this experiment so this gave the 1st kind of evidence in confirmation of crooks is crooks as prediction. I should say that in the time since this experiment was done actually even before.

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And there have been a number of experiments done on various experimental platforms to taste to. Test both of these results as well as other results similar to them so for instance they were tested using a mechanical loss later this is basically a torsional pendulum. A single trap colloidal particle the unfolding of a the giant protein molecule Titan more recently in a in a in a single electron box and as well as a number of other systems I'm not going to go through any of these other experiments in detail but I should say that up to date all of the experimental tests have been in agreement with the theoretical predictions I think at this point it's reasonably fair to say that these result that there's a there's a body of experimental evidence in favor of in favor of these predictions.

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One actually very interesting development in my in my opinion in the last few years is that people have started testing quantum mechanical versions of these results and let me show you some data from from the 1st experiment to have done that this was carried out by a collaboration between 2 groups in Beijing and they used a an ion trap so this ion trap here forms a roughly a harmonic potential for this it terbium ion shown over here.

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We know that the. For a for a quantum particle in a harmonic potential the energy levels are quantized using this very familiar simple formula so the energy of the nth level is H. Baro made the times and plus a half and is the quantum number omega is the natural frequency H. part of course is the reduced Planck's constant and now the experiment went as follows This is so they 1st prepared the system in equilibrium they then went in and measured the initial energy of the of the of this on in in this harmonic trap so when they measure the initial energy according to the laws of quantum mechanics there is a collapse of the wave function the wave in the way function ends up in a given energy eigenstate So let me let ends of I denote the initial quantum number that was measured alright having measured that the the they then allow the system to evolve under the time dependent showed injury question there was no heat bath present here as the trap was just simply translated from one location to another location and it was translated relatively fast in some of the experiments so at the end of this process this is them is now in a superposition of energy eigenstates of the final Hamiltonian the experimentalists then went in and made another measurement of the energy at the final time and again they found the given energy eigenstate of the final Hamiltonian and that eigenstate is quantified by the quantum number let's call it ends of F. for the final So then in this case because there is no exchange of energy with any thermal reservoir there is no formal reservoir in this picture the work is simply defined to be the net change in the energy of the system so it's.

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Times the difference in these 2 measured quantum numbers age borrow times times Delta at OK and one might say that this is kind of an ad hoc definition of of work but it turns out that in the at least currently in the quantum thermodynamics community for these kinds of processes this is the most the most widely accepted definition of definition of work.

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OK So this is the work value that can be measured over and over again to get a histogram in this particular case the free energy difference is 0 because all we're doing to the potential is translating it from one location to another and so the question is if I take these work values that were measured plug them take the average of each of the miners Veda W. is that going to be equal to one and the experimentalists carried out there.

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This experiment at 3 different rates of moving the traffic a slow rate shown over here intermediate speed and the fastest speed shown over here and what are plotted over here are the histograms of Delta N. Delta and again it is the net change in the quantum number and in each case the largest value of the histogram is sorry it got cut off over here when I made this figure the largest value of the histogram occurs at Delta and is equal to 0 so you see that at the slowest speed about 90 percent of the time the final quantum number is the same as the initial quantum number and that's just actually a consequence of the quantum maybe about it theorem if you if you were to move the trap infinitely slowly you would always end up in the same quantum state as you as you started whereas at at faster speeds you get excitations to other 2 other states in this system OK So again these were the experimentally measured distributions of Delta N. and those get converted into work distributions and then taking the average of the exponential of the work they found that the values were equal to 1.032.995.989 which were.

[00:31:37]
Essentially equal to unity within the within this does statistical fluctuations so this gave the 1st this experiment gave the 1st. Piece of experimental evidence in favor of a quantum quantum mechanical version of 11 of these fluctuations theorems and I should mention that in the last year or so my my student and I my student Andrew and I have teamed up with these with these groups in Beijing and they carried out a somewhat modified version of this experiment where decoherence was added to the picture so this is the system was no longer just evolving under the time dependent showed injury question and still it all worked out so we were able to suggest a theoretical reason why it should should work out and they were able to confirm it experimentally and I'll leave you the reference there to the details if you're interested OK Let me now apparently shift gears and seem to talk about maybe an unrelated topic but in a few slides I'll bring this back to what I've been discussing so far so I want to discuss feedback control and in this discussion I'm going to want to make a distinction because between what I call autonomous feedback control and non autonomous feedback control and let me illustrate that distinction with a very famous feedback control system which is the so-called centrifugal governor shown over here so this was an invented in the late seventy's hundreds by Matthew Bolton and James Watt and it was basically used to stabilize the behavior of early steam engines the idea is roughly the following so steam engine of course in a steam engine you have some flow of steam into the engine and that produces rotary motion of some sort of wheel and that that motion is then used for something useful but of course it's hard to control directly the amount exactly the precise flow of steam into a steam engine is even harder 200 some odd years ago to do that than it is today so so the effect was that early steam engines were kind of you know they'd speed up some times then they'd slow down and so forth and you can imagine that this would be both annoying and not particularly practical in terms of of using these steam engines for things like textile mills and so forth.

[00:33:45]
So the way that the centrifugal governor helps to regulate this motion is as follows so the rotational motion that's produced by the steam engine is coupled to this vertical shaft over here so the the governor itself consists of 2 metal spheres that are attached by by by arms by by rigid arms to a vertical shaft and that these spheres can then rotate around that shaft so as the steam engine if the steam engine speeds up.

[00:34:14]
The steam engine speeds up to shaft starts rotating faster and then those arms get lifted by centrifugal force and as they get lifted the vertical location of the arms is coupled to a valve so as the arms go up a valve closes off and reduces the supply of steams of the thing starts spinning too fast the supply of steam gets cut off a little bit and similarly if the if it slows down the arms go down the valve opens and more steam is let into the system and you can imagine if you kind of tinkered carefully with this Istomin if you were good enough engineer and James Watt was probably a pretty good engineer that you can actually get this to to regulate steam engines and in fact for many decades these were a very important practical complement of steam engines so the Center for governor is what is an example of what I am going to refer to as an autonomous feedback control and by that I just mean that device that the system that implementing the feedback control is a is an explicitly physical device of some kind of gadget that's been invented for that purpose so the gov in this case.

[00:35:18]
By non autonomous feedback control I have in mind the situation where instead of some physical device. I imagine some external agent who's not specified so that external agent could be me could be a graduate student that's been assigned to this task it could be some you know if theory all being that we that we imagine we don't really care what the physical composition of this external agent is we just assume that there is some external agent with certain powers of observation and feedback certain powers of observation and it can perform feedback on the on the system.

[00:35:53]
Of course this distinction is a little bit as a little bit artificial but still it helps us to to formulate the kind of questions we're likely to ask so in the autonomous case typically we ask well how do we actually invent a device that gives us the sort of feedback control we desire whereas in the non autonomy case the kind of questions we might ask is given a certain powers of observation and let's say certain certain time timeframe for for performing the feedback what sort of control can this external agent exert over over the system of interest so let's keep that distinction in mind and now let me imagine scaling this all down to the nano scale and let me look at Feed how feedback control might operate for a very small system and this takes us back almost as far as James Watt to the discussion of the famous so-called paradox of of Maxwell's demon so way back in 1967 James Maxwell who did about anything interesting there has to do in science in the in the hundreds except for maybe the theory of natural selection James Maxwell road of letter to his friend Peter Tate and in that letter he described his very nice thought experiment where you have a container filled with a gas of particles there's a partition in the down the middle of the container and let's say that the gas to the left of the partition is hot and the gas to the right of the partition is cold the partition itself has a little hole in it which has a trap of the.

[00:37:18]
Or attached to it and then Maxwell imagine some hypothetical creature that sits there next to that trap door and whenever it sees an unusually fast particle moving from the right towards this opening it opens up the door and lets that that particle through similarly whenever it's seen sees an unusually slow particle moving from left to right it again opens up the trap door and locks that lets that particle through so in this manner there gets to be an accumulation of fast particles faster and faster particles only left slower and slower particles on the right or the way that Maxwell put it the energy in a is increased and that in be diminished that is the hot system has got hotter and the cold colder and yet no work has been done only the intelligence of a very observant and neat fingered being has been employed you can see that Maxwell wrote much more elegantly than we tend to to write in scientific literature these days and I've I've emphasized the words of the intelligence over here because this thought experiment is squarely within the framework of a non autonomous feedback control so I wasn't thinking of some kind of gadget he was just imagining some creature that of course we now call Maxwell's demon that would implement this and he pointed out that such a creature could in effect bring about a violation of the 2nd law of thermodynamics because if this thing operates for long enough it's creating a larger and larger temperature difference without having to pay for that difference in the form of work and that of that's certainly a violation of the 2nd Law So this is why this is sometimes referred to as a paradox but Lot logically it's not a paradox at all of you if you're going to put some sort of magical creature into your thought experiment you shouldn't be surprised if all hell breaks loose so to speak nevertheless it's a very provocative thought experiment and people have been thinking about it for more than 150 years now it's still still pops up in the literature quite a bit especially in recent years and one natural question that you can ask Upon seeing this is well.

[00:39:18]
Could we replace this hypothetical being with some kind of physical device some some kind of gadget that would accomplish the same thing in other words could we in principle maybe not yet in practice because our technology isn't quite there yet but in principle could we design some sort of tiny robot that that sits here or Maxwell's demon was sitting at you know this is the analog of a centrifugal governor this robot would measure the speeds of the molecules as they come by and it would do the sort of actions that Maxwell's demon does to move the hot ones to the left and the cold ones to the right I mean in principle it doesn't seem like there's anything to stop us from making such a robot certainly we can we have robots at the macroscopic scale that do feel measurement and feedback on the other hand if we could create something like that would it be a genuine violation of the 2nd law so this question is a mechanical Maxwell demon possible is not a new one it goes back to over a century there's a very nice paper by Mario on smaller ski in 1912 and there's also a very beautiful discussion in the fine lectures and both small HOF ski and Fineman come up with kind of arguments where they put forth a model systems of a proposed model systems that at 1st blush seem to be able to accomplish something that's analogous to what Maxwell's demon does in other words they seem to somehow be able to rectify thermal fluctuations in a way that gives a violation of the 2nd law but then you know both small and fine men say that well upon closer examination if we analyze this model a little bit more carefully we see that it's not going to work the way it seems to work at 1st so both of them come down firmly on the side of the answer to this question is no matter I can call Maxwell demon is not possible and it would constitute a genuine violation of the 2nd Law.

[00:41:06]
And this of course would be an example of Autonomy's feedback control now beginning in the $1960.00 S. and going up to the early eighty's with the work of Rolf Landauer and Charles Bennett out at I.B.M. and independently all over Penrose in the U.K.. They proposed a somewhat more nuanced answer of what it which I think of as a yes but answer and what they said was the following.

[00:41:29]
In principle there's nothing stopping us from making a mechanical device that would operate as Maxwell's demon but if it is truly a physical device rather than some kind of hypothetical demon if it's a physical device then according to this argument this device is gathering information about its surroundings that information must be stored in some physical memory register right it's not just sitting in some thought bubble sitting up above this creature's head it must be stored in some physical memory register so we can kind of conceptualize the this as follows imagine that there's a stream of bits that's moving past this device as the device makes its measurements it writes information about the outcomes of those measurements onto it onto a stream onto the stream of bits.

[00:42:13]
And so then according to this argument what's happening here is that this stream of bits is being randomized and if we view this randomization of the stream of bits as an increase in it's Shannon entropy and if we think of the Shannon trape as being morally equivalent to the real thermodynamic to the honest thermodynamic entropy then according to this argument that increase in the entropy of this stream of bits could be used in principle to offset the decrease of entropy that's going on over here as we're creating a temperature temperature difference without paying for the cost in work so if in effect we're creating this temperature difference we're not paying for it by doing work but we are paying for it by increasing the entropy of the universe somewhere else namely in this this stream of bits and I would say although there are some people who still have very strong opinions against this argument I would say it's fair by now to say that this is kind of the consensus argument for how Maxwell's demon Maxwell's demon paradox should be should be.

[00:43:14]
Should be understood and actually over the last 8 or so years there's been a lot of there's almost been like a cottage industry of people proposing explicit physical models that would that would carry out that would accomplish this this this kind of result I don't have time to talk about those but I'd be happy to offer offer some to give references to those OK Let me bring this back now to single molecule experiments and let's consider a situation this is the non autonomous case now let's imagine that as we're pulling this we're stretching this single molecule we have some agent again the agent could be me or the agent could be you know I've programmed a computer to make these measurements some agent out here is in real time observing the fluctuations of this beat in this trap and then that agent performs feedbacks maybe it moves the trap a little bit more slowly or a little bit more rapidly and the aim here is to do as little work as possible in stretching the single molecule and basically the aim is to use what we know about the fluctuations the of this bead to minimize the amount of work that's being done on the molecule.

[00:44:20]
In the following discussion I'm going to let capital I denote the information that's gained by by this agent. In the course of the measurements that it does over one realisation of this process I'm not going to give a quantitative definition of that information but I'll provide a reference to where it is quantitatively defined so this was studied by Takahiro.

[00:44:43]
Working in Japan and they showed that they used very general arguments to show that in this particular scenario the average work that is done by this. Instruction the molecule must be greater than or equal to Delta F. minus K.T. times the average information that is gained by by the agent right I'm not going to go through the derivation over here but roughly what this is telling us is that by gaining information we can get away with doing less work than the 2nd law of thermodynamics would would predict and this is very much in line with what Maxwell was showing but this is kind of making it quantitatively.

[00:45:20]
Making it quantitative rather than just sort of qualitative as in as in Maxwell's argument. A couple of years later. We're able to show that in fact these fluctuations you have a fluctuation relation that applies in this situation if we take it to the minus beta W. minus I average that over infinitely many realizations of this process then that is going to be equal to each of the minus beta Delta F. And again this equal of the immediately implies this inequality although they were able to derive this one earlier and in the same year experiments were done by Tejal Bay and colleagues where these predictions were very verified.

[00:46:03]
I'm not going to go through the details of the of the experiment but but to me it's the very fact that people are making these kinds of very precise predictions and that they're being verified in experiments tells me that there suggest to me that we're really getting a very deep understanding of the relationship between information and thermodynamics between and how work and free energy play in there for these for these microscopic systems so we've gone far beyond what Maxwell suggested 10150 years ago OK in the last few minutes of my talk let me.

[00:46:41]
Discuss the arrow of time and let me frame this discussion in the context of a hypothetical guessing game so suppose I show you a movie and in the movie you see a thermodynamic process where some parameter is being changed from A to B. So let's say I show you a movie and what you see in the movie is that a molecule is being stretched and now your task is to guess whether you're seeing the events as they actually occurred or whether I'm trying to trick you and others may be actually contract to the molecule and I'm just being devious and I'm playing the movie backwards for you so it just looks like I stretched the molecule and for the purpose of an idealized thought experiment let's assume that you can you can see every detail of what every single molecule an atom is doing in the picture in fact you can you have all information about what's going on in the movie except for whether the movie is going in a forward direction or in the reverse direction and now the question is how do you how do you go about making your best guess of whether you're seeing the direction of the arrow of time so-to speak.

[00:47:42]
So it turns out that this this. This question can be framed as an exercise in statistical inference so there are 2 hypotheses I'll call it hypothesis F. for forward meaning the molecule was actually in fact stretched as you see in the movie on the other hypothesis are for reversed is the hypothesis that no I'm I'm trying to trick you I contracted the molecule and I'm playing running that backward in time and what can be shown is 2 things 1st of all there's only one piece of information in the movie that can help you make that guess and that's the amount of work that's being done over the course of the movie so if you look at the how all the atoms are moving and use that information to compute how much work was done to stretch the molecule in this movie that you're watching.

[00:48:26]
Then no further details about how any of the atoms or molecules are moving around no further details can help you one bit in determining whether it was the forward process that took place or the reverse process that took place and moreover the likelihood that it was the forward process that took place given the amount of work that you observe in the movie is given by this very simple expression one over one plus either the minus beta W. minus adult bath and we all know what that looks like it's given by this shown by this graph over here so when when W. is much greater than the free energy difference Delta F. then this likelihood is very close to one whereas when W. is much less than the free energy difference Delta af the value this likelihood is very close to 0 in this is kind of what we expect if we have if we see that the work is much greater than for the free energy difference then the events in the movie are very much in agreement with the 2nd law so we're probably seeing the movie being run in the forward direction whereas if we see a value of work that's much much less than the free energy difference then the events we see in the movie are contradictory to the 2nd law so we.

[00:49:34]
In all likelihood the movie is being run backward in time for us and then there's this universal. Transition from the one behavior to the other over a range of a few Katie and what's the what's actually still kind of amazing to me is that this is really a universal universal prediction it has nothing to do with the specific nature of the experiment whether it's a single molecule experiment or or something other as long as what we're doing is driving the system from one equilibrium state to another equilibrium state in the presence of a of a single thermal reservoir and this so this likelihood really quantify is our ability to determine the direction of the arrow of time it says for in particular it says that if we see a value of W. That's equal to Delta F. then we might as well flip a coin there's nothing that we can do to tell.

[00:50:23]
To determine whether it's more likely that it it's being. Stretched or or contract and I should say that this result was 1st derived by Michael shirts and colleagues at Stanford and. Later in a slightly different approach by Paul morrow and colleagues at at Harvard in Berkeley the Inter but they were doing this in the context of developing methods for estimating free energy differences the interpreted Taishan in terms of the arrow of time can be found in this in this review article that I wrote and actually over the last year or 2 experiments were performed on a single quantum dot the name is a little bit of a misnomer there's nothing really quantum about it it's a basically a classical 2 state system but this this hypothesis or this prediction was tested and I'll just show you the results here without going through the details of the experiment the red curve here represents this this function over here there's no no fitted parameters at all it's just just this function and the various circles triangles and squares here denote the the likelihood that the forward process was carried out given the value of work with that likelihood being determined directly from the experiments the experiments that were carried out and you can see that the function that the data lie really well on top of the on top of the curve which gives some which provides experimental evidence in favor of this in favor of this prediction OK So let me let me come to my summary slide one of the messages from this this talk actually probably the biggest message here that is that.

[00:52:05]
When the laws of thermodynamics operate on very small systems fluctuations become important so instead of having their single well defined values of work as in the case of let's say stretching or contracting a rubber band we typically get distributions of work values that will reflect relatively large fluctuations in these quantities and.

[00:52:26]
The main message is really that that these when these fluctuations are properly accounted for then these inequalities that we get at the macroscopic scale become replaced by stronger inequalities such that such as the 2 that I discussed as well as others that I haven't haven't had time to discuss So if you want to follow up let me shamelessly plug my own review article that I wrote about 7 years ago that covers the classical aspects of these predictions there was a very nice colloquium in.

[00:52:57]
By P.Z. handy and talking or that covers quantum aspects and also Takahiro around the same time wrote a very nice article covering the information processing aspects that I've discussed so so with that let me thank you for your attention and I'll be happy too happy to take any questions.

[00:53:18]
Thank you. So that take advantage of these kind of fluctuation theorems. Not directly so these are more these more applying to systems that are are manipulated externally so this has been test these have been tested using biological systems like this in single pieces of single pieces of R.N.A. So.

[00:54:06]
I I'm not aware of specific situations where biological systems would take advantage of these particular predictions although it certainly is possible that they would take advantages of the fluctuations that are there that are there in the works and as I've mentioned these predictions play some constraints on those fluctuations so they tell us about you know the probability of observing violations larger than a certain amount and those constraints certainly apply to any sort of any sort of biological system that's that's operating in that in that context but I can't specifically to point to something where this this machine this bio biomolecular machine takes advantage of these results OK.

[00:54:59]
I don't know the I don't know how that would work I mean I guess in some sense the closest closest one in this was this is more of a model system was this experiment in colleagues where they basically I mean they they observe the fluctuations in real time so they weren't really observing work fluctuations but still they were observing thermal fluctuations in real time and using that knowledge to essentially extract more work from their system than they would have been able to do if you know if they hadn't been observing those fluctuations that's not really directly an answer to your question but it that least there's some resemblance that.

[00:55:48]
Yeah they can be different what. So it's a little bit more subtle that because remember the poem holds free energy in this case it's is the energy minus temperature times the entropy the energy is an average energy it's an average of the energy function over all those confirmations that you just mentioned in equilibrium the entropy is not quite as cleanly expressed as the average of something temperature can be taken as an average but entropy not in a very natural way so entropy is it really defined as the in this case would be minus the some over all possible configurations of the probability of a particular configuration times the log of that probabilities of the the Shannon in Shannon information while free energy is energy minus temperature times entropy so because the entropy term is there the free energy itself is not naturally expressed as the average of some quantity rather free energy is expressed as the log rhythm of the partition function.

[00:57:55]
So I think it's a different different different categorization so so for instance those single molecule pulling experiments I would say those are microscopic systems be that because the fluctuations are very relevant there but I would not say quantum effects are probably negligible there yet so so I'm saying it's a different macroscopic versus microscopic as a in my view it's a different question.

[00:58:20]
It's a different that's not the same difference as the difference between quantum and Cossack all so you can have classical microscopic quantum macroscopic but those are 2 separate kind of separate axes speak. And and these results that I'm that I've talked about here in principle they at least in the classical case in principle they apply both to microscopic and macroscopic systems the problem of that is that for macroscopic systems one would never be able to do sufficiently many realisations to get these this average to converge the exponential average to converge so it's kind of a for macroscopic systems it's a formal result that you could never directly test in and in the experiment in terms of experimental test testing but formally let me get back to the Final 4 formally these results I would say do apply also to macroscopic systems that they're just they don't particularly aren't particularly useful they're.

[00:59:47]
So. Yes. So I would. Say. So I'm not sure about the 2nd answer the 2nd part of your question because I'm not sure how fluctuations in the earth could be put in this kind of paradigm of the transition but but more more generally I would say at least in the classical case let me touch on the quantum in a moment in the classical case these are not predictions that depend on there being an observer OK So so this is them is undergoing fluctuations if I carry out the the single molecule pulling experiment regardless of whether I observe it classically some amount of work is being done and if I do it many times the work is going to be different from one realisation to the next regardless of whether I observe it so I would say that this has nothing to do with the size of the observer at least in the classical case the quantum case becomes a little bit more tricky because of course then you know energy doesn't have a well defined value until you measure it and so forth weren't for any other observer but I think in the quantum case there's still there's still a lot a lot of research to be done.

[01:01:46]
To determine to what extent these really do do apply to the quantum case so in the idealized situation that I described before where you have. Make an observation of energy using kind of the textbook definition of observation where it causes a way function collapse then in that case these results do apply but I think that you know to what extent that's realistic generally dualistic remains to be seen but in the classical case I don't think it depends on the size of the observer in.

[01:02:33]
The eighty's in that case. Yeah. That's that that's a very interesting question but I think the machines you're referring to are probably ones that are copying information so. So I don't think that those are actually creating new entropy in the in the sequence that's being produced but I could be wrong it is it may be that they're creating new entropy in that sequence.

[01:03:34]
But whether they are taking advantage of that is something I guess probably not so formally there's this entry you know formally by writing information to a sequence of bits you are getting some You're creating entropy that could be converted into useful work that useful work is actually going to be very very small even when compared to the amount of work that you get out from Hydra lies in a single single molecule of A.T.P. So so I think so so for a single molecule of A.T.P. I don't remember what the what the free energy changes it's something I think on the order of 20 K.T. or something like that one bit of information will give you K.T. log to of useful work so that's so so it's pretty small so formally these things may apply to.

[01:04:21]
Molecular machines that do things like R.N.A. copying but I would be very surprised if they actually in practice to take advantage of them but I'd be happy to be proven wrong on that too. Thank you.