 equilibrium theorems and linear response so linear response is so as you saw linear responses when you say what happens to a system when I perturb it a little bit and then I can I can use my equilibrium results to compute using to compute how the system responds and the correlations are the correlations by the way let me say once again something I said at the beginning note that there is an apparent paradox which is the following to derive the equilibrium theorems I use that the system is in equilibrium and that its bath is a good one equilibrium one however it is not an equilibrium result in the sense that I can get rid of time it's an equilibrium result that requires for example the fluctuation dissipation theorem requires equilibrium but it requires equilibrium dynamically but it's not a purely static result in other words statistical mechanics cannot tell me what C of TT prime is even if I know that I have a given temperature and that I am in equilibrium statistical mechanics on its own cannot give me this and the way to remember it is imagine you have hard spheres in a box in space with no gravity and they are so you can calculate certain things like the pressure using statistical mechanics now at exactly the same temperature or energy you have the same system moving in water or in honey at the same temperature statistical mechanics that doesn't distinguish these quantities the pressure is going to be the same because statistical mechanics only cares about the equilibrium distribution for the spheres however it's clear that the correlations in time is going to be different because they will move the spheres in a different way or if you want even a simpler example think of a system that is in equilibrium and you can watch the film and you can compute certain things using statistical mechanics now you take the same film and you you stretch time by two and then you you divide all velocities or do something the the dynamics can go faster but the system is the same so the correlations are going to see this but the measure that happens at what temperature there is a phase transition for example will not change so it is important to see this because you see equilibrium also so the the main good thing about equilibrium is that for certain quantities you can forget time but those quantities are one time quantities magnetization now for that one you can use stat mech but if you want the correlation between magnetization now and magnetization in a minute it's different if it's equilibrium or not because I can use the equilibrium theorems but equilibrium on its own doesn't give me that information I have to solve the dynamics okay that's one thing to now we're going to discuss a result that is dates from the beginning of the 90s and it is one of three or four results that are all very closely interconnected and it's one of the let's say remarkable results of out of equilibrium dynamics so we're going to I already mentioned it we are going to for example the original setting was for example I have a liquid and I'm stirring it for example with a spoon and or it could be I have a bar and I have a current that is going in that direct mostly in that direction now when I think of it the easiest way to think of it is imagine that the liquid is made of very few molecules and you are turning and there is a thermal bath so you are turning and most of the time you are giving energy to the particles we will see that this is what I have just said is the second principle of thermodynamics if I disturb a system it will cost me the system is not going to give it to me but every now and then these molecules are going to bang against this wheel and think of it if I am not stirring the wheel will turn in one direction and with and the other depending on the impact of the molecules so now if I make a little force that wants to turn that way mostly I will have to force but now and then so if there is no forcing the force I will feel is something like this no but if there is forcing I will be doing work so let us do the distribution of work but you know it's the same thing but at the higher level so this is time and the quantity we are interesting interested in the sign is conventional for example force times velocity which is the power I'm putting into my system in this case it would be torque time angular velocity and in this case the current again if the temperatures are the same because the electrons are moving randomly with no difference in temperature the current is going to be like this and with a bit of difference in temperature the current is going to do something like this and this is also a situation where the fluctuation theorem will apply so notice how strange these are situations where for a brief time the current is going the heat current sorry current let's denote it with a J so we don't confuse but electric current would do the same so for a short time current is going in the opposite direction and me I am gaining energy from the system which seems to violate the second principle of thermodynamics the second principle says that if I am disturbing a system on average it will cost me and not the system so system will not give me I will have to give energy to the system however for very short times the second principle can be violated you will see that the fluctuation theorem tells you this nicely oh yes sorry thank you very much so this is the temperature which and this is the time so that I get the average no the temperature is there because you want something that looks like an entropy so it's work divided by temperature so so you see this is an average so let us for a moment cancel this one although you can I can give you a reference where this is this problem is studied it's it's technically the same thing so the principle concerns strong potentially strong forcing so it's not a linear response that's what is new it's a strong force applied on the system so it's a truly non-equilibrium thing I am here in contact with a bath a temperature T which is a temperature that appeared there coupled to this and I am allowed to force a lot it is strongly driven that is the nonlinear response regime but the number of degrees of freedom is fewer than say 10 to the 24 well the principle as we will see is holds for 20 10 to the 24 but it's on it completely relevant I mean it's the unobservable why because the more particles you have the more this thing average is out and you see less fluctuations okay so remember the philosophy of large deviations I want to find in an interval T the average of power well it's divided by the temperature so the and so you see one thing that happens is that if I consider an interval or up for a much longer interval it's like throwing a coin more times so if I now do a histogram probability of sigma of that quantity average and over sigma and as I said the second principle says that mostly I have to give the system energy so this will give me something which is the distribution here not a Gaussian something but if I consider a longer time it's like throwing a coin more times so I will get something like this more peaked and even a longer time I will get something even more peaked so this we saw at the beginning when we were doing extensive quantities remember that if we can assume that the time can be broken into pieces that are independent then the result is the product so this suggests that the log of the probability of sigma is a quantity that is additive and this one is going to scale this one is going to scale nicely so this is the quantity I should plot and yes yes so I will ask it in his name yes asking is this negative fluctuation like a negative temperature no no you are at a good temperature but it's working as if you had for a very brief time a negative temperature but it's not the temperature is defined on average on your baths like I meant what I meant is like the derivation when do when we do calculations when we flip the system really abruptly we will for short periods of time we will have a negative temperature yes but temperature is something defined on average so we cannot say negative temperature it's a negative entropy production that's that's what it is and sorry and here we have to divide by the total time because we wanted a sorry not here here we have to divide it by the total time because if not we will see it peaked yes okay so we we let's we we already seen it a bit but let's think in very concrete terms I am doing work on the system I am stirring the system when is it that I am giving the system energy power and when is it that the system is giving me now there is the second principle of thermodynamics that says but I we will this the fluctuation theorem will have this as part inside it the second principle of thermodynamics tells you that if I have a system that has a given temperature and I have the same temperature and I'm doing work on it it's me who's going to do work on it I cannot extract at the same temperature work although there is a lot of energy around if I am at the same temperature as this room I cannot extract energy is this true this is important that it is true on average but during a millisecond maybe I'm lucky and 85 more molecules are crashing on my right side and I get a little bit of energy so as you will see now we're looking at large deviations the second principle concerns the average and and we're going to see what happens as deviations of this and you see that this is a function that is more and more P so we said that P could be scaled as e to the t and then a certain h let's say or whatever of sigma I don't remember what so it means that this becomes more and more peaked as I consider longer and longer times this is exactly because this experiment if I consider more times it's like if they are uncorrelated it's like throwing a coin many times this was a large deviation function okay so now this is the quantity we should plot which is h by definition and you get a kind of universal curve okay which is just this curve but now all these curves fall on to one another because they scale precisely in this form so you get something like this and then for this one my head can go more further away because the system is smaller so you get larger deviations but they with this scaling they fall nicely on to a universal curve which is for this age I am sorry I don't know if I used age but for the notation okay so what does okay so what does the fluctuation theorem tell us it's the following and for this one I'm finished writing because I need your full attention so it tells us the following bizarre thing I mean normally as a physicist you would like to know how they deviate around the mean no this is the reasonable thing to ask but unfortunately we have no theorem for that what does fluctuation theorem tells us is that in this plot if I look at any point here and now I look at negative the negative of that same deviation notice that it's around zero and not around the peak this is the first question everybody asks because everybody reasonable in this world wants to know that deviations around the peak but unfortunately that's not what fluctuation theorem tells us and it tells us that p of sigma sigma is this value this is minus sigma divided by p of minus sigma is equal to e to the no this one is bigger than this one I always get it wrong so I know there is a t the way I define it sorry no no no p of minus minus t h sigma the way I defined yeah yeah yes sorry yes the way I define and the way I drew it it's it's with a plus yes no I defined it sometimes with a minus maybe I did okay yes sorry sorry thank you thank you very much okay so before doing the calculation the I will tell you the history of this but ask me a lot of questions now then we will be drowned into a medium-sized calculation so we better let me say quickly that for t1 t2 we say that t2 is greater than t1 and there is a current here you can write also a fluctuation theorem so the the second principle says if I'm doing this the current has to go from high to low temperature it also says that the probability of of total energy going one way divided by the probability going in the wrong way is e to the minus total energy just like this it's always of this form okay the t the time here is because it's a time average quantity sigma time whether it's here it doesn't matter because it's divided so one question is about so sigma t the way you define it is it's divided by the time divided by the time it's also divided by the temperature yes yes it's also divided by so it's a dimension I it's dimensional sorry sorry sorry thank you it was divided by the temperature so it's not sorry sometimes here we put work if it's work you have to divide it by time if it's already divided by temperature nothing nothing so first remark is this is not a property around the peak which would be what we would like to have but life is not so nice second thing is this is completely model independent totally completely absolutely model independent I didn't say anything the only thing I said is that this system is in contact with a thermal bath with temperature t and this is a relation between how often I get a negative fluctuation and the corresponding positive one sorry I have a question like okay the the equation makes sense because a negative entropy production is exponentially exponentially less probable than a positive one but like you said the fluctuation dissipation tells us these but like how can we get it even with without a full derivation but like I didn't get the question fluctuation like you said the fluctuation dissipation tells us this relation okay this is not fluctuation dissipation this is fluctuation without the word dissipation not to confuse this is called a fluctuation theorem okay okay it's not it's a theorem and it's nothing to do with what we saw but a little to do but not directly to what we saw yesterday sometimes people okay I tell you a little bit the story of this but let me say so this relation is completely independent of the system and the way you're putting energy and it is also when you have conduction of heat or electricity it's all it's the same nothing of what is inside matters and furthermore you can make it quantum and it's the same so about the probability distribution do you assume that the probability distribution is a Laplace distribution or because is that I take the relation that you wrote for that one to happen we should assume that the P follow a Laplace distribution Laplace yeah yes yes we will do something that is very much like a Laplace transform but in a second it's it's a technique of the proof okay yes yes but I want to explain to you the phenomenology if you want before going into embarking into a calculation yes my question is please I'm really confused about the way you draw the hedge because like we study it as a concave up function for the free energy which is minus H so when you have a free energy in probability it's e to the minus the free energy and it's like this but because I was told that I have to put the plus here because this is the way I drew it it's reverse okay but it is concave you're right you're right and should be below the convex sorry it should be below the axis should be negative because it's a problem yeah but this is a log so so this is why can be okay this means that the larger T is the more this thing dominates because this is a universal function but the time is larger okay so okay so this this result has a history more or less like this there is an ancient result that of two Russian names I'm sorry Kuzolev I have it somewhere nobody remembers that result so some people mention it maybe I can find it boho boho boho Bochkov Kuzolev Kuzolev so I never read that paper and nobody has except people who are looking for who to attribute the first theorem of this kind then Australian people Evans well Cohen is not Australian was Dutch and the first and the last are Australian sort of read rediscovered it maybe made it more general but these are the people who taught humanity about this result they were mostly numerical people but they had understood I think they have understood that the main property that matters here what is making this theorem is time reversal which is okay then there was a famous paper Cohen is the same galavoti those of you from Italy surely know which proved it in a context that this is now we are in 94 I think proved it mathematically in a con in a in a very specific context which is the paper is remarkably hard to read so I know one person who read it completely for those of you who know him Francesco and just for the okay it's it's it's tough it's but it's no doubt excellent quality work in that paper they did something important they realize that there is some connection between the fluctuation theorem and the fluctuation dissipation we saw yesterday only that the fluctuation dissipation that we saw yesterday is for equilibrium and this will be kind of a generalization out of equilibrium of the same thing for this this one this one yes this one is the generalization of fluctuation dissipation but applied to so then roughly at the same time because of this paper it is common especially in Europe that people call this thing and sometimes you will hear it mentioned like this as the galavoti Cohen theorem okay a name so now Jarsinski who's American despite the name found a different story which I don't have the time to tell you but very much on the same general frame so the crucial element is time reversal and it involves large deviations of something okay so you will see a lot of literature on the Jarsinski then crooks show that these two which came from let's say these two that came from different worlds so for example galavoti Cohen didn't know anything about her things get the time he realized that they were more or less aspects of the same thing we are going to see more like this kind of thing and then then it was shown for quantum etc my own contribution is to make a proof that is super simple with stochastic dynamics and so for a lot of people including myself this was the first thing they understood of the subject okay very good so sometimes you will hear it mentioned as the galavoti Cohen sometimes you will be here it mentioned as simply the fluctuation theorem or fluctuation relation this this is a similar subject and then there is starting from all these papers which ball the core of them was over by the 2000 there are 2000 at least or more more like three or four thousand papers which is enormous for physics on this subject experimental a lot various theoretical applications and different contexts there is an enormous activity but you see this is the result is this one it is also sometimes said called the probability of violation of the second principle because if I have a temperature gradient that way and I get energy going for some time in the wrong way I am getting energy from from the system in the middle if I am stirring water and for some time short time the water gives me back energy I am getting I am apparently violating the second principle second principle says but what does the second principle say it says on average over time I will give to the energy to the system and not the inverse so the second principle just a second is not a statement that is absolutely it is a statistical principle if you ask me is it true during 10 to the minus 10 seconds that the air cannot give me energy that it is not true for a very short time it is rather possible that I get lucky impacts it's on average over time that the second principle holds yes so isn't it that Jarzynski was about work I mean this yes relation about work yes it's a different context but once you see the proof it's the same things that are in play in fact crooks made a point of view that puts the two together look different because the protocol followed is different at least from an experimental standpoint because Jarzynski requires that you go from any an equilibrium state a to an equilibrium state b relax the system there and then bring it back and you ask what is happening in the forward trajectory backward trajectory whereas Bochko of course of life Evans Cohen Morris Galavoti Cohen they all are talking about stationary state dynamics exactly not all okay but there is equilibrium so yeah these are variants of the same the proofs are all quite similar the proofs are relatively easy imagine I am going to do it now and it's a result which is the most remarkable if you think if you want of the last 30 years of out of equilibrium and yet we can do it in one lesson if I would have to teach you in one lesson the renormalization group it wouldn't be possible so this is a remarkable thing some people okay and what is the main crucial element is time reversal out it's it's good it's going to be time reversal okay questions on this ah why does this have to do with a second principle because you see it's telling you that the probability that you're on the right side second principle is very much larger this is an exponential in time than the probability you get a lucky violation of the second principle also if the times I consider are larger this becomes much more this ratio becomes much rarer so if this time where 10 to the minus 24 there is no problem of violating the second principle during 10 to the minus 24 for the reason I told you in 10 to the 24 I get four impacts of molecules against my body and the chance is that three of them give me energy and one take takes it away is something if the time becomes large as usual it's like throwing a coin many times the probability that I am I get lucky and 10 to the 24 push me one way and give me energy and only 10 to the minus I don't know what push me the other way then it becomes like throwing a coin and getting heads 90% of the time if you do it two times it's easy if you do it well if you do it 10 times it's easy if you do it a million times getting 90% of heads is very rare you have to understand that the second principle is a statistical thing the other thing that you see is that also the size of the system doesn't matter however it doesn't matter for this relation but this curve will depend on the size of the system that the specific form of this curve and if the system is large this curve is much more peaked because it's natural that if you have five molecules the fluctuations are going to be large if you have 10 to the 24 it's again like throwing a coin many times sigma per unit time so yes okay so I think that we are ready so if we draw a horizontal line on that P sigma curve it will in yeah it will intersect at two points right let's say sigma 1 and sigma 2 yeah so I'm not seeing how the bottom relation will hold for both sigma 1 and sigma 2 oh because this sigma 1 has to be compared with this one and this one has to be compared with this one yeah so on right hand side p of plus sigma would be same but p of minus sigma would be slightly different yes no this this relation is it's true or let me put it in another way if you if you take the log of this one and and now you you do the h of sigma minus h of minus sigma you get a straight line okay it's purely linear you are saying if I am here so that's sigma here well it concerns yeah I get here so it's e to the minus t sigma 1 here the ratio e to the minus t sigma 1 this is sigma 1 and here the ratio is e to the minus t sigma 2 right the other value and so these two will accommodate so that you get those two ratios but like numerators are same precisely but the denominators seem different because that curve is increasing yes then how how could the ratios be same it can but let me see so do you say that these two values are same but the two quotients give you the same so these two are different these two these two do not coincide as these two do okay good thanks okay so by this definition it means that if we have a small measurement time we can extract more energy from the system but like in practice how is this t defined because like you can imagine if somehow you keep resetting the system really really fast you can extract a lot of energy good question this so what he's saying is can I be smart and look for these occasions and only connect on those occasions and not on the others this is a variant I think what you're saying or what's called a Maxwell demon and the answer is subtle given by Maxwell the answer is that for you to have the intelligence and the information to do it you are spending the energy whatever so the explanation usually is that the little demon who is connecting and disconnecting gets gets hot himself and then yes but basically it all comes from something very subtle that you have a tendency and then when you do quantum mechanics this becomes really serious you have a tendency to think yourself as if you were out of the physical world and you're observing something that follows the laws of physics the moment you realize that your brain and you are also following the thermodynamics of the thing you realize that if you had described correctly your thermodynamics completely then the whole process would cost you cannot do it that way to get out of this difficult state of mind the best thing is to invent a mechanism that is simple that does the demon and then you realize that you lose where you gained okay so let me remind you of what we said here about a large deviation and how it is done in general so what am I going to erase I think everything so two critics of the fluctuations here and three that have been made the result is amazing first critique it's not about the center of the distribution we have nothing to say about deviations around the peak which would be much more interesting if there were a relation but there isn't one second it is super general but when things are a bit too general they stop being interesting because something you can say about everything begins to be a bit incipid so that's one critique it's so general that it then ends up by being non informative this is the other critique and the third critique which is implicit and nobody says it this way is that these relations are very in general charsinski crooks galavoti coin etc except for the paper of galavoti and coin they're all very quite easy to to derive once you know what you want they are quite easy to derive so there is a there are people who say how can this be brilliant if it's easy okay this is a kind of prejudice because sometimes you have to know what calculation you want to do and then doing it sometimes is not the difficult part the difficult part is having the idea so I'm telling you this because the attitude towards this theorem is variable okay so let us recap a little bit what we said of large deviations so so for example imagine that we want an average quantity over time in our case it's going to be the power but let's do it so you have a fluctuating quantity and you want an average over time this is why I put the time here so how do I calculate this well the trick as we exactly like what we did when we went from canonical to micro canonical as I told you the trick is to put everything in the exponent just the same as we did for space for the free energy I'm doing now over time and now we have to say what is the dynamics of this so so this is POA no sorry this is not POA this is the delta simply that imposes this relation now POA is what first of all I have this integral that I just wrote thank you and now I have to say what is this so what is this this an integral over all the possible trajectories a sum over all possible trajectories of the probability of each trajectory and then of this quantity that is here so I'm just summing over all possible trajectories this is just putting in maths what we are doing in words so for every trajectory so every realization of the noise it will have a certain probability and then I calculate this quantity along the trajectory and I have to sum and of course in this we have to say how we start how we end the trajectory and everything on which ensemble of trajectories we're going to sum and now this is what we did more or less the first day where I when I said to you that large deviations and the thermodynamic formalism are the same thing with time playing the role of space and additivity okay so now that the only new thing that will appear is that now that we know we will imagine that the system follows the Kramers equation so and then I'll tell you why I'm choosing Kramers and not Fokker Planck and then this quantity here we can write it on the basis of the Kramers thing it's remember that when we do something that is averaged over trajectories we can use the Kramers formalism for the probability so this quantity here is in the notation I'm going to decide to start with Gibbs I think I called it like this I decided that way that we will start in equilibrium and the probability of the trajectories is putting e to the minus THK as a Kramers thing but now we want to add this term that we are measuring a on each time so this adds a term because now it's not only the probability of the trajectory which is given by this but we have this thing and this is taken into account by there is a plus here so there is a change in the operator that comes from the fact that this is the operator A this is the operator H and here we put all possible outcomes and this is this term not this one and this one because it's an extensive quantity in time so it's additive in times again we will assume that it's of the form some form of mu this concerns only this piece this is the part that depends on the system so the passage of canonical micro canonical to canonical I did it exactly in the same way I wrote the delta that imposes the micro canonical and then I said things are additive in space and so I suppose that there was a rule here it is exactly the same but only that additivity is now in time so if you want this would be kind of like doing thermodynamics in time and not in space this is what large deviation so sorry so you are replacing the integral between 0 and t of a of t by t times the operator a right yes so I am replacing this thing which is like measuring at each step by putting it in the exponent which does exactly the same thing measure shouldn't you have an integral also in that exponent not because the integral is here this is time independent a okay okay if it depends on time it's an integral I agree and so now now what do we get is the integral of d mu on the appropriate contour e to the mu t a bar and here minus I'm taking the t outside and here g of mu g of mu of course is something that somebody has to calculate for you so this is like in the analogy of canonical micro canonical this would be you fix exactly something and this would be like what you will see and now somebody calculated this for me which I assume that it has the large deviation form so this thing because it's additive in time this part I am assuming that is proportional to t and then just if you you can go back to what we did with the micro canonical to canonical passage now if the time is very large I can solve this by subtle point which means that I have to differentiate this see where the derivative of this is zero and so I get the for large t that point will dominate so the point that will dominate is in the appropriate value of mu so this is playing the role that in thermodynamics is played by temperature this is playing the role that the size plays in thermodynamics this all together is the micro canonical role the role that the micro canonical measure and this thing here with a g only is what we would call canonical sort of I mean I'm using an analogy nobody calls this canonical but it's a conjugate function what did I do wrong so the important thing to bear in mind is that doing large deviations is the same as doing the thermodynamic limit in the first case it's in time in the second case it's in space I didn't get that part like when you write e to the power minus t hk that camp how you introduce that hk from because we learned two days ago that if I want to generate the probability of all trajectories and some over them I get the cloud that is propagating and that this cloud of probability let's say that is propagating is given by hk this is what hk does for a living transform and time the probability this one okay so the cloud is transported by hk okay and at each very short time I measure a in order to calculate this integral yes so to switch between a little evolution and a measurement a little evolution and a measurement a little evolution and a measurement is the same as doing the two things together so this is how I construct this I am telling you in words there is a construction that can you can do more a bit well not rigorously but more completely with the path integral representation of this but I don't want to get into that no no a yes a is fluctuating but that's not taken into account it's taken into account by the probability of the trajectories and that is what hk with its distribution is doing so the fluctuations of a that you see here is simply taking a and measure it with a with a with a with a probability distribution of the trajectory but hk generates that so this is how the fluctuations disappeared this is the operator a added to the operator hk okay I don't get how you got to the violet part did you just like do the brackets and sandwich stuff yes I asked somebody to calculate this for my model this depends on the model and the only thing I assume is that the scaling form of this is of this form that's the only assumption now what why am I assuming this because I see what it is and I see that it is something that it's a longer trajectory a sum over things where the times add so I say okay this is like throwing a coin many times and then it's a product and then in the exponent it's a sum this is what you call normally the large deviation form or the Kramer function this is the assumption of course the form of G somebody has to give it to me and the other thing that I think you are going to say it I don't know if you are why you chose Kramer and not Fokker Planck yes I'm going to say to you now if you want is that I will want to use for my a the power you know because we are talking about power and remember I told you that Lanjevan which gives you Fokker Planck because of the way it's done where you have a Q dot and there is white noise the velocity is really badly defined remember that to do this I threw an inertia term and so what happens is that when I look at the velocity this one really doesn't exist and so it's tricky to find force times velocity when the velocity doesn't exist you can do it you can do it and then you get into the troubles that I told you about ito Stratonovich and things but you don't want to know at this stage so instead when you leave this thing here then eventually you can take it to zero but if you leave it then the velocity is well defined okay so this is why Kramer's is cheaper in this case anyway what we're going to use of Kramer's is very little okay so this it has to do with equilibrium situations only or from like the Gibbs okay okay sorry thank you I am starting in equilibrium okay so I have to go back to history and let me say these guys I don't know Boskov and Kuzolev because as I told you I never read the paper but these ones they did it for starting in equilibrium Galavoti and Cohen instead did it for a stationary state whatever it is so this one starts in equilibrium but then you are forcing it so you're kicking it out of equilibrium this one and it's the easiest one to prove so we start with Gibbs and then we will apply some H will contain some forcing and okay so Galavoti and Cohen did it in a way that you are in a stationary state but there is no equilibrium you were forcing it since time equals minus infinity while in this one this is called a transient because you start in equilibrium and then you apply the forcing and technically it's much easier you will see it's very easy the merit of Galavoti and Cohen is that they say no I apply F the forcing at minus infinity and then I'm in a stationary state so does this make the proof difficult it made their proof very difficult because they worked with a system that is with a very specific thermostat the thermostat that I gave you remember the the one that conserved the energy it's a purely deterministic thermostat so they had to deal with all the issues of ergodic theory and blah blah blah and then the proof is very difficult and then me and later on generalized by Leibovitz and Spohn did it for a stochastic system like the ones we're looking and the proof that was very difficult in Galavoti Cohen becomes almost trivial at least I mean for the people who I mean you are students but for the people who are really into the job it was very easy to read and in that case when the system is stochastic I am not going to do it and I will begin in Gibbs so I'm going to take the cheap away but in the presence of a bath with noise even the stationary context becomes easy so putting noise into your situation makes life easy and this is a generic thing of life when you have a dynamical system and you add noise to it you have a bath technically you get a stochastic equation but everything becomes infinitely easier why because all the question of ergodicity chaos etc you don't have to deal with it the noise does it for you so this is why stochastic thermodynamics as they call it is so successful because you can do things that in ergodic theory would be impossible ergodic theory means without noise you have to explain why the system explores everything with noise you saw that we saw that the Gibbs measure is thick is a stationary thing this is trivial two lines no you see that the Fokker Planck or the Kramer's operator do it for you instead if you want to show this from Hamilton pure without a bath then it's very very difficult one of the very very few examples is the Sinai billiard I gave you okay so that's the so Galawotian coin did the stationary one for a deterministic thermostat when we did the thing with stochastic the proof went from being for most people illegible to being trivial but we resigned one thing that they did we resigned the ergodic theory part because by using a bath we were almost putting it by hand so it wasn't completely for free it was a bit cheating for most people doing their gothic theory or doing it with a bath for many things it's the same for other things where you want to prove ergodic it results no so but if you're doing a system like a typically a molecule DNA or RNA molecule that you're opening like this in water you want a bath I mean you are in a bath what can you do okay maybe Jarsinski is something that starts in equilibrium mostly so you don't really need a bath although you can have it and then there are thousands upon thousands of variants of this I I gave some in my notes but there are thousands more okay I will give you a bit more literature if you want to read there are many things that happen the problem with the literature is not that there isn't a literature is that there is too much okay now so you agree with me that if I can calculate the g and prove something for the g then to get something for this quantity which is the complete probability it's going to be easy okay so now I'm going to prove the result for the g that makes the magic of all okay I'm going to erase this part the history and now okay so now remember that g and this which was h are related through our stand as the micro canonical average with the canonical okay so now the big thing is the following so I'm going to repeat a formula that we wrote already if I find it I will find it eventually so I get the science right yes remember that we did and then I said and this is the important part that precisely this was time reversal and precisely the violation of time reversal is which is the where I got the sigma from and all this is evaluated here is already the seed of the fluctuations here and if you want this is the relation that will tell us the fluctuation theorem it all comes from remember this detailed balance kind of thing but broken because we had a forcing and the term normally I have kind of detailed balance except that I have to change the sign of p but it's broken by precisely the entropy production okay from here and algebra we get the fluctuations here but the point to understand is that this is the important thing so now look what g is in our case a is going to be f dot b I did it in general but the quantity of which I am talking of which I want the large deviations is the power okay I did it with an a so that you saw the what was general but this is what we want now okay so now I'm going to put here a f dot b and I'm going to see now a variant of this how does this guy transform the complete guy here it's a minus and if something happens with a new don't worry okay sorry why are you changing sign in the exponents in the exponents I change sign here yes down it's on the right it's because I made a mistake sorry yeah sorry I'm sorry I have slight variation of this I'm going to conjugate here so but it's the same relation and here I have so conjugation only changes the sign of this sorry when I change the sign but conjugation does nothing to this and the only thing I did is I used the result on top for HK and I just added this term and then it's it's simple how this how this happens I think it's the signs are good because they are in my notes printed notes this way but check it you see it's nothing it's just that you transform this one and you transform this one transforming this one gives you an extra term which is this one over here no the one that is here a one and transforming this one doesn't give you anything so you get mutant start there be a temperature on the left-hand side absolutely thank you okay so I see that you're a bit lost I'm going to repeat it I probably said it in a model way you do first this piece and this should be the thing upstairs up and then you add this thing and it adds this which is basically the same it doesn't do anything to that okay so and here comes the beautiful thing the violation of time reversal is this guy here otherwise the rest is the same and I am looking yes yes yes of course thank you very much again and here of you minus yes thank you it's just the top one is it missing in the top one no okay so the important relation that we get here is the following and this is with this we are almost done if you look at what g of mu does because when you put this in the exponent and your time reverse you can convert it into this and you will get this equation and now we're going to do it you have this here so so you have this in the in the exponent and you have g e to the tg of mu this is you start with Gibbs then you apply e to the hk minus let's say mu f dot v divided by t and then you go anywhere you like and now we do the trick that we've done a thousand times no a thousand yesterday we've done a thousand time we're going to reverse the times and use the the this property of time reversal which is broken by this term so now I apply this thing here and just like yesterday this this will be Gibbs this will become like this and now I only have to conjugate what is on top this is exactly the same trick we did yesterday to get to get on saga no I invert the limits and I take the dagger of what is inside and now I apply this thing here just like we did yesterday and so I can substitute by this so this thing here will be substituted by e to the beta h e to the minus beta h and then here there's going to be this thing here check the on saga relations how we did them we did the same trick only that we did it with Fokker Planck and we didn't do it with kramers but this time we are a bit forced to use Kramer for the reason I told you and exactly like happened yesterday this one together the one this one gives me the flat measure and this one together with this one gives me Gibbs I take the normalization here and put it here and now here I get simply the same thing with but this I look at it and I recognize that I have this so I have proven something about the large deviations now let us recap all the times necessary because we are done basically is what you would call the okay if instead of time we had space you would be the temperature and you would say I fix the energy and conjugate to the energy I have the temperature here we are fixing the work in time this is a bit like doing micro canonical but in time and this is the conjugate thing to the work it doesn't have a name as far as I know it is the conjugate legend variable I don't know how you call it okay or if you want you can say that this is the Laplace transform of the probability and this is the Laplace variable as far as I know it doesn't have a name in thermodynamics you have volume and it gives you pressure and so on but in this sense where it's time the thing it can come up it has to come up look it's fluctuating okay so what have we used what have we done let us assume that I didn't this one is definitely true let us assume that everything I did is correct so let us recap this is the kind of detail balance see thing for grammars I don't call it detail balance because people I don't know if people call detail balance when you have to change the sign of velocities but it plays a role of time reversal and it's morally detail balance but the important thing is that it is time reversal to get a fluctuation theorem you need a forcing so this is the grammars one with with a force no there's nothing to prove if everything derives from a potential now when you apply detail balance there is one thing sorry time reversal that violates time reversal the trick of constructing a fluctuation theorem is that you have to look at precisely the quantity that violates time reversal and look at the fluctuations of those why because when you want to construct your large deviation function with a new now detail balance gives you this one and new is new so this is going to give you some sort of symmetry thanks to this breaking of symmetry and it will reflect but yeah this small calculation where I had to use exactly time reversal exactly as I used it yesterday for the correlations and I get this one directly notice that I used Gibbs to start which has made it particularly easy because these exponential conspired like yesterday to give you the flat measure and Gibbs and so now we have this and now it's easy because remember that it via subtle point there is a direct relation between this and the probability we will do it in a second but and now we go to the integral we had at the beginning and and now things are very easy ah this thing here for reasons I don't really understand it's usually called the label bits spawn relation I don't know why I mean they used it for sure but it's something that was there I mean it's clearly part of the fluctuations here okay so now we have to go back to our calculation of P now it's really very easy I don't even need to do large times here this I don't even need this is from minus I infinity to plus I infinity now I'm going to use here I'm going to use this famous thing that we derived but mu is an integration variable so what I can do is here put one minus mu t sigma here leave t one minus mu g of t minus mu but I have done what I have with a minus so I get the mu but I have added that e and this one that I added it without any explanation so now I have to put it back because otherwise it won't work and this is minus t sigma with a plus with a plus I this is just a change of variables I'm doing nothing and now this integral because mu is a dummy variable you could here put one minus mu and change the this is the same integral as this one but with a change of sign so and it's finished so just by putting inside this famous relation you're in one second in the thing if you want to understand the logic of what we've done in a cheap really cheap example if you go to my notes the six notes there is a little example that I think was done for the first time by Abhishek Dar I think I took it from him it's a very simple example where you see how a discrete symmetry that is broken by a term gives you a large deviation relation just in the easiest possible example so if you want to check it in a very very very easy example you get it there but the idea you I hope you follow the idea it is it is very simple the time reversal property gives you an extra term that breaks it that extra term we said yesterday and it was very important that extra term is the precisely the entropy production so you choose that quantity for the large deviations so that the large deviation you're doing gets together with the extra term that is breaking and this is what gives you this relation okay so after this little tour de force we are in time sorry five minutes late you see that the proof is not terribly complicated you just have to follow the steps and of course you don't have so much experience but it done this way it was accepted by everybody as something very simple and and you have to bear in mind that this is one of the big results of out of equilibrium statistical mechanics which perhaps will convince you how poor we are big results and how small the big results are but this is what we have could you say it again why the minus sign from the change of variable doesn't matter because if you change the sign of a variable there is a Jacobian but then you change the limits and one compensates the other this is a typical mistake one makes always it's an integral of a positive function has to give you a probability so even if you integrated the other way around it has to be positive who of you think that more or less you followed what we did today raising your hands convincingly doing this hmm okay anyway if you like I can so much part of my PhD thesis was an experimental test of the fluctuation that would be nice so I can do a simple 10 or 15 minute tour of how an experiment would go about testing that I think would be give it flesh when would you do that do it now if people are willing to skip the coffee but I think no no okay let's do the following you're supposed to start your presentations at 11 but let's take 15 minute detour to look at how experiments would approach the fluctuation relation what are the requirements that go in from that side so this is again in the in the vein of what I said during my first lecture why is it important for a theorist to know about experiments why is it important for an experimentalist know about here so we reconvene at 11 here good thank you very much my question