Here in Tokora for our lunchtime update for Targa Banvena, third and final day the cars have just come off four special stages this morning, heading out into the monster 67 kilometer stage this afternoon, the penultimate stage of the event before the teams head back to Rotorua. Updates this morning, issues for Bevan Claridge, leader in the middleman classic two-wheel drive class. He broke a diff but managed to change it in between stages and it didn't actually cost him any time. Just done a diff on the last stage and always we've got one in the van so hopefully we'll get started again before we run out of time. Looking at some of the other results and dramas from this morning over the first four stages, looking at the instra.com modern two-wheel drive class, Rob Darrington has unfortunately gone off in the last stage this morning, special stage 19, off road that's confirmed by Simon Clarke who's running second at that stage, so that should put Simon Clarke into the lead of the modern two-wheel drive class, really really disappointing for Rob Darrington because he's done such a good job this weekend. Glenn Inkster, we heard a story that his misfire had returned this morning, they thought they'd cured it last night but then it mysteriously disappeared again in special stage 19 before they came in the head to lunch so they still maintain their lead of around three minutes over Nic de Waal who's putting on such a fantastic display. So we're not too far away from getting out of here, heading back to Rotorua. After the 67 kilometer long stage we've got a small service back at ridges in Rotorua and then it's up to Paradise Valley for the final stage of the event. We'll bring you an end of day wrap up and all of the results from Targa Bambina. We'll talk to you then.


That's the old straight shooter again. You remember me, don't you? From that other TV show, Preventive Maintenance. Yeah, you know, I was a good-luckin' one. Start this here show off with a few words of wisdom. Or something. Anyway, in the last show, we talked about Preventive Maintenance. How to take care of us distributors so we keep working right. In this show, we're going to talk about operation as right. A fella's going to start telling you about that in a minute. But before he starts, I want to say a few words about a couple of things that are pretty important. You know, just because we're talking about operation now, don't forget about Preventive Maintenance. Preventive Maintenance shouldn't stop when operation starts. PM should never stop. Always keep watchin' for things goin' wrong. Always keep checkin' if things are working right. Always keep cleanin' it. Always keep doin' Preventive Maintenance. That's the only way to stay on top of it. So keep thinkin' at PM, even when you're operating. Okay? Think you should always keep thinkin' about two, of course. And that's safety. That's maybe one of them dear devils. If you're in your late car drive, you know, you can ignore safety and have a real blast with a distributor. And that ain't no joke. It's true. A distributor can blow up. Or catch on fire. Guys, gettin' blowed up ain't my idea of a good time.  So take it easy. Use some common sense. Learn to say what you do things. And then do it that way, okay? You're gonna learn a lot about safety right here in this year program. Well, that's all I got. I just think that things are important. And I wanted to get you thinkin' about them, too. Okay. On with the show. It is required for application. The application temperature varies with the different types of asphalt. Obtain this information from your supervisor. Before you start heating the asphalt, though. And I mean each time you do it. There are four safety steps that must be completed. First is to position the distributor so the tongue is facing the wind. This reduces the chance of the burners blowing up. The second preheating safety step is to remove the fire extinguisher from the distributor and place it 50 feet away from the distributor. If you have a fire, the last place you want the fire extinguisher to be is in the middle of the fire. The third safety step you must always perform before heating the asphalt is to check the amount of asphalt that is in the distributor. To do this, unscrew the tank lid and inspect the amount of asphalt in the tank. The asphalt level must be at least six inches above the top burner pipe. Heating asphalt that is less than six inches above the top burner pipe of the distributor is a good way to fill up the distributor. So, don't do it. Close the tank lid and secure it tightly before heating. And the fourth preheating safety step, always check the temperature of the asphalt in the distributor before starting to heat it. This will tell you how much you have to heat the asphalt or whether you have to heat it at all. You check the asphalt temperature by removing the thermometer on the side of the tank and reading it. Remember, as you learned in your preventive maintenance instruction, the thermometer well must contain motor oil for accurate reading. After making your check, replace the thermometer and tighten it completely. Okay, now you're ready to start heating the asphalt safely. To light the burners, first open the valve on the propane tank. Make sure the dampers are all the way open. Then, walk to the main burner valve and open it. Next, open the first burner valve, light it, and adjust the flame. Then do the same on the other burner. And after the burners are lit, recheck your dampers to make sure they are at the full open position. Next, go to the back end of the unit to check and set the three main control valves. Set all three valves to the circulate position. First, the four-way valve. Second, the two-way valve. And third, the three-way or spray bar valve. Now, move to the engine and start it. Next, engage the pump because the asphalt must be circulated anytime it is being heated. Now, return to the thermometer for a second reading. Again, be sure to replace and tighten it when you're done with your check. As the asphalt is being heated, return to the thermometer and check the asphalt temperature every two or three minutes until the desired temperature is reached. Once the desired temperature is reached, shut off the burners in the following sequence. First, turn off both burner valves, then the main burner valve, and last, turn off the propane tank valve. Always turning off the valves in this sequence is the best way to ensure that all valves are turned off, which prevents the chance of raw propane accumulating at the burners when they are relit. Now, you're ready to hook up the distributor to the truck, close to the distributor, and set the trailer tongue slightly above the height of the truck hitch. Then, direct the truck back slowly until the trailer tongue and the truck hitch are aligned. Lower the tongue onto the hitch and latch it. Make sure the hitch is locked or secured. Raise the jack stand and lock it into place. Plug in the trailer lights and the emergency breakaway cable to the truck frame. Next, go to the back end of the distributor and check that all the lights are working. Also, make sure the spray bar is at the highest position. Then, complete a walk around inspection on the chalk block's side of the unit. Finally, remove the chalk blocks and place them on the distributor. Okay, you've got the asphalt heated and the distributor hooked up to the truck. Now, you're ready to go to the job site.  One of those adjustments is the height of the spray bar. The length of the standard hard hat is about the right height for the spray bar on this type of distributor. So your hard hat can be used as a guide in setting the height. This is the pump tachometer or gallon per minute gauge. Before you start shooting, you will have to set the gauge to the proper gallon per minute setting for your job and adjust the engine speed to achieve that setting. The gallon per minute setting varies from job to job and is based on such factors as material type and road design. Another thing you have to do before starting the shooting operation is check the operational condition of the spray bar. Make a couple of test shots. Check all nozzles at least twice. Make sure they're all properly aligned and working. The spray bar is opened and closed with the three-way spray bar valve which is located right next to the operator's chair. You open the valve by pulling the handle to the operator's side. You close it by pushing the handle to the farthest position from the operator and then bringing it up to the center position. This is how the spray bar is operated when making shots on the job. And speaking of making shots on the job, the distributor is now ready for the shooting operation. Every shot you make with the distributor, you must be up to the proper speed when you start the shot so that you are applying the proper amount of asphalt right from the beginning of the shot. The speed needed depends on the amount or application rate of asphalt needed which varies from job to job, so always get a running start. And make sure it's long enough for you to get up to the proper speed by the time you reach the desired starting point of the shot. When you reach the desired starting point of the shot, open the spray bar valve and you're on your way making your shot. Watch the spray throughout the shot to make sure that you're getting a smooth application of material to the surface. To end a shot, just close the spray bar valve. The condition of the spray bar is a critical factor in the shooting operation. If anything is wrong with the spray bar, it will affect your asphalt application. So, you have to treat the spray bar with care and protect it from damage and contamination. Don't just set it and forget it. Any time there's a chance that it might hit something while you're maneuvering the distributor, raise the spray bar to the highest position. In particular, raise it any time you have to take the distributor off the flat driving surface and whenever you transport the distributor. Cleaning the distributor is the final phase of the operation process and a very important one. If you don't keep your distributor clean, it won't keep working very long. It must be cleaned after every job before the asphalt cools. If you let the asphalt cool, you'll have a clogged distributor to feel clean the pump and bars. First, drop the engine RPM as to a high idle and disengage the pump clutch. Then reset the valves. The two-way valve to the fill position, the four-way valve to the drain position, and the three-way valve to the distribute position. Now, reengage the pump clutch to spray asphalt out the pump and bars. Next, open the kerosene valve completely, which allows kerosene to flow through the pump and spray bar and flush them. When they're flushed out, turn off the kerosene valve and close the three-way valve by moving the handle first to the farthest position from the operator and then up to the center position. Disengage the pump clutch and shut down the engine. Complete the field cleaning of the distributor by spraying the spray bars with kerosene while they're still warm. This will keep the bars and nozzles from being coated with cold asphalt. Whoops! Well, that just goes to show that all equipment needs PM, right? I called the cleaning we just saw field cleaning. This means that you do this cleaning out in the field right after you finish the job. After you get back to the yard, you'll have to give the distributor a final flush and cleaning because you probably won't get it as clean as it should be with your field cleaning. You may even wish to leave kerosene in the pump and the spray bars to eliminate clogging in low points. That's all we come to the end. You've just seen the basics of operating a 600-gallon asphalt distributor. We've gone from heating the asphalt, to hooking up the distributor to the truck, to adjusting the distributor for shooting, to the shooting operation, and finally to cleaning the distributor. You have to do all these things when operating an asphalt distributor. Do them like you've seen in this program and you will have many successful and safe shooting operations. Well, that's operation. In these two shows on the distributor, we covered the basics of preventing maintenance and operation. I said the basics, not everything. Everything would take a lot more time than we got. Anyway, you might have some questions that weren't answered in these shows. If you do, ask your instructors. They'll fill you in on the details. Another place you can get more information, especially about asphalt, is the Asphalt Institute. Them folks got a lot of good information. Well, I'd like to stay here and chat with you longer, but I gotta hurry up and get back to the yard for my final flush. So, I'll be signing off now. I'll probably see you up the road sometime. Till then, this is the old strength shooter saying, may all your shots be straight and all your jobs be great.  So long. Okay, get it there, boy. Yeah. Let's go. Yeah. Come on, move it. I gotta get this out of my pipes. It's driving me crazy. Yeah. Yeah.


Holy heck we just checked out Justice League. I'm Ryan and I'm Trent and we are movie nerds If this is your first time here and you want to know a little bit more about movies Make sure you hit that subscribe button and hit the bell notification so you don't miss anything here at movie nerds Okay, guys So the basic premise of the movie Batman and Wonder Woman are putting a team of heroes together to try and save the world from this unspeakable evil from a distant planet in Stepan Wolf and the Parademons That's pretty much it simple as that. Yeah, Superman's dead or Izzy. We don't know But what do you think of the movie? now I actually liked it. This is the first DC movie I've liked ever and It's ever ever From the DC you not the old stuff that's right. I didn't have anything get Michael Keaton. That's fine No, Keaton's fantastic But this is that this is that this is the best of the bunch best of a bad bunch I think sure everything's everything's flowing much better There's still major glaring issues. Yeah, but I really enjoyed it. It was a fun movie awesome I think there was a lot of things wrong with the fact that I mean you can't blame him but Zack Snyder left the film just after the movie had kind of finished production and it was going to pre-production then Joss Whedon took over and Joss decided that you know some of this isn't gonna work some executives apparently had seen it and said this isn't working at all And he came back in and did some reshoots and some of those reshoots in This is a little bit of a spoiler if you don't know he's in the movie then you're just crazy But Superman is obviously in the movie and Those reshoots included Henry Cavill and Henry Cavill at the time had a mustache mustache mustache how do you say it and Legally they couldn't get rid of it So they had to bring in CGI for this thing to delete it and my god. Is it obvious? They're opening of the movie is a sequence where Henry Cavill is talking or sorry Superman Clark Kent is talking to screen and They used a shot that doesn't have the mustache or it's being digitally removed and it's so obvious His lip is ridiculously thin. There's very little facial movements and teeth are weird Yeah, it's just really awkward and stuff like that happens throughout the whole movie and to open it up like that I kind of just had this glaring thing where I would just notice the whole movie that this is a different bit Anyway, look the most the better part of this movie. I quite enjoyed it was just the glaring issues that the different directors came in But you can't blame them. No, just Zach Snyder had family issues and family problems, which are massive and that's cool But overall the movie was good. Yeah, I had issues with it. What were yours? My shoe is Snyder. I don't like Snyder's movies. I never will watch them as terrible 300 everything He has the same You know my feelings a lot of these reviews I'm sorry, I learn a lot about you in them and you're telling things that I don't want to hear and that's fine It's your opinion. I don't agree, but it's your opinion, but I just think he's glossy really muddy murky tone There's a lot of confusing elements and what you just said, but I'll accept it glossy and muddy It's glossy and muddy. Everything's really shiny and CGI. It just doesn't it doesn't blend.  He looked like Aries from Wonder Woman.  It's it's very organic and human, but it's not it's alien Why don't they just make it like a? A bit of prosthetics like rather than making a CGI Just make a prosthetics like there's no reason for him to be CGI other than him being you know eight foot tall But you know they did Gandalf and Lord of the Rings ten years ago. Why can't we do this now? Why are we using CGI when we don't need to and that put me off to I agree what about comedy? I mean this movie was hilarious I mean oil to a degree and we had the flash and he is a really funny character Barry Allen What do you think I loved it Whedon you can tell he's humor straight away. Yeah, absolutely It seemed awkward at times when obviously with the issue Yeah, but it was a breath of fresh air into an otherwise gloomy and murky Yeah, I think I think the the relationship the flash has with all the different characters is really hilarious I love a lot of the sequences within him in the movie And I think he actually made the movie and without him in it. I don't think it would be the same film Ben Affleck is also really good as Batman Wonder Woman is of course amazing as Wonder Woman Sorry, Wonder Woman Gal Gadot is amazing as Wonder Woman. She is Wonder Woman. Yes, and Jason Moore is awesome as Aquaman What about cyborg? That's where I was going. That's where you're going. I didn't like him. He brought nothing to the movie The set is being disabled behind us. That's fine. That's right In the meantime, I need to just pat you on the back.   I would have preferred Marsha manhunter in this But no, we didn't get that.  He doesn't but he could make one I'm sure because remember he had feeling in his toes Right, so surely would have feeling somewhere. We're getting a little bit weird But I liked it nonetheless Cyborg was really weird Yeah, he was a very mopey character and I don't think these milk movies when they're like a big team need it I think I think like Thor Ragnarok really worked because you had Thor and everyone was just trying to be silly They were like big goofy and they were fun And this movie just had old cyborg being mopey all the time and he was a very prominent character He was an old lot. Yeah, it was actually useful. Normally just sits at the watchtower doing absolute stuff. Oh, yeah What was the best part of the movie without spoiling anything for people? Oh post-credit scene. There's two Oh my god, the second post-credit scene is noteworthy. We can't spoil it, but stay for the end of credits Please it is one of the best ends and credit sequences probably I've ever seen anything the Marvel does it beats and destroys This is unreal. It's it's setting up something and it's gonna set up something amazing I can't wait for to see what where it's going But my favorite bit was I mean as we said Superman's obviously in the film Was the connection he made with flash and their repertoire between each other was absolutely great Yeah, and Superman is Superman in this movie. He's he's bright. He's colorful. He's happy Everything that he needs to be right big blue Boy Scout Any I think I think it's the Superman the Joss Whedon made and not that Zack Snyder made No, you could tell there was color scheme differences. Yeah, absolutely But yeah, what was what was the worst part of that? My aside from cyborg my worst part was probably Opening yeah, the start was really slow. It was very trailer-ish Yeah, we did a lot of the things that happened in the start were in the initial trailers So I kind of felt like we were just retrading old ground. Yeah. Yeah, I quite agree Yeah, they could have gotten to the point more and actually Didn't think Mo Moa was as good as Aquaman that I thought he was going to be my man. This guy is No one wants to see this Aquaman He didn't talk to fish So he didn't do any kind of doing fishing. Yeah, he did nothing He did nothing of Aquaman that would normally do you're right? It's very good point I didn't really think about it too much, but you make a very good point My worst part was the CGI over Clark Kent and Superman's lips It was so obvious and I couldn't help but see it every time it happened There was other things that would this happen to like The flash is Ezra Miller. He had a different haircut every reshot. Yeah, Ben Affleck must have gained weight Because every shot that he was in he was fatter. That's fine. It's nothing. Yeah, and his hair went really gray So, you know, I don't know. I guess there's nothing really they could have done, but It was just so obvious to pull me out of the film every time and that's why I didn't like that part of the movie So is it any good? I'm gonna say yes For what it is It's don't go in looking for a really deep plot Steppenwolf talked about mother boxes and kind of stuff that if you're not a comic fan or a new gods fan Kirby fan you wouldn't have a clue what's going on But it is a really fun romp of a movie. It is fun. It's really yeah, it's really enjoyable Look, it's barely gonna get a yes from me But it gets over the line with a lot of the comedy and the action sequences in it It does a lot better than what Batman v Superman did with that kind of action It's poos and shits all over suicide squad, but it probably not as good as Wonder Woman It's in the same line and I didn't love Wonder Woman either, but it's good It's worthwhile seeing it and I would definitely check it out in the cinema My name is Ryan and I'm Trent and we'll check you out next time. We're out the cinema. See you soon


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Yeah, I need you. OK, we can do it like this. And then again, like this. A little higher. A little higher, like this. Perfect. Thank you. Yes, exactly. Thank you very much. OK, so we talked about the assumptions. Now we are going to proceed with the assumptions. So one thing that I want to emphasize before we are going to use a mathematical trick. So in this course, I think up until this point, OK, David gave a derivation of his work, his results on this, you know, KL divergence, the difference between KL divergence and the mismatch costs. But yeah, this was a derivation that was actually smooth, I think, in a sense. In the sense that we didn't have to do really any mathematical tricks, and so on and so forth. This is the first time in the course that we are going to make a proof. And we chose to make this, because I think it's really, I mean, yeah, it gives you an understanding of how we make proofs in stochastic thermodynamics, OK? And so we are going to walk through the proof together. It is, I think, the most comfortable proof of any of the TURs if you don't know statistical inference. And I didn't know statistical inference when I actually started learning about a TUR. So I hope that we will make it true. So yes, we talked about this time reversal symmetry, OK? One of the things that we actually want to do right now is to sort of come up with a mathematical trick that captures it. So we are considering this good old fluctuation theorem, theorem, this strong fluctuation theorem, OK? Now one of the things that we can do is, let's take this, here are your assumptions. Let's go. This is one thing. And the other thing that we, I mean, another way that to express this kind of a thing is just this, right? So when we are dealing with fluctuation theorems, we consider entropy production as a random variable that can take both negative and positive values. Now we talked about time reversal symmetry and so on and so forth. So we are asking, what if we come up with a mathematical trick that actually allows to only consider the positive values? Because we have time reversal symmetry and blah, blah, blah. We need to make some sort of a connection, OK, in between. So based on this kind of an expression, we are going to try and come up with a probability distribution that is different like transformed but identical to the original one, probability distribution that actually describes the behavior of your random variable as entropy production, OK? And the way that we do this is, we know that this, this is normalized, right? This is equal to one. OK, one thing that we can express this, one way to express this is take the integral from minus infinity to infinity, split it into two components that goes, one of them goes from minus infinity to zero, the other one goes from zero to infinity, OK? Take this, so this is still one, right? So now, what we can do is to, what is going on? OK, this one is the iPhone. So now what we can do is to actually use this expression over here and put it here. Because we know that one thing to actually, so these are describing this, you're taking the integral from minus infinity to zero where you have the entropy production random variable where it can take the negative values. So it's actually describing the behavior of P of minus sigma, right? And how we can write P minus sigma by the strong fluctuation theorem, we can write it in this form. So I'm gonna take this, OK? And put it over here, inserted here, OK? And when I do it, we will see that we can do this. But actually I'm cheating here because now the way we wrote this is actually just considering the values of the entropy production. It goes from zero to, that considers like this, all of the like the positive values, OK? So this is like what I'm getting at. If we consider the steps that we have taken so far, we're actually manipulating the probability distribution that originally characterizes this entropy production. And we're trying to obtain a new modified probability distribution that characterizes this entropy production. And the way that we can write it is as follows. You see it on the screen as well, OK? Do you see it? Sorry, Ben? What? From zero to infinity, the limits. Oh, Jesus, yes, of course, sorry. Thank you very much. Exactly so. Thanks a lot, yes. So this is how you get to this expression that we see on the screen. And this allows us to define a new transformed probability distribution, Q of sigma, that is equal to this one. And this is one. So you took your random variable entropy production that can take values from minus infinity to infinity and you have a probability distribution over this random variable that characterizes it. And you took it and then you modified it, you're now treating it as if it's a random variable that can take values between zero and infinity and there's a probability distribution, the Q that we just defined on the screen that allows you to investigate the behavior of the statistics of this random variable in a way that is identical to the original probability distribution, P of sigma. Because what we did here is just mathematical manipulation, nothing else, you're physically describing the same behavior, OK? Let's say that we define this equivalent random variable and so on and so forth. One thing that is important to us in thermodynamic uncertainty relations, it's the average entropy production, right? So let's compute the average entropy production. Now, in terms of this modified probability distribution, Q of sigma. OK, so I'm going to walk you through the steps. What you're doing here is you all know how to write this average entropy production, right? Is there a question? No, OK, it's there. OK, so you're writing this and we saw that basically in a equivalent way of expressing this first part over here is to actually, so let's, OK. Now maybe I should actually use the board here. Sorry about that. This is my mistake on my side. Yeah, one thing that is dangerous is that if you do one thing for like 100 times, like 1,000 times, because I had to do it in like, I don't know, a year ago or so, more than a year ago, I don't know. But yeah, once you do it, it's it looks so trivial to you, but it's when it's not actually so trivial, OK? So yeah, let's take it, split it into two. There's one thing that I want to ask you. So also let's write the remind you this, OK? When we didn't have the sigma term over here, we didn't really have to account for like this minus sign and so on and so forth that would actually follow from the fact that we are considering the random variable that can take values from minus infinity to 0. I mean, the random variable can take values, but the probability distribution itself is like just still the probability distribution itself, so you don't have to account for any kind of like this negative-valued behavior, right? So can we do the same thing here? We have a sigma term. We're not going to be able to do the same exact thing here. Now we need to account for this negative behavior that is like shown by the fact that we're taking the integral from minus infinity to 0. So the way to express these things now, it's a bit different. So let's think about it. No, it's not going to be like that. OK, now I'm going to manipulate it into this one. But yeah, it will come out like. We have this identity 1 from the integral from 0 to infinity, right? But now what we're going to have, because the sigma is taking on the negative values, do you see the difference? There's a minus sign. This is the computer, yeah. OK, I think I can do it. I'm a big person. OK, it's good. OK. OK, so what we did is actually replicating the steps over there. But when we carried out these steps over there, again, we didn't have to account for the negative values that the sigma entropy production can take. But now we have to account for that, because when we're computing the mean sigma, every sigma, we have this sigma term over here that we're multiplying. And it needs to be accounted from this integral, because you're taking this integral from minus infinity to 0. This makes sense, yeah? OK, great. But of course, what we want to always do is to go back to this transformed random variable that is described by this new probability distribution, q of sigma, right? So algebra, OK? Is this good? OK, just multiply. Just put q sigma here. You will recover this one. And the way to write this is hyperbolic. This is hyperbolic tangent. Do we care about it? Not really, but mathematics. OK, did we recover this one? Yes, we did. OK, everything's good. I will make these slides available on Slack. OK, of course, in the same way, you can compute things like, for example, the second moment. And one of the coolest things is that if you're computing the moment n and n is even, you can basically write this. The average, if it's computed with respect to p, like the second moment, if it's computed with respect to p, it's going to be the same thing. It's computed with respect to q. Because when you say, if you don't have a sigma term and if you only have the sigma term with like this nth, what is this?  Order, yes, nth order. Then what you can do is basically take the average with respect to b. It will be the same if you take the average with respect to q. Sorry. OK, so this is good. So now we have something else, one more thing. And then we're going to, things are going to get more, I don't know, more familiar. This is, I don't know, this is not super easy to prove. Again, the main, sort of like the gist of this lecture when we provide this kind of a derivation is not to actually sort of impose on you, oh, you should know how to drive a joint strong fluctuation theorem that we're going to see here, but to know how they are using fluctuation theorems to actually derive some different bonds in thermodynamics, in this case, thermodynamic uncertainty relations. So one thing that we had, remember the other strong fluctuation theorem that we had? Oops. It's this one. Each one includes one random variable. Each one includes entropy production, right? So there is a class of complementary fluctuation theorems that actually describes the statistics of joint random variables. And in our situation, the joint random variables of interest is they actually define, they come in pairs. And it's always mostly like the entropy production and another current. And the reason that we're interested in this is that, again, one thing that I should emphasize is how do we characterize, for example, non-equilibrium behavior, non-equilibrium steady state to start with? Gulce, we have another question from Francesco. Francesco, yes. Why does the average over Q should be any better than the average over P? Any way different? Any, why do you prefer the average over Q with respect to the average over P? Well, yeah, when I started this derivation, I told you that, yeah, there are two things. It's a mathematical theory that we're considering that comes with the second part with an intuition. We are using time reversal symmetry that allows us to speak about the statistics of the entropy production and the values that it can take. So, and when we talked about it, entropy product, we see something, the entropy production is a random variable that can take both negative and positive values. Now we want to treat it because there is time reversal symmetry because it's interesting, because it's working in the proof, actually, mainly. Just as in the, I mean, I take it in the same way of Shannon's typical set. Sometimes you prove things because they work, okay? And with an intuition, of course. Shannon has an intuition about this typical behavior. We have an intuition about how the behavior of the random variable should look like. So what we are doing is defining, modifying this probability distribution so that this entropy random variable can take only positive values, okay? Does this answer your question, Francesca? Okay. Oh, it's a, it's a, it's a cold, okay? Okay, okay, I mean, yeah, thank you, Francesca, okay. So yeah, there is this, did I talk about, okay, we talked about this flag, what it is? Okay, send me a message from the Slack channel. Do you have an insight on this, David? You don't? Okay, he doesn't. We can talk about it. Okay, so yeah, we talked about the stroke fluctuation theorem and I emphasize that now we are going to take care of this. Oh yeah, we had non-viennishing currents to characterize non-equilibrium behavior, non-viennishing currents, they generate entropy, entropy production. So I keep also saying that in stochastic thermodynamics, the core concept that you're interested in is it's not, you know, like, oh, entropy production is a thermodynamic quantity itself and so on and so forth, but like the statistics, the probability distributions and the shape of the probability distributions, how they behave, how they do not behave, we are caring about this. So the reason that people came up with joint stroke fluctuation theorems was to sort of understand the behavior of the probability distributions that are characterizing the joint behavior of currents and the entropy production that is resulting due to these currents. Okay? So, okay. The proofs of the TURs, I don't know of any one of them, maybe you do, that's actually intuitive. They tend to be really messy. This is by far one of the simplest, cleanest ones. In particular, the original ones and many of the subsequent ones use what's called level 2.5, large deviation theory, and it's a horrible mess. You can see how Gould is reacting to it. We're not gonna be talking about any of that stuff. It's basically just, there's a phrase that sometimes applied to certain scientists that they're called algebraic terrorists, that they're able to come in and just spew this stuff and blow everybody away. And that's what these actually almost are. It's amazing that you go on through pages and pages of algebra using all kinds of mathematics that you will not have encountered before to finally, at the end of the day, magically, who knows what the intuition was.  This is one of the simplest. Believe it or not, but there it is. Thank you very much actually, yeah. Because if I say something like this, it's risky. Because, yeah, I think he's the important person in the room that you, yeah, exactly. So, yeah, thank you very much, yeah, okay. So, yeah, but David is right. I think one of the, yeah, they're all messy. They're not really likeable. One of the, I think the most beautiful ones is actually how you drive from statistical inference from Cromerabandt, because then you're treating the fluctuations in the entropy production as if they are errors in your statistical estimator. So, you're even without inserting any kind of thermodynamics. You can find the thermodynamic uncertainty relation. And, yeah, the way that this precision term made sense to me just for the first time ever after months of actually working on TURs, one thing that I realized is that, ah, actually, you know, this variance term, it's coming from the, you know, it's sort of justified by the Cromerabandt itself. So, sometimes to get the sort of like this, this main oldest idea, the profundity that is underlying, it's not really working. If you take one class, two classes, it's not really working. You can feel working on it for six months. Sometimes it's just that someone out there for an ongoing research project, they need to really come up with this, for example, something like a Cromerabandt that actually may renders things even more intuitive and profound. So, don't expect to understand everything in this class. Don't expect to understand most of the things about the TURs. Expect to understand the main idea and some of the points that I'm going to make at the end of the lecture, that's all. Okay, how many minutes though? Oh, I cannot see. Quarter, 15 minutes. Okay, Max, I can do it, I can do it, we can do it. Okay. Yeah. Okay. So, now, believe me when I tell you that these joint random variables, yes. Oh, okay, perfect. Believe me when I tell you, I'm going to put the proof, I mean the article that drives the strong fluctuations and I'm joined strong fluctuations, theorem to slack, but now believe me when I tell you that this is true. This is a fluctuation theorem that works perfectly. If you have this, now what you're doing is you're running the movie forward, but you're keeping track of not only the entropy production, but the currents that are associated with this entropy production, okay? And then, basically, by the same logic of the fluctuation theorems, it's always exponentially more probable that when you go in this direction, you're going to see an increase in this entropy and so on and so forth in this current by this value. Okay, it's exponentially more probable and the exponent is given in terms of entropy production. So, this is satisfied. If you make these assumptions, these assumptions, okay? This is why we have these conditions for F2UR. I see how that it went. I actually told you that this F2UR is working, you can apply this to any kind of a physical system if you satisfy these conditions, but the reason that we can start listing these conditions is just because that the derivation of the F2UR is invoking a fluctuation theorem that imposes these conditions, okay? So, we jumped back.  If you satisfy these conditions, you're satisfying these. And the same for currents and it's not so surprising because as we talked, as we discussed, but didn't prove, it's a really nice exercise for you to do it. Entropy production itself is a current, so you can just write this, and this will be automatically satisfied, okay? So now, what we are gonna do, I'm speeding up, but it's not gonna be, I think, bad or something like that because we got everything that we need up until this point, now we can speed up. So remember, we introduced this new probability distribution cube, modified one, describing the same behavior of the entropy production, but we introduced this cube just for one random variable. So let's remember this. It was the entropy production, yes? Now do the same thing by carrying out the, I actually had the steps carried out here for you, okay? Now do the same thing for the joint probability distribution. Now I have a typo over here because sometimes we use phi, sometimes we use j, these phi's are j's, okay? I'm going to change them before I upload these slides to Slack. So we are defining a new probability distribution, and what we are doing is basically carry out the exact same steps that we just did for this baby cube probability distribution that is characterizing the entropy production for this joint random variables, okay? And if you remember what we had was basically this. For the mean one, yeah, for the mean one, we had this kind of hyperbolic tangent term. So basically if you take this with respect to p, it is transformed to this with respect to q. If you take this with respect to p, this transform, it is the same as the mean square of it, sorry, the second moment with respect to q, okay? So the same thing over here. Now we are doing the same thing instead of for entropy production, we're doing it for the currents, okay? Just see it. This one is equal to this one. Now there is an implicit phi over here, okay? Just if they're identical. And yeah, it's over here, sorry, I didn't even see it. Okay, good. Now one thing, what is the form of the TUR that I erased? It includes the variance of the current, right? Mean square of the current and the average entropy production. So by this formulation, we can compute the average entropy production and we can compute the mean square of the current and just the second moment of the current and then we can compute the variance of the current. So we got all the materials that we need. These are the ingredients that we need to actually see if we can write down the TUR, okay? Now what we have to do, there's gonna be a strong jump. There are lots of mathematical boring steps that we're not gonna carry, I'm giving you as an exercise. So two things, if someone gives you something as an exercise, either it's because it's really easy and it's boring or it's really not easy, but it's sort of boring. I think, I think so. Otherwise they give you as like questions or problems, but if it's an exercise, yeah. So this is the second one, second type of an exercise. So we listed all the quantities that we need. Second moment, mean square variance average of also the mean of the entropy production mean of the current, right? So one of the things that we can do is that always keep in mind that we have a thermodynamic uncertainty relation. We want to bond variance or we want to bond the variance to mean square ratio. So one thing to do, again, this is, I think David just feels free to chime in any time you want to comment on how proofs are done, but you're basically playing around to see whatever you can bond. You're always looking for a bond, okay? And one thing that you can do is to, because you're, I mean, the way that you're expressing variance itself as a mathematical term, it's in terms of the mean square and the second moment, right? So take this mean square it and you can bond it by this term. How do you do it? Do you know how do you do it? Do you know what this is called? Exactly so. Thank you, Carlos. Okay, so you bond this one and then after you bond this term over here, here's your exercise. You need to show that this is possible, okay? Now that I have like about eight minutes left, you're not gonna talk about it. You can do this. I'm going to send a proof in any case on Sunday. Let's say Sunday so that it will be realistic, okay? And once you do this, it's a mathematically marvelous thing.  If you're lucky, you're finding something really succinct and pretty just like this TUR, okay? So I told you that there are two exercises. This is the boring and the hard one. This is the boring and the easy one. Once you have this, you can basically use all of the ingredients that we presented during this lecture that are found on this slide to actually come up with this term over here. And you know how to write the hyperbolic tangent. We did it. And basically just two mathematical operations that you need to carefully carry out. And then you're gonna get this one. What is it? It's a TUR, so yeah. That's how you derive a thermodynamic uncertainty relation, okay? So now I have to talk about just like a review of what we have done so far and what is important for you to learn because there's so much material. So like it's really, it would be crazy, nonsense and so on and so forth to actually expect that you're taking all of this with you. It's not realistic. So what is important? Zero. All the difference between non-equilibrium equilibrium in terms of currents. Know how to write down the current, okay? How to express the current mathematically. I'm saying this is the TA, not myself, okay? And then know what the assumptions are for the TURs. For example, the non-equilibrium steady state and the FTUR and so on and so forth. And just understand why these assumptions matter, okay? And then one thing that you can do is I will send on the Slack channel, the complete slides but also the derivations. Understand how you use fluctuation theorems as resource constraints, as expressions that very formally describe the resource constraints to derive different sets of resource constraints. This is what we are doing here. Think always and always in terms of resource constraints. Okay? These are the take-homes, like messages. That's all. One more thing, and then I'm going to finish. So this is one thing that you're not responsible from in the exam or throughout this course and so on and so forth. But we discussed that, oh, you know, there is something that you can do. There is a method of driving TURs. Might come as a surprise, might not come as a surprise. This kind of a thing is actually a derivation of a TUR from statistical inference. It was then in 2020 and it was then done in 2021. We are in 2022. Like, this is ongoing research, okay? And in the first six years, like in the first five years, where people worked on TURs, I think they also didn't know why we're working with this kind of a precision term of variance to mean square ratio. Where is it coming from? But there is a derivation of TURs from Cromer-Rauband, which basically sort of tells you, I mean, you should probably Google it by yourself, but which basically tells you where this kind of a precision term is coming from. And so there is an interesting history of uncertainty relations that actually can be derived by Cromer-Rauband. It basically tells you how statistical inference might be useful. Good old Heisenberg uncertainty relation, for example, it is something that we can drive from Cromer-Rauband, even though it is really different than thermodynamic uncertainty relations. And I have been thinking that it's really not fortunate to have thermodynamic uncertainty relations as uncertainty relations, because they're not presenting uncertainty in the sense Heisenberg's uncertainty relations present them, right? But still, if you use statistical inference, you can drive both. So this is the part that I'm not going to talk about, but I'm going to upload it on Slack. And if you have a problem with the intuition, we can discuss about it. This is out of the course. This is not in the course material. This is a surprise, okay? But if you want to discuss about it, to understand more about the resource constraints and how we use them and how we drive uncertainty relations, we can talk about them. So that's it. Thanks. Can I make a comment? Of course. Oh, you're going to tell me if I'm bad or not, right? Yeah, from yesterday. No, of course not. No, the comment is that the Cauchy-Schwarz inequalities and equality on differences. So I just want to say that if the integral was in terms of p instead of q, it is not that obvious that you can make such kind of inequalities. Exactly. So in terms of q, it looks like a measure. In terms of p, it wasn't really ugly integral. Exactly. Just a comment. Exactly. This is a perfect comment on why you use some mathematical tricks, because actually they are useful just for you to prove it. And it's exactly as you saw it. Okay, thank you very much. Okay, thank you. So see you at 4 p.m.