so welcome to the first lecture in introduction letters boltzmann method in Winter term 2022-20 3 and this lecture is being recorded so please don't say anything that we later have to edit out yeah but otherwise you're very very welcome to participate here I close the windows because it's a little bit loud outside so if you feel that you get infected then please wear your mask okay introduction to let us Postman method very very briefly in the beginning what um what all of this is about we are um looking at a crazy method that uses particles that are moving on what we call a lattice so this is a regular arrangement of of points in space so particles are moving around on this and we make this lattice very large put this into some geometry like actually this is the building that you see here's the building where you're actually currently in and um then we do this simulation we solve the so-called letters boltzmann equation which is this thing here which unfortunately I don't have the here which is this thing here and which is what we learned in this lecture I mean in this lecture series and this then gives us the movement of the air actually solving the navier Stokes equation okay so this is so now Navia Stock's equation is down here and then you get a crazy flow field around these buildings oh I have to stay close to the mic um otherwise I'm not recorded okay so this is more or less the big the big picture and the graphical abstract of today so this is the today we are talking about the boltzmann equation which is different from the letters Postman equation but we'll have to understand the boltzmann equation first to go to the latest Sportsman equation so the content of today's lecture is um three different ways to understand the world yeah then we look at one of them namely the one in the middle here the boltzmann equation we look how this works with the Collision integral with conserved quantities and in the end we will derive the so-called ether theorem yeah H or is a theory okay so the Easter Serum is the thing that tells us in which direction time progresses yeah because time could go forward or backward and The Ether theorem tells us that what is forward in time and what is backward and time okay so this is something that we learned today and let's jump into the subject so okay the three worlds now to change my pen so there are three ways maybe there are more but they are the ones that we are using three ways to understand the world to see oops two nah sorry to see the world so and the first one is to say there are atoms so there's an atomistic View or there are molecules Maybe which is an assembly of atoms of course and this is just using Newton's Laws yeah for say a molecule I there's a molecule that we call I so this is I as a number and this follows Newton's law so this molecule I has a mass Mi and it has an acceleration a i and this is nothing else than the sum over I and J f i j where J is the number of a different molecule like we have many of them they this here will be our particle one and this here will be our particle two and three and four and then there is this Force if 1 2 going this way there's a force uh two three four then there is a force if three one it's going this way and then if three four going this way and if for 2 going this way and then there are others in between if one four so between all the particles we have forces and they all go in two directions as you should know from axial cry axial equals react action equals reaction yeah so the one-fourth bit acts on particle one from particle two is the inverse of the other one so this Atomic the important thing with this Atomic picture is that we are looking at this tint distinct particles so what do I mean with distinct this thing means that every particle has a name so there's a specific particle so we start with a specific particle particles of this particle for hundreds of years if we say okay one particle came through the window and we wait until this particular particle lived through the door yeah it's not like there are hundreds of uh hundreds I mean trillions of of nitrogen molecules entering in some and they're in there and we don't know which one here we know each one so the particles are distinct and they Exchange they're distinct and they Exchange energy and momentum by the Newton's Laws and they are doing this by binary forces so interaction always between two particles not between several at the same time so this is one way how you could imagine the world so another way to imagine the world is the continuous picture and this is usually described by the Navy's dox equation so I assume that what I what I told what I just told you is absolutely nothing new to you yeah so you know Newton's Laws you know there are molecules and they interact with each other in their binary forces and no nothing nothing else in principle and I would also expect that you have heard about the navier Stokes equation maybe to a lesser extent but you still should have heard about this somebody never heard about another Stock's equation okay nobody is reacting so the navier Stokes equation is an equation of a velocity field the time the the time change in the velocity U that comes with something that we call an addiction term so this thing here is attraction and the entire thing so what is how do you call this entire thing somebody give me some interaction no the material derivative do you hear the material you you heard about material derivative okay so this is the material derivative change it will change in time in a frame of reference moving with the stuff by the way we are not supposed to drink in this room that there's a there's a sign in in the uh in the in the entrance so just okay so this is this is what we usually put on the left hand side the material derivative and on the right hand side we have um the potential term so this is usually given by the pressure pressure difference and there is a diffusion term diffusion term that in the simple case of a Newtonian fluid looks like this could be more more involved so this is the diffusion and then there is something that we actually derived the name of all of this of this is the time derivative of the density which is rule um what the derivative of you and this is a continuity equation so this is actually the incompressible continuity equation now and con um this is this is the thing that gives the name to this view of the world so if if you like to see the world like this then you are very much okay so you are in principle in Good Company but you think you also think a little bit 19th century 19th century way of thinking they have flowing matter is a continuous is a continuous field now and also these fields is obeying Newton's laws and conservation and this leads to conservation of Mars momentum so one thing that is often forget forgotten that this conserves is angular momentum so this also conserves angular momentum not beautiful and it also conserves energy potentially so if we write down the incompressible navier Stokes equation we have forgotten about temperature and therefore this navigation does not conserve energy yeah so this is this is why I put this into brackets yeah but in in principle we could also write an energy equation on top of this and then we also would conserve energy foreign 19th century like since when do people believe in atoms who was the first to believe in atoms John Dalton or when was John Dalton I mean there was somebody before John told believing in atoms certainly though in in Old in increase three thousand years ago they believed in atoms no this is and this was of course no yeah but many things that were believed in three thousand years ago of course we are wrong and were later not believed anymore so the old Greeks they believed in atoms yeah fire earth water somehow elements atoms so when did people when did we like the the in the Modern Age started to believe in atoms where's this idea coming from okay I I tell you in in the middle of the of the 19th century people had this this kind of theory but [Music] um that there was actually a proof that real physicists would start to believe that this makes sense was only in the beginning of the 20th century so the navier Stokes equation is much older than the belief in in networks yeah so there's no this is why we don't need atoms here we can describe the world like this and this is a continue yeah so the atomistic view of the world this we need for understanding the postman equation we need an atomistic view of the world but the atomistic view of the world is accepted only actually there was a paper by Albert Einstein in in one of these papers by Albert Einstein in 19 in 1905. um that actually proved that the viscosity term this is this year is incompatible with the Assumption of a continuum yeah and through this it could actually prove the size of an atom yeah so so you can you can take the viscosity and from the viscosity you can derive how large uh you cannot derive how large an atom is but you can derive how far an atom has to fly to hit the next atom this you can derive from the viscosity and this is what Einstein did in like 115 years ago like 117 years ago so this is not this is not too old yeah so I I knew some people who lived before this time okay this takes us now to the um third picture which is the kinetic which is the one that we are dealing with in this lecture series which is called the kinetic and sometimes it's called the mesoscopic so I'm not a big fan of um saying it's mesoscopic because this has a little bit sometimes wrong um implications but this is the way it's usually called okay and this is probably one that you have not heard of this the Bossman transport equation which we will call BT BTE so this is an beast on its own and we should actually also have a complete lecture on the boltzmann equation but we have here a lecture on the latest Postman equation so we'll have a simplification of this thing but now let us have a look what is the boltzmann equation actually so the boltzmann equation has a Time derivative of something that is called if which is um a probability density function Plus something that now turned out very ugly this is supposed to be a vector C which is a vector of microscopic velocities time spatial derivative of f so this again is an attraction of an advection operator and here okay comes something that I just right now notice I left out of the navius dox equation yeah okay but it's not so important and there is a forcing there's a forcing term which is a force at times a derivative in a vector derivative in the velocity space in the microscopic velocities based on F and this is equal to an Collision operator so this all together is the boltzmann transports equation so now what is the difference between the postman transport equation and the other two and why is it somehow considered to be in in between so the idea behind this kinetic approach is that molecules so we still have molecules this is important we don't have a Continuum we have molecules but there's something drastically different namely these molecules are now undistinguishable so this is very important and this thing do I write it with each I misspelled it on my so they they cannot be distinguished particles and they are still following Newton's Laws though there is conservation of mass energy and momentum and so on okay so what is the difference so previously I said there's one molecule entering this room and then we are observing this one molecule now what we say is there is maybe nitrogen and there is oxygen yeah and all nitrogen molecules are the same the only thing that we're dealing that we are carrying off is there's nitrogen and not there is this nitrogen so they are now not counted so we forget we forget if we forget completely about the individuality yeah and only have a statistic and this statistic is expressed by this um by this probability density function f this is the probability density of so what does this mean this is the probability of finding a particle in now unfortunately seven seven dimensional so we have seven Dimension here face space as it is called and this phase space is composed of the location so location we write r r yeah location this is the uh the odds Vector location for this yeah x x y z so it's X y and C so this is the position so these are three of these seven variables and then there is the microscopic velocity which unfortunately we use Creek small Creek letter so this is a small c yeah and okay so maybe a little bit um okay I don't know whether this this normal clutter is completely consistent but we will write a small C Vector for all of them and this is composed of for the X Direction it is a small C without Vector yeah and then okay now comes the letters that you never use otherwise small oops you alone and so this is the the last three letters in the correct alphabet I'm very sorry yeah so because you have usually and the microscopic velocities you um no they're not the last so okay it's very very high in the in the back yeah so C oopsilon and chatter Okay so anybody has difficulties with remembering what is C and what is settle you know the the the thing that you need the only thing that you have to remember in order to know what is C and what is Setter is that um so it's header is the snake is the living snake that's and C is the there's the snake where you make it not in the in in their naked yeah and if you remember this you will never have difficulties distinguishing C and set up okay so so these are the three velocity components and then we still have um the time which is then number seven so time is T and this is now the statistics of particles instead of individual particles okay and one one warning this mesoscopic the mesoscopic in the name does not mean that this is now only usable to cases where you are somewhere in between atoms and you know the Continuum yeah that's not that's not what it is so it's not it's not dependent on the size that does not mean that the theory would be applicable only to a certain size in principle they should all be applicable to all sizes but of course the um just what do I write so but of course please measles okay miso is in between mesoamerica is in between South and North America yeah so and the the idea the idea here is that this is a description that is in between the description of a Continuum where you don't know anything about molecules at all and between the individualized molecules yeah so you do something in between yeah you you um you still have molecules but these molecules are now no longer distinguishable yeah so you have just the statistic of the molecules that restrict the workings okay so we should not we should not be tricked by thinking that there is a certain size of the system where you do molecular Dynamics yeah so if if the thing that we want to understand is so and so small then we have to use molecular Dynamics if it's so and so large we have to use the navier Stokes equation and then there is a gap in between somewhere where you have to use the boltzmann equation yeah that's not this is two simple um a few of the world but you will certainly if you look in any book or something you will certainly read this somewhere okay let's let's look at these these parts of the um of our equation now so we have the time derivative F of f of our probability distribution function this is of course very simple the change with time then we have this Plus C dot f sorry napla F so this is the usual now plus or the derivative in in space and this guy here is the advection so in the navier Stokes equation we have an addiction and here we also have an addiction then we have here a false term this is something that's I forgot to write down in the Navy stocks equations in the Navy stocks equation you could also have a force term but I left it out unfortunately so what does force term mean here what we mean by these forces here are the long range forces long range forces is something that um you know it doesn't doesn't necessarily [Music] um in in the real physics it doesn't really play play a big role or only the degravity probably so long range forces are these forces that that you assume somewhere yeah um which are not balanced therefore you say okay we are in a gravity field and there's gravity this gives us a force this from from very far away yeah and this acting on all of my on all of my particles so these forces I put here yeah in order to distinguish them from the short range forces so short range forces are the forces that act between particles which come in contact to each other so these are the binary forces that we previously spoke about in in the atomistic view so and this is here in the Collision operator so this thing here does the collision and this is the shorts range forces so these are the forces that act when things come into contact meaning they have Collision so there's something very interesting with this advection thing so first of all let's let's look at it we have here of course a particle and this this particle is its time T it is here and at time t plus DT it is there so it has moved in this direction and here it is at position R and up here it is at position r Plus b r yeah and now it is of course moving with its velocity saying that the velocity C this is the change in its position over time okay so this is the microscopic velocity how does one particle change its position over time so a particle particles move in the direction of the velocity of course so and remember that age is a function h f is a function of the velocity the microscopic velocity so this is very important we always have to distinguish between microscopic velocity which is the thing that we have in the boltzmann equation and the microscopic velocity that we have in the navier Stokes equation so and now there is this um very interesting thing we have a ction operator in the boltzmann equation and we have an adduction operator in the navier Stokes equation yeah so let's let's write them let's write them down again I leave here a little bit space because I want to write something on top so the change of the macroscopic Velocity in the navier Stokes equation yeah so this now we should write Navia stocks equation here and it's e plus u dots U so this is the advection of the Velocity the velocity the microscopic velocity is self at Vector okay so it is self-advicting so we have multiplication of U and U with something um related to U squares so this thing here is non-linear and the non-linearity of this thing is one of the big troubles that we have in solving the navigation equation now look at this terrible boltzmann equation DT f plus c into those scalar product on on napla F okay Bossman transport equation this thing is linear so there is no no non-linearity in the addiction operator which is one of the big advantages of the boltzmann equation even though the Navigator's equation is only in four dimensions so it is space and time the boltzmann equation is in six in seven dimensions yeah but therefore we have a linear um so but then we have to deal with the Collision which is the thing that now becomes more complicated and what we will assume is that there's only binary Collision so usually if there would be um collisions of more than two particles at the same time you could decompose them into collisions of two particle each yeah so this is why we only look at two particle collisions but this is important that we only um that everything so DC read that we built is built on binary collisions so binary collisions means two particle meeting so when two particles when two particles collide so they go either this or this way yeah so then they come in they change their Direction and they go out again so they do this in three-dimensional space um okay okay so no I I okay let me try let me try it first so we have particles in now in in velocity space so this is this is supposed this is supposed to be now C and and oopsilon so now trons two-dimensional way obviously yeah and there is one particle coming in this we call C1 as is particle one coming in and another particle add another angle comes C two and then they leave they come with in this angle and then they leave at that angle at a different angle say this is now C one um Dash and this is C2 after Collision yeah so we call this here so the black one I hope you can distinguish this this is pre-call and the blue one is post called post Collision state so and they will end up on this circle and the reason why they will end up on this circle is because and the Collision has to conserve energy and this is probably better better understood when we draw this in a different in a different frame so now this here is in the so-called laboratory frame what is the laboratory frame can we expect can somebody explain what the laboratory frame is not you frames of reference inertia of Franklin with this this you love where the laboratory frame is a frame where I said okay we're in the laboratory I'm sitting on my desk and I Define myself as being at rest yeah and everything else is moving relative to me yeah so this is not related to the problem there's something else this is a center of Mars Center of Mass frame is one that we need quite often so we can draw the same picture in the center of Mass frame of reference so Center of mass is means a is is we are we are now moving with the average velocity of these particles yeah so when we are moving with the average velocity of these particles there's something very interesting happens they never they never come at a strange angle right because they come at a strange angle only because they have a they have a common component in their velocity so if we put us in the frame of their average velocity they can only hit head on yeah so if we are in the center of Mars frame of reference they will always come this way so c one NC 2 will always come this way this is not a very beautiful C and also because they have to conserve momentum they have to go out the same way so they always go out heads so they come in head to head and they go out tail to tail so to speak so then this here Greek is denxi one dash in the CSC 2 Dash so this is what happens um so this is how it looks in the center of Mass frame of reference and what can we say now about these collisions these collisions have some conservation laws I should write in Black to be consistent conservation what what is conserved in these in these collisions everybody gives me something that is conserved starting from you what is conserved in the Collision energy okay energy is conserved um okay I probably because I should probably write this down here to be energy so what is conserved momentum yep momentum okay sorry I left you out okay now okay then there is Mass I put here so now you have to okay so [Music] would also be conserved in in principle because there is no yeah this is also it's not on my list but it is conserved so okay angular momentum is still conserved what do we still have and we have okay I have only one more thing on the list now I have to something very simple probably too simple the number of particles is conserved yes there's two coming into going out okay there I create the other ones have been more obvious yeah um so these these things these things are all conserved in here and we look at now we look at two in particular namely the momentum so we can express this in an equation namely that there's a mass m let's say the particles have the same mass so that we just call them M okay M we do not distinguish them though they would have the same mass so C1 Plus C2 is equal so this is the incoming pre-collision state is equal to the Mass the sum of the um momenta at of the outgoing case and energy energy is one half m and the velocity is square so now the velocity is square is C1 times scalar product itself yeah plus C two scalar product itself would be equal to one half M so this is something that you can erase and see one after Collision times c one after Collision Plus c 2 after Collision times itself so this is the definition of [Music] yeah of of energy and this is conserved so this is something that that happens to be um that gives us information on on what is coming in and what is going out important invention of Postman in regard to evaluate these things and this is the so-called molecular chaos molecular chaos so molecular chaos is the proper English word but even in the English literature you will still [Music] um read the Austrian the real Austrian word that is foreign this is a proper scientific term which those challenges this is how boltzmann called it molecular chaos is something that sounds too too much 20th century and that's the assumption that Postman made because he does not distinguish between particles he also does not teach degrees between collisions of particles so the only thing he is carrying for is how many collisions happen yeah and the Assumption behind this is that the um that everything depends only on the number of the collisions and not on anything else and for this to be true the particles should not know anything about each other beforehand which mathematically expressed is colliding particles are uncorrelated so what does it mean is uncorrelated uncorrelated means that if you know something so I give you one particle and I tell you everything about this one particle and you know everything about this one particle and then you still know nothing about every other particle this means uncorrelated yeah and the the thing um okay so when particles Collide they are in the same position okay so this is probably something that we could no yeah but we do not know anything about their velocity so this means that the velocity of particle one is not influenced by the Party by the velocity of particle by the way this is completely an assumption because this is not this is not clear and is most likely not even true right but it works very well yeah okay so this was his assumption and um what what we then do in in order to simplify things if we use a parametrization of this Collision with the direction and this direction is expressed by a vector which we call small Sigma supposed to be a small Sigma and the sigma gives us just a direction which means it has length one so this this Sigma just tells us so how do I rotate um if I very very briefly I don't like to do this but I just um very briefly go back to this this picture of the center of Mass Collision yeah the sigma tells us um by by what Vector do we rotate these um this this outgoing particles yeah in order to get the the new velocity so okay let's let's go back here so the particle after Collision which is written with this Dash here so there's nothing to do with the derivative um particle one can be expressed first of all by the center of Mars of the velocity so this is the center of mass velocity obviously Plus now comes the um the the magnitude of the of the difference between these velocities divided by two and this multiplied with a direction vector the outgoing Direction Vector so this is particle one and in the center of mass system so the center of mass we put in front right this is the center of mass and in the center of mass system the part the other particle has to go exactly the other way so we already know that then that c two after Collision is again the center of mass now minus because going exactly the other way C1 so this is the incoming the difference between the incoming velocities times Sigma Sigma Sigma is Sigma is the direction it's a vector that just gives the direction the parameter yeah so could now could be anything so we know particles go in but we don't know where they are going yeah so there could be anything the only thing that it has to be it has to be it has to have magnitude one okay and with this Postman derived um the so-called Collision integral Collision integral which is a long thing to write so I have to start very much here in one corner so the Collision integral can also be expressed as a Time derivative of the distribution C of the distribution C all for only the Collision no addiction yeah so this is um time derivative of the effect of of f due to Collision and this is now a double integral over capital Sigma capital Sigma Sigma is the direction yeah so the small Sigma is the direction so this means all directions an integral over all possible directions because it could come out in any direction and integral over all speed so very um over all speeds of the particle 2. so this um I write the capital Sigma over all directions because Sigma is the direct small Sigma has the direction yeah velocity is small XC so I write a large c meaning all the velocities the entire space so this is a large C if you are not completely familiar with the correct alphabet and now we have something that's called the Collision kernel that is a function B which we currently do not specify specifically that is but this function B is dependent only only on the difference between the particle velocities not on the absolute velocity and the direction and now this has to be multiplied with the probabilities of particles actually meeting each other yeah f is the probability density function of particles being in a certain in a certain space so the probability that two particles meeting each other means they have to this would be the product of these two and distribution function so the probability of one particle being there and the probability of the other particle being there the two being independent of each other by definition through the postmaster style answers then the probability of the both being together in the same space has to be their product yeah and this would be now if off and I write the outgoing the outgoing guy first so particles outgoing going out is F1 outgoing times f 2 outgoing I write these first because they they emerged through Collision so they are positive this is the gain term yeah so there are new particles that is the the particles that that are so true so the the probabilities so to speak they are generated yeah they come into the system they are this is a game tool on the other side the particles that were going in so the state that was going in is destroyed yeah so they are going out of the system and therefore I have to write them with a minus so then there's if C1 so that was in going times it's meeting a guy another guy with this other velocity and this is um this is the loss so this is the term that is lost and all of this has to be integrated over over the velocity space of the particle 2 and all the directions so and now I have obviously forgot to write here in the beginning yeah so this is the velocity C1 yeah the Collision acting on velocity C1 so now the important thing about this what I've already said I have to write down the Collision kernel as we call it B is a function of the relative velocity and the direction only so another function of the absolute velocities oops writing a sigma and we have to distinguish between this gain thing there's other particles as we say scattered into this particular velocity and the laws is the particles scattered out of a particular velocity state so and these things have conservation laws of course now we come back to our conservation from the so B has to fulfill some conservation laws and okay so this will be the Collision integral has to fulfill all the the things that we have stated before namely Mars momentum and energy so if they are so the thing is if there are two particles and there are Point masses they also conserve automatically angular momentum in that case yeah so this is why we usually do not mention just explicitly um so they do this by more or less by their by by their fall so this means so how do we express this conservation laws in in the Collision operator we do this by recognizing that if we integrate everything now over velocity one over the entire velocity is base one now remember the Collision integral is an integral over velocity Space 2 of the other particle now we integrate over particle one and in this particle one in this integral we put the Collision operator where we have already integrated over particle two now we integrate also the particle one and we multiply this by this Phi and to the integration on particle on the particle velocity one with Phi being um a kernel function that comes out of out of these that for example could look like this oh no it would look like this so because I have three and I have I have three conservation laws yeah and these are expressed with this integral so if I integrate everything um if I integrate the entire Collision operator with v equal to 1 what I had what I would have is the mass of the particles and the mass of the particles does not change in the Collision therefore the integral must be what it must be zero right so this because this here is Mass and the next gives me momentum so if I integrate If I multiply the Collision Operator by the first microscopic velocity integrate everything I get the momentum change of the Collision operator it has no momentum changes it has to be zero when they create with velocity Square I have the energy it does not change energy yeah so it must be zero so this is how I express um and so this is this is how I expressed the conservation laws in in this way in the Collision operator and by the way something is very important these things what we just introduced so an integral of a distribution of a distribution say call this if now it could be any distribution but now it's the one that we are using here at the moment the particle distribution an integral of distribution f with a kernel Phi Phi is called a moment so for one strange reason um here in civil engineering people people often believe moment there's only one moment which is the bending moment yeah so if you if you say moment they always think about bending moment a moment is neither this is a moment because it's an integral with a function over over some space the bending moment is a moment but never say Just moment because what you learn here is we'll have zillions of moments yeah so everything we're dealing with the rest in the here we are we are dealing with moments what is the question yeah yeah it's a first okay so the bending moment X means it's a moment in space Here's a moment in velocity yeah so you can have moments in different moments in space moments in time whatever but in here we have moments moments in time okay um but here we have moment in velocity space so and they are okay uh yeah maybe this this I have to write first so the inter the Collision integral has five has five conserved moments the reason why it's five is because the velocity each velocity component is is a moment by itself is conserved by itself yeah so the velocity the velocity moment has three components so there's three of them together with mass together with energy there together five and they are all conserved and these conserved moments they have an um very interesting relationship to the Primitive variables in the NS in of the NS Navy stocks equation in contrino [Music] Theory so because these guys are moments okay they are to be a little bit more specific than I have on my on my sheet here um they are mainly um okay so they are moments okay I don't write this because it's not consistent their moments the one little difference is that most of them are um intensive moments yeah so we have one extensive moment this is why I don't write this an extensive moment what is an extensive variable so if I integrate F just over XC with no kernel or with kernel being one and we call this the zero moment you know when the kernel is one we call it the zero moment well because it's the velocity to order to order to order zero yeah because the order the the exponents yeah so we see this we see this at velocity velocity U so we multiply this now our kernel our kernel is C1 and there's a little difference here namely that I divide through the density through rho which makes this an intensive quantity yeah an intensive quantity is one or can can you explain because you didn't get the can you explain what an intensive quantity is yeah so if you if you add two intensive quantities together you don't get the sum of of the other so the intensity um the density is an extensive quantity because we put more things into it you get um uh you you get a larger you make the system larger you get more more Mass okay okay Mouse yeah and you get more mass and um the other one um is is now in in so velocity is the Intensive quantity that corresponds to momentum and what is the Intensive quantity that corresponds to energy some ideas so actually what we have here is we have to subtract we have to subtract the microscopic velocity and we have to divide through rho times d and and what what will this give nozzle this corresponds to the energy but we have taken out the center of mass velocity and we've divided through the mass and D is the number of dimensions then it doesn't make so then then we have the total okay that would be the total energy and if we do not divide it by the mass yeah but we want we do not want to to have the total energy but we want to have the energy just distinguish the the energy um actually the thing that gives us the temperature yeah so if we if we subtract the um so because you a microscope EQ microscopic velocity that would be the kinetic energy of say the air or the yeah that we are looking at the kinetic energy and the rest if we subtract this then it will be the thermal energy and the terminal energy of course it gives us the temperature if we take it um intensively so I'm getting let me see I hope that I can deal with okay we have we have something like 17 minutes and we have the most interesting thing in the world still to learn today and this is the this is the so-called ether theorem that Postman proved and which is one of these these very important um things here in the world so it is not known to science because we have no voice recordings of of Bosman how to read this letter so in English literature most of them think this is an H but an H looks exactly like a creek ether and it is it is believed that for Postman meant and because he was talking about something related to energy it should not be an age it should be in either so this is why we believe is an easel theorem but you will also hear people say this is an age Theory okay and now the idea the idea behind this is um to look where where the universe or where where particles go go to and actually where the entire Universe goes to so and I now write this sentence when gain is equal to loss and even though I'm German I purposely say when and not if yeah so when gain equals to loss Collision because eventually this will happen um when gain equals to loss Collision does no longer change the distribution and then the distribution is said to be in equilibrium I said is said to have reached equilibrium now one of the things that Baltimore wanted to do is to derive how this equilibrium looks like this is um on the way there he discovered something that's even more interesting so what the equilibrium is we learn only next week now we and we look at what he discovered on the way and arriving it so um say I put here an equation 1.1 and this is our Collision operator and this we have decided so I write the integral without without the bounds yeah because otherwise I have to write a lot so we have F1 Dash now being an equilibrium times f 2 Dash also an equilibrium minus f 1 equilibrium times F2 equilibrium so I have not I have not written and the the arguments of of um of B so there's a multiplication in in this area yeah yeah um what is that distribution signal right the distribution this is the distribution we've previously used the particle distribution yeah uh just a a let me finish this equation then I so we are integrating over xc2 in so what's so if they are in equilibrium then there will be no further change so this means that this here would be equal to zero let me briefly so this if what I call F1 maybe that's your question yeah it's a shorthand because now I have to write a lot of these things now is is if is fc1 and if I write f 1 dash this is F1 Dash so and the equilibrium and that I've written up here this is okay maybe mathematically not completely consistent I just want to imply that now this is the stage so when they are in equilibrium yeah then um by definition this is how we defined it then nothing changed anymore and then the Collision integral becomes zero right um did I somehow also answer your question yeah okay okay now both none 10 minutes boltzmann defined a strange function that will only become clear after some calculations a functional so e sub of T ISO of t to help identifying the equilibrium and the thing that he defines h now I said H myself ether of T so it's now the integral over f F1 times the logarithm of f 1 and all of this integrated over C 1 and the entire space when I say the entire space I mean the universe okay so we take everything all velocities all space each molecule in the universe yeah is included in this why is there logarithm so this is a big mysterious so he he defines this and we'll later see what this leads to yeah so he he defined this function yeah mysteriously and looked at what happens with this function okay we'll see um I have to write integral integral Over All micro velocities and all coordinates so remember this is the entire universe here otherwise it's not correct now coming to equation 2. 1.2 now what he did is he derived he did the derivative of H with time and he put this derivative into the integral which is now on F of C1 of okay I noticed I have written it just in a different way I hope you forgive me um okay no no I write it in the shorthand so that is consistent if F1 yeah lock F1 so but remember there is a c inside yeah the C is inside of this so um this integrated over c one and over the entire universe so now we can do the um what is it called the chain rule on this integral sorry not the integral but on the derivative which means we first take out D DT F1 and this then the wave put this on the first one so this goes into the logarithm of F1 and if we take the other part then the and then the derivative of f of F1 would be 1 over F1 yeah times df1 which we do not take the derivative of gives us an f 1 divided by F1 yeah roughly clear okay so MD take C1 Dr and now the important thing is that this guy here is our Collision operator now we can put the Collision operator in this in order to show the esa theorem okay I'm I'm going to finish this it's even though I only have five minutes left but so in order to show that this derivative is smaller or equal than zero and the equilibrium would be the case the equilibrium we have reached if there is no change anymore and this is when the time derivative is equal to zero and now the question of course is why should this be the case and for this we have to put the Collision operator into 1.2 so if we to do this we get to one point 1.3 D Isa by DT is equal now comes the Collision operator times F1 Dash F2 Dash minus F1 F2 and this goes into one plus log F1 and now we have to integrate over a lot of guys namely C1 C2 so C2 comes from the Collision operator the r and the sigma so D Sigma also comes from the Collision operator now we do not distinguish between particles which means that F1 and F2 are interchangeable and the integral goes over both of them meaning that if this is true for equation 1 F4 for F1 then it will also be true for F2 which means we can write the equation 1.4 which looks exactly the same with one small difference so we have again one f one dash F2 Dash minus F1 F2 and here one plus now there's the only difference is that now we have the logarithm of F2 and here we have the entire stuff again now what we can do is we can simply sum these two things together 1.3 plus 1.4 and you see because there's the only difference here it's giving us the equation 1.4 we sum them together so on the left hand side we have twice DH by DT is equal to the integral so most of this guy is the same so if just to rewrite it again and here we have just 2 plus the logarithm so if we sum up two logarithms we get the product of the two guys so it's F1 times F2 and again the integral over the entire space B is Multiplied yes as I said in the beginning it's always multiplied foreign now there is another thing which is very important yeah the Collision is symmetric yeah so it can go in like this and go go out like this and it can also happen exactly the other way around so it can go in like this and go out like this so um and I mean this is this is the most amazing thing because you have to keep this in mind we now assume that the Collision is symmetric in time so it happens forward the same way as backwards and from this will arise that it only goes in One Direction the assumption is forward and backward is the same yeah Collision is symmetric can go forward and backwards which means we can exchange F1 F2 Dash after collision with F1 F2 before Collision so we get another equation one point 1.6 which is the same chest with the guys exchanged so again twice Thrice and the left hand side is the integral B okay B multiplied by now I exchange yeah these these distributions now they come out before Collision minus the one with the dash here times two and now comes log b f 1 dash F2 Dash and all of this is still integrated over the entire velocity space in both directions and over the entire universe and over all directions now we are almost done we are also almost done in time the last thing that we do now after we exchanged also the direction of the particles with some equation 1.5 which is the forward Direction together with the backboard Direction 1.6 which gives us four times four times this [Music] um 4 times d h DT the integral B now I put them together f um in in a common in a common Direction and I take this common direction from 1.5 yeah if one F2 so because because they have different sign that the tool that comes in both of this equation this will be eliminated due to this inverse sign and the other one will go log F1 F2 this is for coming from 1.5 minus look F1 after Collision F1 after Collision times F2 after collision and the entire thing DXE 1D C2 D Sigma and d rho and this thing here is smaller than zero okay I'm doing two minutes over time but I have to but I have to explain this the the proof is done this is the proof you only have to see it yeah this thing here has to be smaller than zero because of Y the law the logarithm is monotonic and logarithm is monotonic so this means that if F1 F2 would be larger than f 1 dash f 2 Dash yeah then if this is the case or smaller if this is the case and the guy is here so this bracket here would be negative right but in the same in the same way the logarithm would then also behave the same way right and here on the other side we have the logarithms but the other way around so this is why we put the logarithms into it so meaning that if the first bracket is negative the second bracket will be positive and if the first bracket is positive the second bracket will be negative and so this is always negative so this is always either smaller than zero or it is zero and this means because this is a Time derivative the time derivative is always negative so this means that either provides a direction in which things happens so everything in the universe um changes such that ether becomes um smaller and the the very interesting thing is that the assumption that everything here is based on is that the Collision is symmetric in time that it can go One Way exactly the same as um the other way so the from the Assumption of reversibility see okay let me reverse the bill reverse the sorry build collisions arises the irreversibility of the system so if parts are reversible then the system is irreversible so this was the in tune okay okay I'm I'm five minutes I think over time um I have two more cents that I just write okay I'll write them down so the system goes closer and closer so it's approaching to equilibrium equilibrium is the states where entropy does not change anymore so if this is zero and this here is it is the scaled negative so here's unfortunately got them the wrong sign to it the scale negative entropy okay sorry for doing overtime today um yeah any any further questions so I you know in this um you have to do a homework for this for this course and I have sent you I believe I sent you an email that you have to choose one of these 10 um 10 assignments and have to write a little bit of code in order um yeah to do this here we of course start first of all with with theory in order to understand how things actually work okay but get your assignment so please form some troops or groups up until up to three persons if you really really like you can also do it alone but I do not suggest this okay then see you next week