This presentation introduces you to the latest option of studying on the Virtual Linguistics Campus, the MOOC option, which provides quality assured content to anyone from anywhere at any time. In the following, I will first define the principles behind our MOOCs and will then let you know how you can register and how you can become part of one of the VLC MOOCs. MOOC stands for Massive Open Online Course. Studying online means from anywhere at any time without any in-class meeting. Traditional online courses charge tuition, they carry credit, and they limit enrollment to a few dozen students to ensure interaction with their instructors. A MOOC, by contrast, is usually free, at least on the Virtual Linguistics Campus it is. And it doesn't involve credits and, well, it is massive. Massive means there is no limit as far as the number of participants is concerned. On the Virtual Linguistics Campus, these two MOOCs are ready. Introduction to Linguistics and Phonetics, Phonology and Transcription. And they will now regularly be offered. This group of MOOCs, Morphology and Syntax, Semantics and Pragmatics, is under construction and both MOOCs will start later this year. Our spring term MOOCs start in April and end in May, and our fall term MOOCs start in September and end in October. That is, within these periods the content of a MOOC should be mastered. Well, and then there are no fees. Our MOOCs are free. Interested? Okay, but before you start, you have to register. However, registration is really simple. Just load the VOC homepage, www.LinguisticsMinusOnline.com and create your free account. After your account has been created, you can log in with your access data, and then you should visit the VLC registration office, where you can register for one or several classes via a few mouse clicks. After you submitted your class registration, you will receive a confirmation via email. That's it. Now, when you log in, you will first see your personalized VLC start site. On this start site, you have permanent access to many other options, such as the language index, for example, the e-lecture library, and a few more. These options are available to all VLC community members. Furthermore, you have additional options that concern general aspects, for example, your account and your profile. Most importantly, however, you will see your MOOC, or if you have registered for several MOOCs, your MOOCs. In our case, this will be the MOOC for netics, phonology, and transcription. A simple mouse click on this class name leads you to the so-called class start site. On the class start site, you have several links that allow you to access additional options, such as the link section and the class-related forum where you can discuss problems with your fellow students. So here is the class-related forum, and this is the link section. Just click on them, and then you will see what happens. Central to each class, however, are the e-learning units. Each class consists of a predefined number of units. In the case of this class, we have one introductory unit and 12 e-learning units. The introductory unit is mainly a textual repetition of what I am explaining in this video. The e-learning units contain the content of the class. Now, after class activation, the current unit, that is, the unit that is currently in focus, is marked red. So let's assume the unit phonetics is in focus, then it will be marked red. All units are represented by hyperlinks that open their unit start site. So let us illustrate this on the basis of this first unit of the phonetics, phonology, and transcription class, the unit phonetics. Here is the corresponding unit start site. All e-learning units on the virtual linguistics campus have the same structure as far as their unit start site is concerned. Thus, they have the same navigation menu on the left. So here is the navigation menu. Within the navigation menu, you have a link introduction that refers to the start screen where the central goals and topics of a unit are listed. The introduction is always displayed when you load a class start site. So here it is. The virtual session is the heart of an e-learning unit that presents the content. So a click on virtual session opens a new site which contains the content of the unit. If you click on Worksheet, an interactive electronic test that is represented and evaluated by the computer will be shown. And the link practical loads a set of questions and tasks about the virtual session that are associated with model solutions. Let us look at these main components of an e-learning unit in detail. Here is the virtual session for the unit phonetics. The virtual sessions are for you. They supply you with highly interactive multimedia content. In each virtual session, you find a link to an e-lecture which can also be directly seen via YouTube. As a special service for class members, you can also access the board content that was generated on the active board by Promethean behind me. And then there are additional integrated exercises that can be loaded via the interactive tutor. The central question you might ask yourselves at this point is how do I study all this? Well, studying online simply means go through each e-learning unit at your own pace. Study the content of the virtual session using the guiding questions. Watch the corresponding e-lecture video and optionally use the integrated tests in the interactive tutor. And if you wish, you can discuss further problems using the class related forum, which is open for all registered users of your class. And the important point is there is no pressure. You can do everything as often as you like and wherever you like. And once you're ready, then you tackle the worksheet and do the practical. But before we look at the worksheet and the practical, let us say something about the VLC assessment principles. All assessment within a MOOC is carried out on the basis of electronic tests, one per e-learning unit. Here is the electronic test, a so-called worksheet for the unit phonetics. Well, it is a relatively simple multiple choice task. Worksheets have to be passed. That is, you have to score at least 60%. If you do not reach 60% or more in a first attempt, don't worry. You can try as often as you like and no one will notice. Only if you pass, your score will be stored in the VLC database. And if you're not satisfied with your score, you can do it again and try to raise your score until you're satisfied. Last but not least, each e-learning unit is connected with a practical sheet where additional questions and problems are presented. These questions are paired with model solutions that can be retrieved according to a predefined rhythm. Whenever the red indicator on the class start side, you remember this tiny little thing here. Whenever that red indicator moves on to the next e-learning unit, you have access to the practical model solutions of its predecessors via these little symbols which will then show up. Well, and the model solution that can be retrieved via these symbols can be a textual explanation, a diagram, or even a video that explains the problem. Well, and at the end of a MOOC, you will be sent a certificate. Each participant will receive a certificate confirming their active participation. If you want a qualified certificate, you have to submit and pass the worksheets and reach an overall average of 60% or more. That's it. And what about the role of the class instructor? Well, first of all, all class instructors on the Virtual Linguistics Campus are professional linguists with a profound background in teaching linguistics. In a MOOC, however, you won't see us. And we do not really teach, so this scenario is definitely out. Rather, we supervise your activities from a distance. We try to motivate you to do your tasks in time, and we make sure that everyone is served as well as possible. Well, and from time to time, we also check the class-related forum to look at your questions and your threads. We do not use standard social media in order to keep the discussion straight and topic as well as class-related. And what does it cost? Nothing. The VLC MOOCs are free. They are our contribution to the world of learning in linguistics. And for free, you get MOOCs that are quality-assured and up-to-date and combine all modern digital media, MOOCs that support each learner type most efficiently, learners who enjoy watching video material and learners who like using interactive multimedia. So, we invite you to join us in one or several of our VLC MOOCs. The material has been created by the Marburg Linguistic Engineering Team, and it is there. The rest is up to you. We're waiting for your registration, so see you soon.


And we will do that here at a hat trick. And Greg, as you say, the hat, he gets in the interview. So that's right. I'll let you lead it off, my friend. Well, I mean, you're doing it all, man. You're playing offense. You're playing defense. And you're on the wings. To my midi. I'm just going to ask you, what do you like more, scoring goals or cross-checking guys on defense? Definitely scoring goals. All right, I like it. What did you see out there? What were your opportunities out there tonight? Just the defense is not sliding as much. We're taking our time, spacing on offense. Cam and Coleman are going to get a lot of the attention at midfield. So coach told me when I need to be aggressive. So I took that chance tonight. What about coming back off the Gettysburg loss? Coach was talking about that during the week. How was the week of practice? What did he tell you guys? And how did you get ready for today? It was tough. I'm not going to lie. But we were aggressive. We stayed the course. We did our job. We got our heads in the right place. We forgot about last weekend. That's pretty much it. OK. All right, awesome. And look, what does it do to have a goalie like Paus, right? When teams are getting some pretty good looks that they probably should finish. And Paus just standing on his head and making those saves. What does that do for the team? I mean, it's definitely game changing. Obviously, the score would not be what it was if he made two last saves. But yeah, he really stood on his head tonight and bailed us out a lot of times. Absolutely. So Jack Scolidi gets the hat trick. Also Lachman gets a hat trick. Paus has 13 saves and 22 chances tonight for Stevenson. You added it all up, Greg. It's a 10-9 win, a big win for the Mustangs. Yeah, it was a fun one. I'll take it. Jack, thank you. Thank you. Go MustangsSports.tv.


Hold the wall place everybody forced here rip in 2122 tops chrome Bundesliga Hobby six box number 11 random team stop you can always check the results of late in sports cards calm Thanks for joining everybody and good luck Water here All I do is cheese takes fries Hold the wall place Those are the ones But I wanted the first time I didn't know where to order there's like there's like the counter is like so tiny All just like concrete building They might get one tomorrow Here we go stuck here refractor rookie Georgina Rootser nice start for Haufenheim Rootser should transfer to a decent club pretty soon here Got a lot of hype with him Golden generation Is that the Davi let's get pristine filled up next everybody Got a magenta way Jude Bellingham Dortmund 170 of 399 Nice card there Refractor of Leon Goretzka for Bayern. Yeah, we break we break a lot. We break a lot. That's why a coach have heroes of tomorrow Just watch this week Last night's gonna have to Sam probably Marvin Friedrich refractor Munchen Gladbach do a lot of breaks. That's what happens, man. Astor Frank heroes of tomorrow Wolfsburg Refractor of Brian Lasmay Bielfeld the rookie Luthe Union Berlin On golden generation there Musiala heroes of tomorrow for Bayern refractor Milos Pantovich Bochum Hey Munchen Gladbach Florian Newhouse 77 to 150 a prism refractor Otto Ricardo Pepe Augsburg nice hit 16 of 175 go Augsburg refractor rookie Abiyama for greater for it And then you got to try Ellie Lou's got to try Daniel's yeah Prism refractor Julian Green of Groucher fourth 25 of 199 Guys we got pristine up next pristine baseball 11 to go I'm gonna tweet that out box two moving on moving on We should give him a haircut That's what I'm saying. Give him some cool Or just get the sides a little lower again. He can start working on his uh his mullet Well cocoa heroes of tomorrow I was gonna say a hockey mullet with like some stripes on the side That's a good look.  That's what we call them jake mario lemieux Jake macho Man, I'll remind golden generation. That'd be awesome. It would be awesome Runs around. Oh, yeah, magenta way rookie brian lasme for bielfeld 363 of 399 in the joder mullet Yeah, feathered like bangs and stuff. Oh man green hariguchi Union berlin 21 of 99 What's up guys? Everybody watching baseball right now Today has been awesome. Uh kone heroes of tomorrow mungengladbach Back to back to back to back to back playoff baseball games. Love it refractor giovanni reyna for borussia dortmans refractor bielfeld stefan ortega Ryan, what's up, man? Refractor julian green greitzer fourth. You guys know why it's also a great day That's because we got a 10 off members only coupon in the community tab youtube members 10 off order expires midnight eastern Got breaks listed all the way through monday daytime lsc baseball breaks monday daytime We got breaks here tomorrow and sunday And it's good for personals too. Let's go for anything on the site 10 off an order brian munich refractor alfonso davies You guys somewhere on two and I didn't even know Oh, I slept in a bit today and I was watching the stream. Hasebe golden generation J rod bass hit heroes of tomorrow. Um, everybody has stuttgart refractor rookie for life saying mohammed sima khan sima khan J rod and blaze in each other And a blue rookie auto timothy tilman for greitzer fourth 42 of 299 Last england 2 1 that's tough. England's a good opponent though. So I'm sure that's probably debated it's such a Simple sport. I'm sure there's a lot of different iterations of it Top 11 insert lars stindl barusha munching gladbach those are two per case on the top 11s Heroes of tomorrow is berghards for mines refractor hertha berlin lucas to start golden generation entrees luther purple refractor rookie life sig sydney raviger Nice car there 263 of 299 refractor rookie cool timo huber's A stuttgart refractor autosilus katompa mvmpa Not numbered on the mvmpa refractor rookie wolfsburg aster vrunk Vronks Acklin camp heroes of tomorrow for hertha berlin refractor rookie stuttgart borna sosa Nine to go now on the Uh pristine breakup next There's a link for you in chat Prison refractor giovanni reina dortmund 28 of 199 blue refractor bayramunik jamal musiala 100 of 150 nice musiala good box here Refractor yosui kimik for bayramunik Niklas start golden generation Refractor denny zachariah for munching gladbach. All right, so moving on the box for refractor marcus thorum munching gladbach golden generation yonas hector one sec guys a Uh makmu dahud borosha dortmund magenta 82 of 299 refractor hertha berlin malita rookie Steel is dealing refractor rookie coach hat for bosham a couple of base packs Heroes of tomorrow day on lubitschitzch for comb refractor mvmpa for stuttgart Emil forceburg golden generation Leipzig Aster vrunk's heroes of tomorrow wolfsburg refractor jamie leveling greater forth A phillip leinhardt's rookie green auto 35 and 99 for fryberg Solid hit there See you got refractor alfonso davies bayramunik an aqua rookie mark marco meyerhofer 106 of 275 greater forth A musiala heroes of tomorrow and some five card back here. Oh, uh, coach hat heroes of tomorrow as well Prison refractor daio open micano 74 of 199 bayramunik Zidane set to mere heroes of tomorrow labor kuzan Georgina router rookie refractor hoffenheim magenta wave hoffenheim andre komirich 339 of 399 levendowski refractor bayramunik Heroes of tomorrow nico schladerbeck refractor rookie nicolas dorsch augsburg Brian last may be a failed rookie refractor angelo stiller hoffenheim heroes of tomorrow Refractor colm yon thielman Come on big card. Come on a daishi kamara magenta for frangford 222 of 399 Nice rookie prism auto georgina router for hoffenheim 80 of 175. That's an awesome hit You go nice router sign with it Sharpie it looks like Doesn't look bad. It just looks like Refractor is abiyama for greater forth Nice one hoffenheim. Georgina router All right Oni siwo insert Let's get pristine fill up guys. It's breaking next Still at nine Still at nine to go purple refractor frunkford timothy chandler 203 of 299 Danny blum refractor for bochum Heroes of tomorrow aster frank wolfsburg refractor vicenzo grifo fryberg Refractor jude belling and borussia dormant Golden generation to mass molar Refractor has sebe frunkford Heroes of tomorrow day on lubechitch for colm Florian new house refractor moogen gladbach. Come on big card. Come on Last few packs here everybody Magenta wave rookie for levy kuzan. I mean it oddly 221 of 399 Last hit is a colm auto rookie day on lubechitch 147 of 150 refractor laquaf or wolfsburg And lubelling heroes of tomorrow great your fourth Yeah, no really low numbers decent autos though. You got a colm refractor auto day on lubechitch or uh magenta auto 147 of 150 Georgina router rookie prism auto for hoffenheim nice hit 80 of 175 green rookie auto fryberg fill up leinhardt 35 of 99 refractor auto in boom buff or stuttgart A blue auto of regret for timothy tillman to 299 and a riccardo peppy prism refractor auto for augsburg 16 of 175 that's gonna do it. Thanks again everybody. We got baseball coming up


Continuing with the Brownian motion theory, we showed in the previous lecture that the velocity of the particle starting with its initial velocity v naught in the presence of a fluctuation acceleration called a t prime evolves according to this formula v v t equal to v naught e to the power minus beta t plus e to the power minus beta t and integral over basically the fluctuating acceleration. So, very first principle solution to the first order differential equation it is Newton's law for v. So, then when you do ensemble averaging, ensemble average of a is 0. So, we get the mean velocity just decaying of the original velocity and we also did an exercise to find the v square average as a function of time and if you subtract this v bar from that, we basically got sigma of in v the variance that would be gamma by 2 beta into 1 minus e to the power minus 2 beta t. So, this evolves with the rate of 2 beta as opposed to v bar alone which evolves with the rate of beta. Very interestingly, we then showed that as the time goes to infinity v square average because v bar will tend to 0. So, v square average becomes gamma by 2 beta and then using thermo dynamical arguments that is the theory of equipartition of energy, we established the relationship between the fluctuation coefficient gamma and the friction coefficient beta and that relationship was quite consistent with a relationship that we derived using gas kinetic arguments for the for the coefficient gamma. So, that shows an internal consistency of the whole process that we are discussing. We continue with this how the velocity fluctuates in a Brownian theory. We have to understand the velocity fluctuations well. So, that it is from velocity fluctuation that the random walk originates. The real variation of x the randomness in the position variable comes as a kind of integral over the fluctuation of the velocity variable. So, that we need to understand the velocity autocorrelation function. So, velocity autocorrelation function v autocorrelation. Now, when we say autocorrelation it is a the correlation of the same quantity at 2 different time scales 2 different times. So, by definition it is nothing, but velocity at some time say t 1 correlated with velocity at some time t 2 and its ensemble average. We also can write it as v t 1 v t 2 and over bar we can use because it is sometimes it is more convenient to write. So, this is the definition of an autocorrelation function 2 time correlation. So, we try to now establish from all that we have done how this quantity between 2 different times evolves with respect to their difference in the times. How does it depend on t 1 itself and how does it depend on t 2? t 2 can be greater than t 1 can be less than t 1 we will establish all this. So, we first make use of the expression that we have given here for v t. So, we start from first principles. So, v t is given. So, I know v t 1 and then I can write v t 2 and putting them together I will have the expression v t 1 v t 2 I am not still ensemble averaging I am just writing the two quantities that is e to the power minus beta t 1 plus t 2 e to the power minus beta is a common in this expression. So, it can be taken out and that will be t 1 plus t 2 then there will be a term v naught plus integral 0 to t 1 e to the power beta let us say t prime is the dummy variable a t prime d t prime this is 1 with respect to t 1 and with respect to t 2 again it will have v naught plus 0 to t 2 let us say e to the power beta t double prime let us distinguish because these are dummy later when we put them together there should be no confusion. So, this time we will call it as a t double prime d t double prime and this bracket closes. So, it is a product this is common. So, this is v t 1 and v t 2 let us expand this it is because there will be now four terms. So, all the four terms we can expand.  Then there will be two terms multiplying v naught 1 will be an integral over 0 to t 1 of e to the power beta t prime a t prime d t prime and another one will be that integral over d 2 e to the power beta t double prime a t double prime d t double prime only brackets close. Then there will be the double integral which will be 0 to t 1 and 0 to t 2 which will involve e to the power beta first with respect to t prime and then with respect to t double prime and that will be a t prime a t double prime d t prime d t double prime and here we close square bracket. So, the four terms are like this it looks quite messy, but when we take the expectation or take the ensemble average many things drop out. As you can see when I take the expectation of v t 1 and v t 2 overall realizations or ensemble average the first term does not depend on any statistical variable it is original velocity. So, it will remain the same the second terms within the curly bracket. Since a t prime expectation will be 0 a t double prime also the ensemble average expectation when I say I actually mean the ensemble average that will be 0 and here of course, this is the two time correlation of the acceleration function which is a delta function. So, we input this to arrive at an expression v t 1 v t 2 we use over bar for ensemble average that will be e to the power minus beta t 1 plus t 2 will remain the same. So, v naught square will remain the same because it is not a statistical quantity thus this term within v naught will become 0. So, just to remember that we will call it as 0 the next term also will be 0 and the third term will have gamma. So, gamma we can take out. So, I will write it as gamma 0 to t 1 0 to t 2 e to the power beta t prime plus t double prime delta of t prime minus t double prime dt prime dt double prime square bracket closed. So, for future purposes we write it more neatly remove all the zeros. So, this is to create space e to the power minus beta t 1 plus t 2 into v naught square plus gamma double integral 0 to t 2 e to the power beta t prime plus t double prime delta of t prime minus t double prime dt prime dt double prime. So, this part we call it as the first term there is nothing more to do here this whole part we call it as the second term. So, we first focus on the second term because first term is just a simple term. So, the second term when we look at we have the second term by definition it was gamma e to the power minus beta t 1 plus t 2 and a double integral 0 to t 1 and 0 to t 2 e to the power beta t prime plus t double prime and it has to be integrated over a delta function which involves the difference between t prime and t double prime and one important property of delta function we should never forget that it selects the value when its argument is 0 providing the integral passes through 0 somehow. So, that means, if I am integrating over t double prime if I am integrating here over t double prime from 0 to t 2 then that t double prime when it becomes t prime only it will have a value which means t double prime when it runs it should run beyond 0 beyond t prime actually. Now, since we have not so far ordered t 1 and t 2 we cannot say. So, for the next step we make an assumption that let t 2 be greater than t 1 this is just an assumption because t 1 and t 2 are arbitrary let us say we have selected t 2 greater than t 1. So, if we do that then the inner integral t 2 which goes beyond t 1 can be split as 1 up to t 1 and the next one between t 1 and t 2. So, with this assumption the second term can be written as gamma e to the power minus t 1 plus t 2 the first integral is up to t 1. So, we can take that coefficient that is beta t prime will be out the dt prime is the first integral. So, the second integral if I put a curly bracket it is I split it as 0 to t 1 first of whatever is there that is e to the power beta t double prime delta t prime minus t double prime dt double prime and the second part will be t 1 to t 2 because t 2 is larger. So, it will be still a positive integral it will be e to the power beta same thing double prime delta t prime minus t double prime dt double prime and this curly bracket closes here. So, now let us look at the first integral and the second integral again further. So, if you look at this integral I call it as a and this as integral b. So, we have already selected 2 and in the second one now we look at a and we look at b. If you look at a the integral a here the time interval for between both for t double prime as well as t prime they run between 0 to t 1. You can see the upper limits are t 1 which means that when I am first integrating over t double prime somewhere it will it will have to cross t prime since t prime also runs between 0 to t 1 which means t double prime is allowed now to select the remaining part of the function corresponding to the value t prime since they cross each other that is why this diagram is drawn. So, we can see that the a integral therefore, the a integral will be simply 0 to t 1 e to the power beta t prime dt prime and t double prime is going to be t prime. So, it will be e to the power beta again t prime only. So, we have this part of the integral can be easily evaluated because it is e to the power 2 beta t prime. So, it is integral 0 to t 1 e to the power 2 beta d prime dt prime which is going to be e to the power x dx. So, it is e to the power 2 beta t 1 minus 1 divided by 2 beta. So, the a part of the integral is evaluated. Let us look at the b part of the integral. Now, for b part of the integral if you look at this here the inner integral varies from t 1 to t 2 and we have assumed that our t 2 is larger than t 1 whereas, the outer integral varies from 0 to t 1 only. So, you can see that the t prime varies from 0 to t 1 and t double prime varies from t 1 to t 2. So, what does it mean that the in the variable of the inner integral that is the t double prime never crosses t prime they are exclusive intervals. Hence in b t double prime does not cross t prime. Hence that whole integral b will be 0. So, we have put together the various terms now the 2 integral therefore, is only the a part with the of course, this pre factor. So, we write neatly the second integral therefore, second integral or second term we called is going to be here gamma e to the power minus gamma e to the power minus beta t 1 plus t 2 into the a which is e to the power 2 beta 1 minus 1 by 2 beta. So, we write it down as e to the power minus beta t 1 plus t 2 will go back gamma of course, being outside e to the power 2 beta t 1 minus 1 by 2 beta. This can be simplified as the 2 term we take this e to the power minus beta t 1 plus t 2 we can take inside. So, this will lead to gamma by 2 beta here there is 2 beta t 1 and here there is a minus beta t 1. So, it is going to be e to the power since t 2 is larger we keep in that form it will be minus beta into t 2 minus t 1 which is a positive quantity. So, it will be plus t 1 and minus t 2 that is maintained and the second one of course, e to the power minus beta t 1 plus t 2 here it does not matter which is larger and we got it under the assumption that t 2 is greater than t 1. Now, if we had if you go back and see the assumption we made that is t 2 greater than t 1 and then followed the steps we got this result. But supposing we had started with the t 1 greater than t 2 then the whole thing would have been in the converse form that is the integral would have been then t 2 to t 1 here and the same logic would have applied the difference being that if t 1 was greater than t 2 then this term this term would have been only changed because t 1 plus t 2 does not change here it would have been minus beta into t 1 minus t 2 this term would have become that that is the only difference. So, on the whole whether t 1 is greater than t 2 or t 2 is greater than t 1 all that matters is there a difference should be positive which is possible when you call it a mod function. Hence, in general for any combination of t 1 t 2 the second term will be gamma by 2 beta e to the power minus beta modulus of say t 2 minus t 1. Now, it does not matter of course, minus e to the power minus beta t 1 plus t 2 for all t 1 t 2 regardless of which is greater. So, we can now put the first and the second terms together you can see here the first term is simply v naught square e to the power minus beta into t 1 plus t 2. So, we combining all that we have we can write v t 1 v t 2 bar will be we can write it this way v naught square minus gamma by 2 beta multiplying e to the power minus beta t 1 plus t 2 and another term gamma by 2 beta into e to the power minus beta mod t 2 minus t 1. This is only a rearrangement you can note it from the combination of the previous result. See here you have a term minus gamma by 2 beta e to the power minus beta t 1 plus t 2 and in the first you had v naught square e to the power minus beta t 1 plus t 2. So, the exponential term being common we can subtract that and this modulus term can be kept separately. So, that is what we have done. Now let us say let us say that our times are large both t 1 t 2 are large. So, of course, when it is large we must give a unit and that implies what we meant is both beta t 1 is and beta t 2 are greater than much greater than 1. Now we precisely know what we mean by saying large t 1 and t 2 are much larger than the relaxation time of the system that is 1 by beta or the relaxation relaxation rate into time is much larger than 1. However, it is possible that the difference beta of t 2 minus t 1 can still be of the order 1 it could become parable t 2 can be large supposing t 1 is 100 units or beta t 1 is let us say 10 beta t 2 is 11 then both are large, but their difference 11 minus 10 may not be large it is comparable to beta. So, under that condition we can note from this expression that the first term t 1 and t 2 beta t 1 beta t 2 are very large. So, this exponential decays and we are left with the second term which depends only on the difference of the time which need not be large. So, under this assumption so as t 1 tends to infinity t 2 tends to infinity, but t 2 minus t 1 finite we have the autocorrelation function v t 1 v t 2 bar will be gamma by 2 beta e to the power minus beta mod t 2 minus t 1. It is a very important result that the velocity autocorrelation function is finite it is not it is a exponentially decaying function. We can write some in a way sometimes this is written in a better way or in a more transparent way that if t 2 is t 1 plus tau and t 1 we call as t then often it is written as the autocorrelation between velocities at time t and t plus tau is a simply gamma by 2 beta e to the power minus beta mod tau. So, if tau is always positive then we do not need to give mod, but it because of the modness it basically implies the velocities you measure at time t plus tau and correlate with the velocity at time t decays e to the as e to the power minus beta into tau if tau is positive or negative that is the prior or later correlations necessarily have the same functional form both of them decay exponentially, but with the mod argument. So, we can represent such a function in the following fashion supposing tau is my time and this is the autocorrelation v t v t plus tau bar then it will have a form a cusp like behavior because both are exponentials. So, there will be a nondifferentiable property here. So, this is an exponential e to the power minus beta plus tau and this we can say e to the power beta plus tau if tau is 0 here. So, this is an exponentially decaying correlation. In stochastic mechanics literature correlation between quantities if they decay in any form other than through a delta function it is called as a colored noise color noise problem that is the velocity correlations have the form of a color noise as opposed to the acceleration correlation which we started which was basically white noise. To understand the meaning of this color perhaps it is instructive to understand the meaning of white. When we say white what we mean is the autocorrelation is basically delta correlated and if you look at the frequency spectra we contribute to the time domain and if you convert it into the frequency domain via Fourier transforms and we know that e to the power i k t of delta t is 1 if t goes from minus infinity to infinity. This is basically the Fourier spectrum with respect to k the conjugate variable and the conjugate in time is the frequency. Conjugate in space is the kind of a reciprocal length whereas, a conjugate in time. So, k now will be having the dimension of a frequency and this says that the correlation since it is a delta correlated it is made up of all frequencies with the equal intensity equal strength and that is what is supposed to be a white color. When you say white although strictly it is not equal intensity, but it contains all colors in a mix. So, carrying forward that terminology a situation when the Fourier spectrum has all frequency components it is called as the white noise. From that perspective the color noise emerges as that correlation in which it is not a delta correlated, but there is some persistence of correlation for certain period of time. Delta correlation means the force force imparted by the molecule at one point in time is not correlated to that at the next instance. But to velocity the particle if you measure velocity now there is a persistence of velocity because of viscous drag and other things in spite of the noise for a certain period of time. But after times of the order of much larger than 1 by beta that correlation will decay and particle will have another velocity. So, that is the implication of the velocity autocorrelation function. Interestingly the whatever we derived for tau equal to 0 case implies that v t 1 v t plus tau will be simply v square average and that comes back to gamma by 2 beta which we already proved. This is already proved by thermodynamics very important relationship. I now remember to tell you that in statistical physics literature this relationship relating the strength of random fluctuations to the decay constant is called as the first fluctuation dissipation theorem. First fluctuation dissipation theorem because it relates the fluctuating force to a dissipation velocity. So, there is a connection between fluctuations and dissipations via thermal equilibrium because we just said to v square equal to v square average equal to k t by m. We now delve a little more into it obtain more instructive and deeper results especially the so called Stokes-Einstein relationship between the diffusion coefficient and the friction coefficient by continuing with these arguments. Thank you.


Recorded books and one-click digital present A Change of Climate by Hilary Mantell One-Click Digital provides free audiobook downloads in partnership with thousands of public libraries. To see if One-Click Digital's extensive catalog is available to you, visit your library's website. The author opens with two epigraphs. The first is a quote from The Descent of Man by Charles Darwin. We are not here concerned with hopes and fears, only with the truth as far as our reason allows us to discover it. I have given the evidence to the best of my ability. The second quote that begins this novel is from Job, chapter 4, verse 7. Consider what innocent ever perished, or where have their righteous been destroyed? And now, A Change of Climate. 1970. Sad Cases, Good Souls. One day when Kit was ten years old, a visitor cut her wrists in the kitchen. She was just beginning on this cold, difficult form of death when Kit came in to get a glass of milk. The woman, Joan, was sixty years old and wore a polyester dress from a charity shop. A house-wifely type. She had chosen to drip her blood into the kitchen sink. When Kit touched her on the elbow, she threw down the knife onto the draining board and attempted with her good hand to cover Kit's eyes. By this stage in her life, Kit was not much surprised by anything. As she ducked under the woman's arm, she thought, that's our bread knife, if you don't mind. But she said, you shouldn't be doing that, Joan. Why don't you come away from the sink?  The woman allowed herself to be led to a chair at the kitchen table. Kit pulled a clean tea-tile out of a drawer and wrapped it around Joan's wrist. The towel was a checked one, red and white. Joan's reluctant blood seeped black against the cloth. Her cuts were light, early, indecisive. The practice cuts. Just wiggle your fingers, Kit said. Make sure you haven't done any damage. The woman looked down at her hand and dry-eyed dread, while the child scrambled on a stool and brought down a box from a cupboard. Lucky it's half-term, Kit said, unpacking the bandages and the round-ended scissors. Otherwise I wouldn't have been here. I was upstairs, I was reading a book. It's called Children in the New Forest. Have you read it? It's about a family like us. Sample complete. Ready to continue?