Okay, so this is a quick reminder that I'm going to do just to remind you so that you see the connection, the formal connection with what we're going to do later. So Schrodinger equation tells you this is, today it's going to be one of the most technical days, but if you find it hard to follow the technique, at least try to, or ask me and you cannot see, it's hard, why? Because it's yellow, because it's small, it's small and yellow, okay, I'll make it larger. So as I said, if you are, if you find that the technique is beyond you, try to hop to the conclusions and, okay, so, okay, I'll keep this as my own. So, for those of you, I think that did physics, the Schrodinger, you have the Schrodinger equation is an equation for the evolution of a wave function, which is a vector, which we denote as vectors like this, with a bracket notation, and it's an equation that says this, and where H is the Hamiltonian, for example, okay, and is this okay now, yes, okay, and so how do we solve this problem? It's a linear problem, so we first construct a basis of eigenvalues, eigenvectors and eigenvalues of H. Next, we write our initial condition as a linear combination of these things, which is, it's a basis, I can do it, and then I know that this equation simply in this basis takes the simple form, this simple form, and so, and in this way, oops, sorry, I can, I can solve exactly the equation by the fact that I have diagonalized this. So, those that, so, okay, so why am I telling you this? We are not going to do quantum this time, but many of you, or I suppose all of you who did physics have studied this in the course, and have studied in particular the notation of brackets where you denote vectors like this instead of denoting them like this, but it's a usual notation that we use in quantum mechanics, and this instead of denoting it like this, dagger means that the vector is horizontal instead of being vertical, and the scalar product instead of, so, what will happen to us is that we will find in our, in the course of what we're going to do things that are basically, technically the same, although the interpretation is completely different, and I think that it's useful to connect things, and so, again, the notation that you find in all, but I think, well, my notes for sure, but I think that Kadanov Swift did the same, but for historical reasons, this kind of formalism is applied by physicists, but not by mathematical physicists or mathematicians, and so, what happens is that the same physicist, without realizing, is exposed to two different literatures to do things that, in many senses, are the same. I told you already this happened before, and with the idea of large deviations, again, here, this is what happens. We will see now the stochastic problem, which is not a quantum problem, but usually because you study in your life stochastic problems later after studying quantum, at least unless the courses have changed a lot, it's, I think, like this in every country. So, you study a whole lot of techniques that you use for elementary quantum mechanics, and then you go to stochastic and you change gears, because the history of the world is different, and then you do the same things, and because they have been in the hands of a different set of people. So, I don't like this, and so I try to use as much as possible the notation and the analogy with quantum mechanics, also because it is important. So, I keep them here. Now that you know them, you can perhaps recognize them, but I cannot leave them in the big version, because I'm not, I am not going to be able to do anything in the black equation. So, this equation, sorry, in terms of the eigenvalues and eigenvectors becomes trivial and can be solved, and so we can study the evolution of any wave function just by diagonalizing H. So, with this said, let's do it with what we're doing. So, we had arrived to two different equations. One is, okay, so when we did the, with the noise, Markovian and, so, delta function and proportional to the temperature. So, these were the two equations that we did last time, and F means it's an optional. Sometimes we will want the system to be driven by forces that perhaps don't derive from a potential. So, I say it explicitly here. Okay, sometimes I will omit it, and then the system will be in contact with forces that do derive from a potential. Okay, and what we said was that, okay, so this one is the one we arrived to first. Remember what was the logic? We did the bath, we got a Langevin equation with memory, then we said, okay, let's imagine that the memory of the noise is very short, then we arrived at this one, and then at the end we said, okay, but suppose that the inertia can be neglected and we are over-damped, then the second derivatives can be neglected and we fall into the over-damped equation. So, notice that this one is an equation that happens in phase space, and this one is an equation that happens only in space. So, this one, if I want to write it like this first order, like in Hamilton's equation, I need to make this happen in the whole space of coordinates and momenta. If I want to eliminate p, then I get a second derivative thing, but second derivatives in time we don't like, because we don't know how to solve easily the equation. So, just like when you do Hamilton, this one lives in phase space, this one lives in ordinary space.  Now, the point is that this equation is an equation, it's called stochastic, because those, the two of them, because they have a forcing, let's say, that is a function that is a random function of time, which is delta-correlated, white noise. So, if I put this equation, let's say this one in the computer, I have to launch a simulation. Let's take this one as an example. There is a force, there could be an extra forcing or not, and then I have to generate random numbers, and p, p, p, p, p follow the thing. Now, when I do this, I'm going to get a trajectory. I can do that and it's okay, but I can do another strategy, which is, because this is the random that corresponds to the randomness of the bar. I can do this, but I can also repeat with another realization of noises. In the computer, you do that by changing the seed, and another, and another, and so, at the end of the day, for example, at time t, I'm going to have a certain, in my space, a certain distribution of probability. This distribution of probability is considering the ensemble of possible noises. And so, what we're going to do today, which is going to be our job today, is, okay, now imagine that instead of doing this, I want to understand how does this work? So, it's the probability distribution when I do the sampling of all the possible trajectories, and then I collect my data, and then at each time, I have a cloud of probability that is expanding. So, can I get an equation directly for this one that derives from this one? And these two equations are going to be sort of counterparts of the same phenomenon. Of course, if you're doing here, your space is PQ, and if you're doing here, your space is only Q. Okay? This is, it's PQ here because I don't want a second derivative, so I do like when you do Hamilton. Okay? So now, you're going to derive the equations that are satisfied by P of Q, and in this case, by P of Q, P and the time. Is this okay? Okay. So, let us, one word, and this is quite a horrible thing, but I have to tell you this. It's not nice, but suppose that you have this equation. Look what happens. You have Q dot, which is a velocity, and here there is white noise. Peel the RRL things. So, you see that Q dot has a component that comes from this. This white noise is something that looks like this. At each point, it's completely uncorrelated. So, you already see that I would want to take a derivative of this and find the acceleration. It's going to be awful. So, this equation has problems. This one less so because here it is the acceleration that has the problem, thanks to the fact that this has inertia. There is a mass, and so the velocity is well defined, and it's the acceleration whose derivative is problematic. Now, these things bring a set of problems in mathematics. So, I'm going to be brief on this, but it's something that you will need to know. It's a chapter I dislike profoundly of this subject, but it's inevitable. So, imagine you have this equation, let's say, without the forcing, and you want to simulate it. So, you take intervals of time, T, and so you're going to do like you do in a computer. You simulate this to get the next time. Here you take a random number, and here you have to normalize it like this. I have to explain to you this bizarre fact here. So, this, you know, when you're solving step by step an equation, you discretize it and you move by steps, no? In the computer. Now, why is it here that I have a delta T to the one-half? You see, normally this is the kind of thing you have. Why do I have a delta T to the one-half? So, when you think of it, the reason for this is that if my delta T is this size, and I'm finding one random number every delta T, I'm picking up a random number, and I do it n times, n times delta T. So, I have n, let's say, m random numbers, no? Eta here, eta here, eta here, eta here, eta here, eta here. Now, and of course, m is the true time divided by delta T, the time interval divided by how many times I... Now, imagine that I take half so that m goes to 2m, okay? Now, I'm taking 2m random numbers, but what is the size of... So, that is a random number, but what is the size of 2m random numbers? When you think of it, it's not twice, but it's the intensity of it grows by square root of 2, because, you know, when you sum variables, it's the variances, the square of the standard deviation that is added, and when you want to see the typical size, you take the square root of that. So, if you have, for example, a score that goes between zero and one randomly, and you do it 10 times, your expectation of this... So, sorry, a score that can be plus 1 or minus 1, and you do it 10 times, it's called... The average is zero, but what is the amplitude? Well, the trick is you have to do the sum of the squares and then take the square root. So, when you make 10 steps at random, your typical size is square root of 10 and not 10, and this is because the signs are random. And when you do 100, it's the square root of 100. So, the noises add as a square root, and this is the explanation of this fact. But this is very concrete. When you're going to put in your computer this thing, you will choose your delta T. But if you make delta T so small that the number of steps is multiplied by 10, then this has to be scaled, not with 10, but with square root of 10. It's a tricky business. Remember it. It's awful, and I will continue a bit more.  So, you will find this problem. On top of it, I tend to say that it's not very important. You will find that in books it's considered to be more important. Okay, so another problem you have is how am I going to discretize? And there is another way of writing this equation, which is taking the intervals in a different way. So, I am doing the same in the computer. Here, the V. I'm not going to take it. What has changed in this equation is that I have taken things, not evaluated before, but before and after in the interval. This one is not very nice from many points of view. For a computer program it's a bit of a disaster, because you want to find out this, but you have it also here, and you have it in a way here. So, it's not very nice for making a step-by-step computer program. But mathematically, this form has better properties. I'm not going to get into the details, but so, the words you have to remember is that this is called the Ito Convention, and this one is called the Stratonovich. I promise you that if you're going to do all sorts of complex system dynamics, you will fall into these two names. This is just a reminder. I don't think it is something that is too important conceptually. Why do I think it is not too important conceptually? Because all the problem here comes from having taken the Markovian limit. Remember that we had this memory function that was like this, and then we said, okay, we're going to the limit because we prefer a Markovian, a local in time thing, and this is going to be delta of t minus t prime, and this is how we got these equations, and this is how we got that eta eta was delta. So, this is eta eta of t, eta of t prime of 0 or of t prime, okay? And this is t minus t prime. It all comes from having done this limit. So, Ito Stratonovich, how you normalize the delta t to the minus a half, all of this is a headache that we have gained because we insisted that we wanted a noise that is local in time, that is, has delta correlate, that is white noise. And what happens is that when you do white noise, this function is really horrible because at each point it takes a completely random variable number. Now, I personally think that conceptually, this is not so important. Why? Because at least conceptually, I can always imagine that I am solving this problem which is what life is truly about.  All noise will have, as Amakhesh asked the other day, when you are being crushed by molecules, they take some time to replace themselves. So, before the second one comes, there is an interval. That interval is very short, but it's not completely instantaneous. So, the noise is not white ever. But, you know, for certain things, and for every time you use this equation in your life, you use a thousand times this one. So, unfortunately, this is life. But, okay, so this is a technical problem, but you have to simply be very careful because if you don't discretize correctly, your results are different. So, it's a, it's a, let me put it this way. It's not a nice fact of life, this problem, but you have to bear it in mind, okay? Yes. So, about the Itto Convention, isn't it dimensionally incorrect? Like, the two sides of the equation initially has same dimension. And then square root of delta t is multiplying with delta t. Yes, but remember that eta eta is a delta. And then the delta, when you rescale, it has the dimensions inside. So, at the end of the day, don't worry. It's, there are some dimensions hidden here, which is the correlation time with which you regularize your delta. So, yes, it's... So, the, and the difference between us in Stratonovich, we are taking the midpoint convention. The midpoint convention, yes. The idea is that the same physical situation can be written in Itto or in Stratonovich. Now, if you write an equation in Stratonovich convention, you can transform it to Itto so that the physics is the same. But, if you did the wrong thing and you wrote it thinking that you're writing it in one convention and not in the other, you can change your physics. So, it's either two things that represent the same. But then, if you write an equation and the same equation, you interpret it ala Itto and ala Stratonovich, then you can have different results. Especially, this happens when here there is a multiplicative noise, other things. But, the whole point is this, and the same happens in a lot of domains of physics. You write things that are continuous, like this equation, but they don't have a real meaning in a continuous. You need at some point to say, wait, what do I mean by this? I should discretize. And, and this is the recipe to discretize correctly. So, to tell you personal thing, there is one such thing as Itto calculus. Mr. Itto made an entire calculus out of this and did it properly. And, there are experts on this. And, I, I swore to myself at some point of my life that I was going to die without learning it. And, I'm in a good direction to reach that ambition. One thing, there should be equal sign in the second equation. Before square root of delta. Here, there is an equal sign, yes. But, it doesn't really matter. Second. Is it an equal or is it equal to zero? Okay. Doesn't matter because the sign of this is random. Okay. Okay, good. After telling you this, remember this. Be careful with this. I don't think it is enormously, ah, one, one, one extra comment. Maybe we're going to do, maybe. I'm not sure. The distribution of work. Now, work, as I said many times up to here is force. Let's say it's a force. Scalar product with velocity. Okay. This is the work per, the power you're doing. No force times velocity is the power you're injecting. Now, look what happens.  You already see it's a horrible thing here because the velocity is equal to something that might be smooth, but time plus white noise.  It's just hopping madly from one point to the other. So, the velocity is, has this velocity is like this. And you're multiplying it by force, but then it's a mess because it's a force at the same time or delta T more or delta T less. And you have, you run into all of these problems. In here, thanks to inertia, something lovely happens is that acceleration is horrible, but the velocity not so much because the velocity is an integral of, of the acceleration. So, the velocity is continuous in this equation and not in this one. Thanks to the mass. So, this object is horrible for over damped, but it's okay for when you have a mass, when, when you have the top equation. Okay. So, you see something that comes from all these questions from having white noise that seems so terrible. Just add a mass and the problem goes away. So, this is why, and sometimes it is better to leave the mass there and then eventually send it to zero at the very end. So, you see, this gives you a measure of how conceptual these stuffs are. But if you read, for example, books of economy, econophysics and economy, I think, too, they start with a very long introduction on this one very mathematically on this problem. Okay. Having said this, let's go back to our, our problem. And of course, if you're going to put stochastic equations into the computer, you have to be very careful with how you discretize. Okay. So, now I'm going to, so, as I said, these equations, you can consider them as single trajectory by single trajectory. I mean, for one realization of noise or the distribution. And there is always a back and forth between one thing and the other. One thing I would like to do for you next week is to do a couple of things that have been done recently, quite recently for single trajectories that are very cute where you really don't work on the average or better. You work at the same time with the average and with the single trajectory. And this has been one of the big developments in the last years which have thousands of papers. So, and those we will see next time, next week. Okay. So, let me take this one as an example. And I would like to write an equation for the probability distribution now, forgetting about individual trajectories, just considering a sampling of the whole thing. I will use a method that is, I think, nice, but there is not the one you find in the books. So, imagine that this term would be absent and that you only had this in one dimension. By the way, we don't gain anything by doing many dimensions, so most of the time I will work with one. What you learn is okay for. Okay. So, what is this? This is just a random walk. You are kicked randomly to the right and to the left. So, what would the distribution be in this case? With time, it would be, it would be diffusion equation. No? So, that if I consider all the possible trajectories, this is time. At a given time, my distribution is a Gaussian. This is just, okay. So, this is, and of course I can solve this, but it's a diffusion equation. Okay. Sorry. If I'm in one dimension, if we are in one dimension, it's simply this. Okay. Now, imagine that on the contrary, I had this other equation without the noise term. So, I'm doing separately this case and this case. Okay. The solution to this, it's just a probability that is being advected by this force field. And this is a standard exercise. In many dimensions, it would be like this. And in one dimension, it would simply be. This is the equation when you have a force field, how a probability is carried over by this force field. And in fact, because I could, even in a computer, simulate this by switching from one to the other by very small intervals, what the result of the whole evolution is the, is the addition of the two. And this is the equation to which we want to get in many dimensions. Oh, I'm changing notations all the time. I'm sorry. Can I go back to Q? Sorry. But if not, I'm going, this is going to become, if we allow for a possible, okay, so we get to this point and we were, we're going to talk a lot now. This equation is alternatively called Fokker Planck, which I think is unfair. Perhaps the correct name would be Smoluchowski. But mostly, I suppose, you're going to see it referred to as Fokker Planck. And this one is the one that is related to this one. Okay. So I want to pause a little bit and make sure that you understand, even if you didn't understand how I got there, but that you understand the meaning of this equation. First of all, first of all, this forcing, I have added it, often it is not there, or sometimes it is the only thing I have and I don't have this one. This depends on your problem. You can be forced by a force that does not derive from a potential or only from a potential or any combination you want. Okay. This depends on your problem. Okay. So when we're going to do an example of active matter next week, there's going to be only a force that doesn't derive from a potential. Okay. Okay. So that's easy. Then the next thing you should see is that there is a temperature here that comes from, remember what we did before. This is the second derivative. It's the diffusion term. Okay. So you are being driven by this force and at the same time you're diffusing and the measure of this is the temperature. If you send the temperature to zero, for example, and f is zero, this is just a thing that is sliding down the hill with a potential. Okay. What else? Things to notice. There are second derivatives. So in this sense, it resembles a lot Schrodinger's equation. However, it doesn't have exactly the same form because there is a term that has a first derivative, which Schrodinger equation doesn't have. There is another thing that is important comparing it to Schrodinger equation is you see it's a linear equation in p. No, it's nonlinear here, but as a function of p, it's a linear equation. So I should be able to solve it just like I solved the Schrodinger equation if I can develop in eigenvalues and eigenvectors of this thing. So I am going to do now already one thing just to make the analogies a bit stronger. I'm going to call this guy here. I put a hat because it's an operator, fp for Fokker Planck, so that just to underline the analogy with a Schrodinger equation. This is a linear operator that is a bit of the flavor of Schrodinger's equation, although as you see it has second derivatives, not in the same places, and it has a linear first derivative, which Schrodinger doesn't have. How did you obtain the equation for the probability from those equations? Can you just tell a bit? How did you obtain the equation for the probability? So what I did is I broke it in two. I said what would I have if I only had this term, and then I used, I didn't prove to you, but I used the fact that you know that this is ordinary diffusion with a random force. Okay? And then this gives you this term. And then I forgot this one and did this one. Okay? And this is the equation of q dot equals some force field, and I suppose that one knows that this is the simple advection, which is this equation here. But I didn't completely do it, in the sense that I didn't do the whole derivation in detail. If you want the whole derivation, you can find it in many places. The references are in my notes, but in general the Riesken book is a good place to find it. Okay? At any rate, at this stage, if you take it as half proven, but the important thing at this stage is to see, because if I do the whole proven, I'm going to lose you completely. So the important point here is to see what this means. Okay? What is this equation doing for you? Okay? So can I suggest that maybe we do a tutorial on this thing? I mean stochastic differential equation to Fokker Planck. I can do it this afternoon at 2 p.m. There's an opinion down there at the end of the room. End of the room. I just want to say that in the diploma course we have some lectures online that we talk a lot about Hito Stadanovich. They are available. So I have six lectures on that online. About, sorry, can you say it again? Stochastic differential equation. Perfect. Hito Stadanovich, et cetera. And you have the person who did them in person. Yes, exactly. So I can find which lectures online if it's useful. Of course it's useful, but it's unequally useful, I think, if I... Okay, so then we'll send you an email pointing to these lectures, okay? Or maybe... Okay, excellent. And if there is a not online tutorial eventually, of course I will participate, except if you want to make me an expert in Hito Calculus which, as I explained before, is something I would... But, yes, okay. But, okay, but for the purposes of what we're going to do now, it's going to be all right if you understand perfectly well what this equation does, okay? What it is doing for you. Okay, I doubt a little bit the order in which we're going to... Okay, I will do now even more briefly the equation that corresponds to this one. Okay, it's a different equation. So I'm going to write the result and we will discuss two minutes about it. Why am I doing it? Because I want to discuss two minutes about it and because it's probable that next week we're going to use that one. So what happens if I want to do this equation now and study the probability of a cloud, but now instead of having a space where this is only q, now it will have q and p. So now my equation is a kind of probability cloud that moves in this space of q and p. Okay, so the equation that satisfies is very similar. So this one is for overdamped and now it's going to look like this but now of q and p and instead of having this operator it's going to have another one and the equation is going to be, is called the Kramer's equation and now I have to tell you what HK is. Okay, it has the same, the identical interpretation, it's just a cloud of probability that is moving this time in p and q given by the cloud is constituted by the ensemble of trajectories of this kind and now the Kramer's operator is even uglier. I have to say that I'm going to denote H, whole Hamiltonian, the whole classical energy and it's calligraphic because I don't want you to confuse it with the operator. It's awful, it's what it is. But the important thing here again and again and again is that you understand what this thing is doing. So I let you finish and then we can about the different terms of this equation which have a nice interpretation. Sorry, the last term is minus the derivative with respect to Pi of fi of q, isn't this? You could put it here, yes, I agree. You could commute it if you want. No, yes, but this is applied to, okay, sorry if you want me to, so let me move once again my formula. H is p square over 2 plus v and here if you prefer to be more clear you can put it here. Okay, I just commuted them. Okay, it is a linear operator. It contains derivatives, some of them second derivatives, some of them first derivatives just like the Smoluchowski Fokker Planck one and now let us discuss a little bit the different terms. First of all, the temperature in red is here and always as you would expect the temperature is together with a second derivative because it's a diffusion term. It's the only diffusion term in this equation.  This term is a Poisson bracket by definition.  Yeah, this is the evolution of Hamilton or Newton or you will or whatever. So the other way to see this is if there is no dissipation and there is no force then that's Hamilton. This is Hamilton. It's a clever way of writing Hamilton, a la Liouville in the sense that sorry the passage from a trajectory to a cloud that is moving in phase space is also possible in Hamilton's equation. You do this and then it acts on a cloud of initial points, the famous ink blot I'm telling you all the time that distorts and it's ruled in the pure classical case by this equation. Then to make take advantage of Matteo's remark, here this is the bath and notice that it's and it's not a coincidence very similar to this equation but only that it's acting on the piece but you see that it has a form that is very similar to this one which is okay. If this term is absent there is no diffusion no friction and you're you have what you have and finally this term which could be there or not is the forcing. The only thing that makes it forcing is the fact that I'm saying that perhaps the force doesn't derive from a potential. I'll give you will give you a special microphone. It's good. It's good. Don't make him feel ashamed. Why isn't there a diffusion term with a special coordinate? Okay because I decided unilaterally and arbitrarily that my noise is going to act on the on this equation. Could I have put some noise here too? Yes yes but people don't do it because usually you don't expect a bath to talk directly to the velocity you would expect it to talk to the coordinates like a potential no but you could have put the noise here also some noise here in this equation and then you would have diffusion in both of course you do you do not have to you have to go back remember ah sorry and where does this where where was this bias arbitrary bias where was it born that remember the equation we did the exercise we did the other day so I'm going to erase here it's a nice question I remember that there was an interaction term when we had the bath and the oscillators and I decided to do q which was the coordinate of my system and then I here I did sum over ci xi and these were both coordinates and there was a term like this which was the interaction between system and bath here already I chose because I could have added for the same price and even I could have added for example p other ci's or even pi i's here I didn't do this but I could have done it and if I do everything redo the whole thing I will get a term that will couple to the p and then at the end of the day it will become a noise that will be here and this is perfect perfectly legitimate I don't know maybe Edgar or Matteo know of any place where this is explicitly done but it could very easily be done um can we get Fokker Planck operator from this Kramer's operator ha that's a nasty question um yes but it's a mess okay um okay that's a delicate and and very delicate question because remember that I said remember how we got here we we said okay let me try the case where the mass goes to zero the mass you can put it here if you want and then boom boom I throw away a second derivative and land up here okay remember it was very easy last time from here to here easy now your question is how do I go from here to here or let's say from here to here no it's the same thing I'm changing operators and so this is uh usually a very difficult problem that we face a lot in physics and in maths it's the case where you are neglecting a derivative because it's the acceleration here that you are neglecting now neglecting a derivative is a nasty thing because you can neglect a term but if you neglect a derivative how do you know that it's small so in our case what we neglected is mass times the second derivative with respect to time of the position and we said this is small I throw it away but do I have a right to do that well more or less because imagine that this function changes very fast in time then even though this is small this can be very big so neglecting derivatives is a tricky business always okay so that's one consideration I think the idea is that you can neglect it when you look at long times right so you can neglect it when you look at long times but you can also I mean if you have to check time to or you have to check a posteriori when you solve your new equation that the new equation doesn't have a variation that is fast because then it feeds back into your original problem so to be you will find in risk and this thing done and let me tell you how it goes it's complicated and tricky if you want to do it at the level of operator you should be able of course you your question is good if there is any justice in this world you should be able to do it at the level of but what is the problem the problem is that first of all you have to change spaces here you have a space made of peas and queues and you want to get rid of the queues how do you approximate an equation so that you lose coordinates okay let me give you a quick hint of how this is done because it's worthwhile you will find it in a risk and well done so imagine this kind of approximation that leads you from here to here in the well done is imagine you have a quantum problem now for those of you for example a channel like this with periodic boundary conditions and this is very narrow okay and you want to study the dynamics so this other this coordinate I'm going to call it q because for and this other coordinate I'm going to call it p it's a name okay how do I use the approximation that this is a very narrow channel my wave function is a function of p and q so I want to lose I would like to do the approximation that p is super narrow with respect to the length how do I do it well the way to do it is for every slice you first solve it as if q didn't move because you say that p is moving super fast and then completely solve it eliminate q p and then this you get an effective equation for q for every p this is I'm going fast this is called uh bone open Heimer approximation it's an approximation where you say I have a very fast variable I completely solve it at each time and then I carry it over adiabatically adiabatically would mean it's a tricky business not super difficult but you have to take some time to read it and risk and does it essentially for here so what does that mean imagine that the mass is very small so you cannot neglect the velocity but what you can say is that the rattling of the velocity is very very fast with respect to the motion of q and so you do bone open Heimer analogous to this one in the sense that you solve the this this equation but for given q and you let p move and then once you integrate away p you move along q and you end up with the other equation for those of you who are interested look for it in risk and but it's a word of warning that yes throwing derivatives in equations is a tricky business okay everything okay so now I think I sort of made you suffer a bit so we will continue next week but we have to say a couple of things that are important and these with this we will finish today so it's not okay well anyway the analogy with Schrodinger we will come back okay so I leave you as an exercise that if you are not very memorious of how you solve Schrodinger like this in general it's the first three pages of quantum mechanics have a look at it okay so that you go back to the bracket notation and so okay so what is the nice thing of these equations that both have the form dp d time equals some operator which could be Fokker Planck Smoluchowski or Kramers a minus is because I like a minus there p okay and this is going to make an evolution p of q and eventually p in its kramers and the time so what these equations have that is very nice I think I can erase this and this okay what these equations have that is super nice is that if you start from a distribution so you you start in your space with some now initial ink stain but now there is noise so the the stain will will move and if we let us think two seconds at the case where there is no forcing okay so from now this what I'm going to say is without forcing what happens is that the p will evolve and one can show and we will show that it eventually settles into the Gibbs measure so p of in kramers q p t equals infinity is simply the normalization e to the minus beta h of pq that this is the ordinary energy function divided by the normalization and p of q t this is for kramers problem and for Fokker Planck where we have is simply e to the minus beta v of q over the normalization so t equals infinity so this has the lovely property when f is not there this is super important if f is there okay this is when this is not there and this is when this is not there what happens if there is an f that doesn't derive from a potential well you're on your own there is no nothing I mean you have to solve the equation the brutish way there's nothing you can do nothing really nothing there is no general rule for solving for the target distribution there may be a target this there going to be a target distribution but you have to compute it brutally with you know solving the equations what about perturbation theory or perturbation that's brutal yes things like perturbation I mean you're on your own it's you're in the world of physics no perturbation approximations of various kinds etc but an a priori formula like this one or this one forget it then you could ask even more do you do is that what I really want a formula like that for an out of equilibrium problem is this really the information I want I would say not really but that's another question okay so this is an important thing to bear in mind because all problems like when you have two temperatures that you are kicking the system out of equilibrium or when your particles are motorized like in active matter they are sort of driven they they have their own they have an F implicitly they are out of equilibrium and there is no way that you can in a closed manner solve for the distribution yeah so to me the left one the last time solution of the focal length dynamic looks like the Boltzmann distribution but without the kinetic term exactly but like how can I make sense of that because I mean the particle is obviously moving it's a tricky business related to the question that he asked before you have integrated away the velocity but then when you threw away the mass and you said it was zero zero the velocity is infinite all the time because if you have particles that have zero mass their velocity so if you ask the question what happened with the velocity you shouldn't ask that question you see you have made in throwing the mass away you have made a mess this not with standing this this is the equation you will find more often but with respect to velocities you have made a real mess and you pay it you gain because you have a simpler equation than that one that looks really awful but you pay it because then you have all the ito stratone of each mess which is a a remembrance of this fact that you you have been playing dirty of course remember that if I had left the kernel of memory as we had it the first day with a generalized Langevin equation everything would be sweet and innocent but of course the equation is nasty okay let's check it it's very easy let me plug this one in here now let me be a bit more precise so what is this Gibbs measure it's e to the minus we are in the Kramer's case beta p squared over 2 plus v okay this is this divided by z but you see that already this term will kill this one because this term you you you see it's the Poisson bracket of h with e to the minus beta h so it's zero or if you want calculate it explicitly apply this to this thing this thing and you will see that you get zero but it's not no coincidence because this is the Poisson bracket of something with respect to itself okay and this is normal the Hamilton equation this is up to here pure Newton of course will leave this thing stationary because it doesn't do anything to it it just keeps it this is by the way you will see okay if you want to do it explicitly you just compute this derivative acting on here so h will come down and you will have a term dhdq and here you have dhdq dhdp and you have exactly the same thing here and here and they kill each other okay so this is normal it's this term we said was absent so let's cancel it and cancel it and let's cancel the one here so that you know our discussion we are clear now this term is more interesting this is the bath and the bath the way it is you see that it kills it will kill this part of the term because the derivative will kick a p down with a beta that will be cancelled by this t and it will have a minus and a beta where m and here you will have the same thing with the opposite sign so already this term here you don't even need to take this derivative this term kills this one and because this is a product it kills everything check it look at it funny no because it's only killing the kinetic part you can be worried about this but if you're not I'm not going to worry you okay good so we found out that when h acts on p Boltzmann it kills it so the time derivative is zero so so the Boltzmann Gibbs distribution is stationary then we will have to discuss if it's the only stationary and we will just we will say we will discover that roughly unless you do something really nasty yes but we have to decide what nasty means okay and now let's look at this one this one is even easier so we want to show that this gives you zero so the way of giving zero is that all this thing applied to p gives zero but it's much easier than that each one of these terms kills this one you don't even need to take this derivative why because dvdq here where are we what happened with the Boltzmann okay so p in this case is e to the minus beta v so when you differentiate one time a q will come down here and with a beta that will be killed by this one and with a minus because of the minus and then dvdq is the same thing so this will this will kill the thing so already this little bracket kills the measure and each one of them kill the measure so zero okay so you have proven that the Gibbs distribution is stationary so now think a bit what we have discovered is that all the exercise of getting rid of the bath etc etc that we have been doing for three days we have now we have an ergodic theory we have a system with noise that we let it go and it goes to equilibrium as as one likes okay if there is no forcing again also in this equation if there is a forcing that doesn't derive from a potential you're on your own even for the simpler case you have to there's nothing you can do which means that you have the things of theoretical physics perturbations and things like that but there is no clever thing that you can do like this and get a generic formula like this one okay okay questions even if you didn't follow every step I hope that you see the strategy okay yeah ah yeah I think I should have said perhaps if you have a forcing term you're putting energy into the system potentially you can put energy into the system and and then you're taking it out with this term which has friction and in those cases if the potential is reasonably bounding etc you can show that you have a stationary distribution but it has to be shown because sometimes your but if your forcing term is too cruel and it overcomes the frictional term then you have a problem I want to go back when you wrote the Stratonovic discretization the noise term has to be compute from the next time step is there meaning in there or it's just the definition no no it's all constructed Stratonovic has an advantage technically that if you do things alastratonovic then you can apply the rules of calculus with a chain rule it's constructed for that so for theoretical things it is more powerful but in a simulation it makes your life completely miserable because you need to know what the next step to be able to make the next step so it's bad but it's it's it's it's a formal thing did did Edgar did you give this ito Stratonovic okay it's it's a subtle thing and it's a nasty thing but it's life yeah sorry you need the so I don't want to create controversy but ito on the other hand has also some advantages for calculating our average use yeah in terms of yeah you can find this in the lectures that yeah Stratonovic is better to you want to do calculations like derivatives because as I was explaining you can use the standard rules of of calculus in it or no you have to you cannot do a derivative with a chain rule for example but it's I would say it's not that much you have to add it's one extra term in the derivatives but it has the advantage of averages to compute averages ito is very convenient your details I can explain you and let me add in in when you do things with path integrals and such we always use ito I have never seen but it is a subtle problem and and I think that the thing is that the risk is that the trees don't let you see the the forest because when you face this ito Stratonovic question it's rough and and and for a moment you forget what you were doing so this is why I I insist it a lot okay okay so thanks okay so we recommend at 11 in in the computer lab okay


Good afternoon everybody. I'm George Putnam. I work for Scandium International Mining Corp. There's a picture of our project in New South Wales. And I'm going to talk to you today about delivering on the Scandium Promise. I'll make some forward-looking statements. All right. So project status. This business, this project, we have one project in this company. It's the Ningen Scandium Project. It's located in New South Wales, Australia. We own 80% of it. We've got a definitive feasibility study on file on CDAR now. It's got some great numbers in it. It's got a 33% internal rate of return. The capital cost on this project is pretty reasonable, I'll say modest, for a mining investment. It's $87 million U.S. And we are planning a nameplate production from this project of 38.5 tons of Scandium Oxide, 38,500 kilograms. It's a surface-minable resource. It's low strip ratio. It's a lateritic clay. So it's free dig. We've got a demonstrated flow sheet on this project now. We spent over $3 million U.S. dollars in developing our flow sheet. Time table and milestones of production. We just finished our last significant milestone towards being allowed to build this project in New South Wales last week. On Wednesday of last week we were granted a mining lease on the project. We've got off-take and construction financing discussions underway presently. And we've got one off-take agreement in place. A little bit about our capital structure. We've got approximately 230 million shares outstanding. We've got about $800,000 of cash in the bank, which is walking around money, but that's what we need. This is the stage we're at. We're in the customer acquisition mode at this point. We've got no debt. Current price, 38 cents on Thursday. We were sitting just under 40 today. Market cap, $86 million Canadian. And you can see our stock range there and some volume numbers for us, which are picking up. All right. Let me beat my chest for a moment here. Really the starting point for this project for us came in June 2014. That was the point when we retained control of the project and we were able to move forward to begin to do metallurgical work in earnest and publish that. That was the point where we sold down 20% of the project in June 2014 and moved forward with an 80% ownership and no debt. So in October, we followed that up fairly quickly with the PEA, with a great IRR as you can see. It was a high pressure acid leach design, our first. In May 2016, we followed that up again with a full definitive feasibility study, which is on our website, plus 15-5% error bars with again a 33% internal rate return. In June, just after we finished the definitive feasibility study, we filed an environmental impact statement in EIS and we filed a mining lease application with the government in New South Wales. We got our development consent in November of last year, which is probably the bigger of the two achievements, that's probably 85% of the pathway towards gaining approval and gaining the trust of the state of New South Wales to develop a project. And then as I said last week, we got our mining lease grant from New South Wales government. Ladies and gentlemen, that's a 35-month start to finish point on what is a great deal of work and design. That's moving very fast and we're very proud of that. All right, so a few more details on our project. You have to start with the map, right? We are located in the center of New South Wales, Australia, on the eastern seaboard of Australia. We are in the center of a laddered clay belt that runs up and down the New South Wales area where Ingelmers has his project, Dubbo Zirconium. There is a lot of activity in nickel, cobalt, scandium and platinum in this area. We've got a resource approximately 17 million tons, grading about 230 average PPM. We've got a limonite top layer that's over 300 PPM and that's what we're going to mine in our definitive feasibility study plan for 20 years. In fact, the PPM in that project for 20 years is 409. In an excellent location, we've got good infrastructure. We've got all the benefits of Australia, which is really a mining-friendly environment and it's a great international jurisdiction. The project has significant upside potential. The DFS uses about a little less than 10% of the total resource that we have established now, measured and indicated. So let's talk about scandium's place in the specialty metals marketplace today. Where is scandium currently made? Almost all of the scandium that we know about is coming from co-product production. It's coming from production in China. It's coming from production in Russia. The Russian production is basically from uranium projects. The Chinese production is on the tail end of TIO2 pigment plant production. So you can see some of the issues here. There's not very much scandium available today in the marketplace. We estimate about 15,000 kilograms, 15 tons. If that number's wrong, it's probably a little more than that, but it's still not much. What's the value proposition for scandium? Well, scandium makes any aluminum alloy lighter. If you learn one thing about scandium today in this room, understand that scandium changes. It's a grain refiner for aluminum. Scandium particles sit inside the aluminum crystal itself and they sit on the grain boundaries and they add strength. And they do that without spoiling all the other properties in the alloy that they happen to be introduced to. That's unusual. You can make aluminum stronger, but you often have tradeoffs in terms of loss of properties and particularly brittleness. So there are many other properties that scandium enhances in certain alloy families. Beyond weight savings, obviously, you've got corrosion resistance improvements, flow rates in extrusion processes, and full weldability. You've heard that discussed before, too. Scandium is also a terrific heat stabilizer in a solid oxide fuel cell. Probably half of the scandium being used today is being used by bloom energy in solid oxide fuel cells. So there's a big market there already. So we talk about improvements in scandium strength. How dramatic is it? Well, it depends. It's always good. Sometimes it's pretty exceptional. And you can see here the one series and the five series alloys have pretty substantial strength pushes. The pink bars are changed to strength levels in all seven alloy families plus the aluminum lithium variants of those. So what do we do with scandium today? So marketing slide around where scandium goes on the left side of that slide you see in yellow. You see sporting goods, aircraft, solid oxide fuel cells where I said about half of it goes and lighting. Down the center, you can see where that growth is. And these are natural extensions in many cases of the applications that are in use today. You can see they're substantial. And they are really supply constrained in those circumstances. In terms of new applications that aren't much in use today, you can see automotive, marine, high speed trains, computer cases, and electronic enclosures. All good applications for scandium, but you need a lot more scandium than we have today to get started in those areas. So before you go selling scandium and I'm taking you down into our marketing conversation a bit further, you've got to figure out who your customer is. Well, in the scandium space when you're selling properties, you're selling properties to the end-use customer. It's the customer, the end-use customer that's going to tell you what scandium is worth and how much he wants based on the properties you can deliver to him. Now that's over on the right side of that slide. At the green end, we're on the, obviously we're on the other side of that, and there's a large supply chain in between us and them. The answer we found is that the customer is everywhere along that chain. You've got to talk to all levels. You've really got to think about a pull-through strategy for scandium because you're selling properties when you're selling an improved aluminum alloy, aluminum scandium alloy. We'll drill down a little further. So this sounds pretty complex. How do you tackle this kind of a conversation in a number of different markets? I've picked one here, automotive. We've talked a lot about aircraft and there's been a lot of good conversation around aircraft applications. It's obviously a place for aluminum scandium alloys. Anything you lift off the ground, lightweighting is the imperative. So it plays an automotive as well, too. So what we do is we talk to automotive folks about the different properties and we prioritize those from top to bottom in terms of which property that automotive group is going to value the most. We talk about the cost impacts of those things and then we talk about the scandium advantage in each case. And it becomes a very powerful dialogue with folks that engineers are very receptive to it. I went to the Lightweighting Conference in Birmingham, Automotive Lightweighting Conference two weeks ago and it was really stunning. This slide is a Volkswagen slide and it really shows in the first instance, the blue bars on the left are steel and steel improvements based on engineering over the past 15 years to lightweight steel cars. Then you can see aluminum next to it and you can see a 40% improvement. We got between a five and a 25% improvement engineering improvement in steel body redesign and then we picked up a 40% improvement in aluminum. Notice also that the benchmark is now aluminum. It's not steel anymore. Magnesium, carbon fiber, they're benchmarked against aluminum now and you can see that blue line going upwards, that's cost. That's the cost implication of those trade-offs. So you can see the Lightweighting imperative is there and you can see aluminum is very much the forefront of that. This is the other mantra from the Lightweighting Conference. The right material, the right place, the right time. They're saying properties. Properties, properties, properties. An aluminum scandium carries properties that allow engineers to use it in more places in ways they haven't been able to before. This is an old slide. I leave it in for you to look at on the website or on the presentations that we have for you. Aluminum Outloys in many applications don't represent a significant portion of the value of the product. It's the properties that will sell the scandium advantage. I'll read you to finish. This project that we have is construction ready now. In fact, I hope to announce relatively soon an EPCM contractor award on that project. Our financials, we think are quite good. Our resource is a world game changer in the space. We're deep into the customer acquisition phase now. We're delivering on the scandium progress. Exciting times.


Good morning everyone, thank you. Success, Georgia, Anita, Siddi Kenu, Lubega and Ilya, Lama for joining class this morning. Good to see all of you. Hope all of you are doing well. Yes, no response. Okay. Yes, we are doing well. Thank you. Thank you, Jafeera, for joining. Good morning, welcome to class. Okay, can I ask one of you to... Good morning, Success. Can I ask one of you to please lead us in prayer? Please? In Jesus' name. Let's pray. In Jesus' name, the ancient of this, who want to say thank you this morning, I'll give you all the glory because you are worthy to be praised. Thank you because you wake us up this morning to learn through your daughter. My father may go, we ask the Lord to take up man control in the name of Jesus. Lord, release your grace and your nighting through your daughter to us today in the name of Jesus. I see she speak to Lord, she will speak your mind. In Jesus' name we pray. Amen. Amen, amen. Thank you, Success. Yeah, it's a holiday today, but I think the Bible colleges are all on a working mode. So, yeah. I think some of the others are on holiday. Anita, you on a holiday today? No, ma'am. Okay, you're working. So I think Bible college of Japan has also... Siddhika, no, you're on a holiday today? Yes ma'am, I'm on a holiday today. Okay, good to have you in class. Thank you for joining us. Yeah, it's a festive season here in India. So everyone's pretty excited because I think after the lockdown for the last two years, they hadn't had the access of just meeting together and having fun. So everybody's in a great mood of celebration. Anyway, we're celebrating God, we're celebrating Jesus, we're celebrating all that he's done in our lives. And this morning we are looking at the second chapter in Kingdom Builders. We're looking at, we look at how, you know, what the kingdom of God is, the different aspects of the kingdom of God, and now we're going to look at how we should be kingdom builders. Okay, so what is the framework that we work within? What are the things that we keep in mind? How do we, what are the things that we put on? What are the things that we put off? And how do we build God's kingdom? How should we be, you know, kingdom builders? Good kingdom builders, okay? So we looked at chapter one, which was just a review of what we had learned in the kingdom of God, the publication kingdom of God. Now we're looking at chapter two, Christ, the king of the kingdom. We also studied this in the publication, the kingdom of God. So we're just going to go through some of this. And the first aspect we looked at was, you know, that we need to be coworkers with the king. We need to partner along with him. We also need to partner along with others. And we spoke about the importance of working together, the importance of unity, and how, you know, God uses one man to sow, the other to water, yet the other to leap. But, you know, one man lays the foundation, the other one builds upon it, the other one takes it further.  We are all important to edify and build the body of Christ. Each one of us are roles, the talents, the gifts that God has given to us. And we looked at Romans chapter 12, was supposed to six regarding this in Ephesians chapter four, verses 11 and 12 and 15 to 16, and that is where we stopped. Okay, so this morning we're going to continue with chapter two, you know, where it says that our relation with the king of the kingdom is of prime importance.  Well, all that is important. We need to build a healthy relationship with our fellow ministers, but what really qualifies us to be kingdom builders is our relationship with the king. That is of prime importance. Okay, because if you look at Colossians chapter one, verse 16 and 18, a very, very powerful passage of scripture, you know, can one of you please read Colossians chapter one verses 16 to 18, please? Colossians one, 16 to 18. Colossians chapter one verses 16 to 18. Colossians chapter one verses 16 to 18. For by him, all things were created that taught in heaven and that all of heaven was civil and invisible, whether thrones or dominions or principalities or paths. All things were created to whom important. And he is before all things. And in him, all things coexist. And he's the head of the body, the church, who's the beginning, the first born from the dead, that in all things he may have the premonitions. Yeah, so thank you, Jeffita. So here we see that, you know, in who's the head of the church? Who's the head of the church? Verse 18, who's the he who's the head of the church? Christ, please. Sorry? Christ, yes. Christ is the head of the church.  Preeminence means we give him, his preeminence means he's more important than anything else because he's the head of the church. So, you know, in the kingdom of God, we need to remember this thing with this powerful truth. We need to keep telling ourselves again and again that, you know, we are here to build God's kingdom as kingdom builders and the head is not me. You know, it's not I, but it is Christ.  We will always be mindful that we are building the kingdom of God and the head is Christ. He receives all the glory and the honor, okay? So Christ is more important in all that we do, you know, even as we are considering to build his kingdom, we are building not our kingdom, but Christ's kingdom. And so that is very important for us to be mindful of.  He says, your kingdom come. So it's not my kingdom come, or it's not my pastor's kingdom to come, or it's not my father who earthly father who has a church, not his kingdom or his church to come, but it says your kingdom, which is talking about the father's kingdom, the heavenly father, your kingdom come, your will be done on earth as it is in heaven. So we need to be mindful of this fact and this truth that it's not about me, my ministry, my church, my organization, you know, but it's about building God's kingdom, okay? And we should desire to build his kingdom and we desire to see his kingdom come and his will be done and not our will be done and not see our ministry come, but his ministry come and his will be done. And we are here to do what God wills, you know, not what we will, what we feel comfortable, what we like, we're here to do what God is asking us to do. And also we need to give him the glory because he deserves all the glory, honor, and power.  He said, yours is the kingdom, the power and the glory forever, amen. So all the glory and honor, you know, has to be ascribed to God. Even as we are building God's kingdom, we need to be constantly pointing out, pointing others to Jesus.  So even in small things, you know, we need to give God the glory. And sometimes we can, you know, this can come very subtly. You know, we can give God 75% of the glory and we would like to keep 25% of the glory for ourselves. And that is why, you know, Paul says, you know, we need to crucify our flesh daily. Okay, we need to crucify the flesh daily because, you know, even as 25% can slowly become, 50% can become 75% and 100% where we are looking for self-glory and we, you know, we're not giving the glory and honor that's due to God. So John chapter seven was 18 says that he who speaks for himself seeks his own glory, but he who seeks the glory of the one who sent him is true and no unrighteousness is in him.  You know, there is no unrighteousness in us. But if you're seeking for self-glory, then, you know, there is a measure of unrighteousness. We are not pure in our thoughts and our motives of serving God. And it's very important because Jesus says in the Beatitudes, blessed are the pure in heart for they shall see God. So if you want to receive more revelations, if you want to see the move of God in your church, your organization, then you need to ensure that you're in a place where, you know, you are giving all the glory to God ascribing him all the glory and honor because it is him who's doing everything. It is his power, his authority. It is his kingdom. He's the king. He's the one who's enabling you to do that. And so all of these truths, though, they are very simple and something that we already know, it's very important to reiterate these truths to our minds, you know, time and again, because, you know, subtly they can just come in and, you know, the I me, you know, can take precedence and become the center of our lives. And that's when we see the downfall of our ministry or, you know, the work of God or the work of God gets hindered and we don't see the move of God. We don't see, you know, the church throwing or people coming into the church or we don't see, you know, science miracles and wonders, okay? And we need to ascribe the glory and honor because to God, because he deserves our glory and honor. You know, Isaiah chapter 42 verse eight, the Lord says, I am the Lord, that is my name and my glory I will give, I will not give to another or my praise to carved images. So we know that God is a generous God. He's a zealous God. He does not tolerate his glory to be given to anyone else.   That I am the pastor, I am the senior pastor, I am the, you know, apostle or I am the prophet.  So it's important that, you know, we are making sure that, you know, Christ increases in us and we decrease and people see Jesus in us and appointed to the Father. Just like, you know, when Jesus did all the science miracles and wonders, we see that, you know, people don't fall down at Jesus' feet and worship him as God, but they glorify the Father. That's what we read in the Gospels. So we see that Jesus very clearly is pointing them to the Father. He says, I only do what my Father asked me to do. I only say what my Father asked me to say. I only, you know, go where my Father's asking me to go. So he's constantly pointing to the Father. He's showing them the Father. He's revealing the Father heart of God to them. And that is what we need to do. We need to reveal the heart of God. We need to reveal the Father heart of God. We need to reveal the nature of God. And we need to tell God that it's people, that it's his power, his revelation is anointing is what they are receiving in and through you. Like this example in Acts chapter three, when, you know, Peter and John go to the temple to pray and there's layman who was sitting there begging and he just looks at Peter and John and, you know, he's begging and, you know, and he, Peter raises him up and he's able to walk and, you know, he's dancing and praising God. And everyone will look at this miracle, seeing this layman, you know, healed.   So why not so intently at us as though, you know, it's by our own power or godliness that we made this man to walk. And it says it's not us, but it's in the name of Jesus and by the power of Jesus that this man was able to rise up and walk. So we see that they pointed people to the living God and to Jesus Christ. John chapter five verse 41, Jesus says, I do not receive honor from men. So we need to come to a place, you know, where we don't desire honor from men. But that means, that does not mean that we do dishonoring acts or dishonoring deeds or behave in dishonoring ways. It does not mean that. It just means that, you know, we do things that are honoring in such excellence and such perfection that, and we preach in such excellence and perfection to pursue the heart of God, the will of God for our ministries or the places where God has appointed us in that offices. We also, you know, preach the word of God, just waiting on God, praying, fasting and praying, receiving revelations. But then we're saying, you know, it's not me. It's not my fasting or my praying. It's not my excellence. It's not my style, but we are pointing to Jesus saying, you know, this is what he told me. This is what he revealed to me. And so, you know, we need to point people to Jesus, give him all the glory. That's what makes us true kingdom builders. And Jesus also, you know, he said, if I honor myself, my honor is nothing. So honor that is a self-bestowed honor is of no value, absolutely no value, okay? In John chapter five, verse 44, Jesus says, how can you believe who received honor from one another and do not seek the honor that comes from the only God? So here he's basically talking about the Pharisees and the Sadducees, the Pharisees and Sadducees, in Matthew chapter six, verses one to six and Matthew chapter 23, verse five is basically talking about these, you know, these Sadducees and Sadducees, you know, basically when they're doing their charitable deeds before men to be seen by them.  Your motives are, you know, not to do it for the father in heaven or to give him the glory, but, you know, to be praised among men. And so he's saying that, you know, they actually did it with no purity in their heart. And, you know, the righteous deeds is actually, you know, not pleasing in God's sight because they're just doing it to get praise from men. And so, you know, Jesus telling us not to do these kind of acts, to display ourselves or to get a good image. And then he's also goes on to say that, you know, when you do this, you will get your reward from people and you'll have no reward from the father in heaven. So, the idea is when we do righteous deeds for the, to get attention and applause of men, their attention and their applause is already become our reward. But it's much better to receive a reward from our father in heaven, okay? And he goes on to also talk about, you know, when you pray, you shall not be like the hypocrites for their love to stand praying in the synagogues and on the corners of the streets that they may be seen by men.   So here basically when he says, when you pray, you shall not be like the hypocrites. He's basically talking about these men who stand in the synagogues during the prayer time. And, you know, they just pray in a very hypocritical manner or they might also be praying, you know, during the public prayer on the streets which happens at nine a.m. in the morning at noon and at 3 p.m.  And these hypocrites prayed not to be heard by God but to be seen by men. And so, you know, Jesus is saying that there is no reward for them. And also he talks about in the same passage, these same passages, he talks about how, you know, they enlarge the borders of their garment. They love the best place in the feast, the best seats in the synagogue, and they love the, you know, for the people to greet them as a rabbi. They like to be called rabbis and also the phylectries, you know, which they put on themselves. You know, all is just going to show that Jesus is saying all of this is just a show. It is, it's not something that, you know, they're saying that they're doing all this because, you know, it's told in the Old Testament, like Jesus said in the Old, in, sorry, God, told the Israelites in the Old Testament in Deuteronomy chapter 11 was 18 and Numbers chapter 15 verses 38 to 40 that, you know, when they wear their garments, they have to, you know, leave tassels and put up tight the corners with a blue cord just to remind them, constantly keep reminding them of the commandments that God has given to them. And also these phylectries are like small leather boxes with tiny scrolls in them, with the laws written in them, the scriptures written on them and they tie it to their arm or to their head with leather straps. And, you know, all of these things, you know, God had told them to do in accordance with the law and how they need to keep the law, but all of these religious leaders, you know, they were doing it, you know, to show themselves to be more spiritual than the rest, okay? So the idea of wearing these phylectries and, you know, and tying these phylectries and the special borders in their garments was in obedience to what God commanded the Israelites when he gave them the covenant at Mount Sinai, but they were using this as, you know, to promote an image that they're super spiritual and, you know, that others are sinful, they are not, and also commanding a kind of respect for themselves so that, you know, they like the great places of honor in the synagogues and people to look upon them and just, you know, kind of showing other people that, you know, you are not that spiritual like compared to us.  So, you know, whatever we do, whether we are praying or even singing or worshiping God, we need to always check our motives.  We want people to know the truth. We want to lead people in meaningful worship. We want to lead them in meaningful prayer for them to grow in their prayer work with God, or are we doing it just to, you know, magnify ourselves or to point them to Jesus or help them to grow in the ways of the Lord. Okay, so the real test, you know, whether our heart is in the praise of men or, you know, to receive praises from God is, you know, what do we do when we are in a situation where we have to choose between God and men? So here, you know, in John chapter 12, verses 42 and 43, it says that many people, they actually believed in Jesus, but they did not confess that they believed in Him because of the fear of the Pharisees that they will be put out of the synagogues. And it says in verse John chapter 12, verse 43, for they love the praises of men more than the praises of God. So when we choose to do things in situations where we want to be praised by men rather than, you know, doing what is honoring in God's sight or doing things that will get a reward or praise from God, you know, because we do it because we don't want to face rejection, we don't want to lose our job, we don't want to lose our position of leadership roles, whatever, or we want to please our, you know, so-called senior pastor or our boss or whatever, you know, and we compromise in such situations, you know, you know, we are not honoring God, we're not giving Him the glory and honor that is due. Okay, so Galatians chapter one verse 10 says, you know, Paul says, for I do, for do I now persuade men or God or do I seek to please men? For if I still please men, I would not be a born servant of Christ. You can't be a servant of God.  So our motivation should always be to please God and not men. Our motivation should be that God receives all the glory and, you know, our heart is pure in God's sight, okay? And Psalm chapter 115 verse one is a very beautiful prayer that we can pray to keep our hearts pure and to set our hearts in the right direction. It says, not unto us, O Lord, not unto us, but your name give glory because your mercy and because of your truth. I think we can pray this prayer and say, God, you know, let me always give you your name, your kingdom be glorified in and through everything that I do, okay? The next thing as a kingdom builders, we need to be mindful of is, even as Christ is the king of the kingdom that our authority on earth is dependent on our submission to the king, okay? So, you know, we see that in the Garden of Eden when Adam and Eve obeyed God, as long as they obeyed God, they were in dominion of the earth, you know, they had a relationship with God. They also had the access to the tree of life. But, you know, that the minute they disobeyed God, you know, they lost their dominion on the earth. They also lost their access to the tree of life. They also lost their relationship with God and, you know, everything what happened after the fall. So, you know, the important keys for kingdom building and kingdom authority and kingdom power. If you want to have more authority and power in the way you minister or for science, miracles and wonders, then the key is obedience and submission to the king, that is to God, okay? Our spiritual authority is simple, his dominion in me determines his dominion in and through me. To the extent we submit and surrender our lives to God and, you know, it's the extent that we experience spiritual authority and, you know, we will see Christ's dominion in us and his dominion in and through us, you know, influencing the environment and the demonics put it around us to the extent that Christ reigns in us, you know, to the extent he can reign in and through me, okay? So when we submitted and obedient to Christ, you know, that's when we can exercise our God-given authority and power and, you know, and we can move mightily, we can extend the kingdom of God here on earth, okay? The other point is, you know, let us not glory in man. In 1 Corinthians chapter one was 11 and 13, all says, you know, he's talking to the church at Corinth. He's saying, you know, there is a division in the church that is happening. And he's saying, some of you are saying that, you know, I am of Paul, some of you are saying I am of Apollos, some of you are saying I'm of Cephas, or some of you are saying I'm of Christ. They don't want to take any man's position. They're saying, okay, we are not belonging to any, you know, group of men that are there, leaders, but we are Christ. We are of Christ. So he says, is Christ divided, you know, and Paul asked a very, you know, very powerful question to them. He says, was Paul crucified for you, or were you baptized in the name of Paul?  So the answer is basically no, okay? It's like a question that he asked, and the answer is very evident. It's a rhetorical question, because the answer is there, the question is there, but the answer is also very, very evident. And the answer is no, who died on the cross, who was crucified for their sins? It was Jesus Christ. And whose name are they baptized? The name of the Father, Son, and the Holy Spirit. And so Paul is saying, how can you say I belong to this, I belong to that person, I belong to this person, you know, we all belong to Christ. So here we see that, you know, Paul is leading them to a very important truth, and it's not men, you know, but it's Christ that has to be honored, and, you know, who we belong to, who died on the cross for our sins, and whose name we are baptized, and whose name, you know, the demon shudder and shiver, and whose name a sickness and all demonic authorities and powers bend their knees or bow down before. So it's very important that we also are mindful that we don't bring about this kind of division or this groupism in the kingdom of God, because, you know, when we elevate man or we elevate individuals or we put ourselves about others, then we come to a place where we are very subtly bringing about divisions in the kingdom of God, and then at that moment we cease to be kingdom builders, instead we become kingdom dividers, okay? So also, you know, when we begin to elevate ourselves, thinking all this is happening because of me, because of my skills, my hard work, my commitment, you know, all of this, then, you know, what you are doing, you are glorifying, bringing glory to man, but not for being glory to God, okay? So even when you elevate yourself, thinking that you are more spiritual or more sensitive to God, more prayerful, more anointed, and others, you are actually glorifying man and not glorifying God. So it's important to know where are we receiving this power, this authority, the calling, you know, who's giving us this power to heal and to cast out demons and who's giving us the revelation. So Holy Spirit is Jesus. It's what Jesus has done on the cross. So it's not my preaching, it's not my praying for a sinner and leading them into sinner's prayer, it's what is causing them to come to salvation, but it's what Jesus has done on the cross that is cutting their hearts and it's the Holy Spirit that is working in them. So we basically know all of these truths, but pride just fills our heart at times and we need to take stock of that. And so we need to be very careful and not give room to the evil one. And in any small area of our life, pride can come in order where our egos are growing. We need to starve those areas of our life so that God can be elevated and he can be glorified in our lives, okay? And it's asking the builders, we need to know that, you know, Christ is the king of the kingdom and even as he's king, they're accountable to him because he is going to judge all things, okay? So it says in 1 Corinthians chapter four versus three to five and 2 Corinthians chapter five versus nine to 11. So can two of you please read those scripture passages, please? 1 Corinthians chapter four versus three to five and 2 Corinthians five, nine to 11.  In fact, I do not even judge myself for I know nothing against myself. And I'm not justified by this, but he who judges me is the Lord. Therefore, judge nothing before the time until the Lord comes who will bring to the light the hidden things of darkness and reveal the counsels of the hearts that each one's praise will come to God. Thank you, Jeffina. Can someone else please read 2 Corinthians five, nine to 11? 2 Corinthians five, nine to 11. It says therefore we make it our aim whether present or absent to be well pleasing to him for we must all appear before the judgment seat of Christ that each one may receive the things done in the body according to what he has done whether good or bad. Knowing therefore the terror of the Lord we persuade men but we all, but we are well known to God and I also trust are well known in your conscience. So Paul is saying this that, you know, yes, it's important for us to be accountable to men the way we live, the way we act. You know, we can't just do anything and say I'm not accountable to anybody I'm just accountable to God because he's the one who judges. So we're accountable to men that God is placed because we saw that we learned that, you know, we are, you know, God wants us to be accountable to the authority structures or the authority governments that he's placed in our life, whether it's home, you know, the workplace and the church and the society, the government that is there, we are accountable to them and we need to respect them. And, but here it says also that, you know, but even more, we are accountable to God. So ultimately we are accountable to God in all things because he's the one who's going to judge all men. We're all going to appear before the judgment seat of Christ, second Corinthians five was 10 and we will receive what we have done in our body. And it says, you know, we will, each one's praise will come from God, first Corinthians chapter four was five, the last phrase. And so it's important that, you know, we do things that are honorable in the sight of men, honorable in the sight of God, also that is honorable in our conscience and we do things so that we will be well known by God.  Why is it so? Because we did what we wanted to do in our own will, in our own understanding to receive praises from men but not what God wanted us to do and not giving him the glory and honor, okay? So we need to be accountable both to man but we are more accountable to God who is the one who's going to judge. So accountability is simply putting, you know, is being true to God and true to ourselves. When we are true to God and then we are true to ourselves, we'll be true to our family, we'll be true to those we serve and we will be true to those who watch over our lives. So, you know, we can only be accountable to our spouses, to our children, to our bosses, our pastors, our fellow workers, believers, only when we are true to God and true to ourselves. And that is what Paul very beautifully says, you know, but we are well known to God.  It's important that God knows what you're doing is his will and what you're doing is right. He says, and also trust are well known in your conscience. So it's your conscience. You know, it's a God given conscience where Paul says in Romans chapter one, he says, it's the law written in our hearts. What's the law written in our hearts? It's our conscience. So, you know, the Jews are saying, we have the law, we'll be judged by the law and the Gentiles don't have the law. So how will they be judged? You know, they will be judged by their conscience or those who do not know God. How will they judge? They will be judged by their conscience because it's the law written in their hearts and minds. God has put their conscience in a voice. He's pre-designed it. He's pre-programmed our conscience to know what is right and what is wrong. So it's our conscience. But Paul ultimately says it's not the law, it's not the conscience, but you will be judged by the gospel of Jesus Christ. That is Romans chapter one. So, you know, it's important that, you know, to know that, you know, when we're true to God and to ourselves, then we know we will be true or accountable to the rest of our relationships, the list of people God has put in our lives. But if we fail in the first two, being true to God and true to ourselves, then very likely that we will fall in the rest of the other three areas that I mentioned. So God will judge us not for the amount of work that we accomplished, number of churches we preached, where the places we preached, you know, the number of countries we visited and preached the gospel. No, no, that's not what God is going to judge us on. He's going to judge our motives for why we did what we did. He will not judge us for the greatness of what the things that we have done, how many people we healed, how many people we raised back from the dead, how many people we led to Christ, or how many people we led into Holy Spirit baptism, but he's going to just look at, you know, if we were obedient to the Father's will, okay? And he will also judge us for the significance of our calling and not for the significance of our calling and gifting, but the faithfulness in which we carry them out, okay? So on these areas that God is going to judge us on, it's going to judge us on our motives, our intentions, obedience to the Father's will and our faithfulness in which we carry each one of them. In Revelation chapter three, verses one and two, you know, John is talking about the church of Sardis. He's saying, you know, I know your works in that you have a name and that you are alive.  Everyone knows, you know, you have a name that means your church is well known, you know, in Sardis and you are alive.   Revelation chapter one says, I know your works that you have a name and you are alive, but you are dead. And then he goes on to say, be watchful and strengthen the things which remain that are ready to die, but I have not found your works perfect before God, okay? So this is the angel of the church writing in Sardis writes. Okay, so the angel is saying beware, be watchful. You know, at least the areas where you're about to die, you know, be watchful in those areas. And it says, you know, God is saying, I have not found your works perfect. Works are not perfect before God. The angel is telling to the church at Sardis.   It's I, me, myself, and we need to, you know, be careful.  So it's possible that, you know, we can look anointed and spiritual before men, but God can be disappointed with us. So our desire should be that our work and ministry be perfect before God. So we need to develop a heart of a kingdom builder. So what is a heart of a kingdom builder? A heart of a kingdom builder is a heart that is devoted to Christ the King. You know, it has Christ the King as a center in our hearts. It's a heart of a kingdom builder is a heart that seeks to glorify Christ alone. The heart of a kingdom builder is a heart that does not receive honor or glory from a man. It's a heart that is looking to glorify and give honor back that is due to God that's reserved only for him. It's the heart of a kingdom builder is a heart that is pure in its motives. There's no unrighteousness in them.  Say, God, give me a heart of a kingdom builder. Where I'm seeking Christ alone. Where my heart is not looking for glory and honor and praise from man, but it's looking at pointing out, giving you all the glory and honor and give me a heart that is pure God, a heart that is righteous.  Okay? So this is chapter two. Anyone has any questions, doubts, anything? Any questions you want to ask? Any comments you want to make? Anyone knows this while you're thinking about, you know, questions or doubts and you want to just, you know, write it in the chat section or you can unmute your mics and ask. Anyone knows this hymn on page number 25 or hail the power of Jesus' name? It's a very powerful hymn. Anyone knows this hymn? Jefina? You know this hymn? Paul, do you know this John Paul? Do you know this hymn? No, I don't know this hymn. Okay, it's a beautiful hymn. Jefina? Yes, I know this hymn. Sorry? Yes, I know it. You know, okay. Want to sing? Sure. Okay. Maybe just one or two verses because we just have very little time. Yeah. Oh, hail the power of Jesus' name, let angels prostrate, let angels prostrate, forth bringing forth the world. He chose emcee, though his reigns risen from the fold. He runs from... Thank you, Jefina. It's a very beautiful hymn. Just love this hymn so much. It just talks about, you know, who has saved us. You know, it's Jesus Christ who saved us by his grace and hence we need to crown him because he alone is king and there's no other king besides it. We meant, okay. Anyone has any questions, any doubts? No? Okay, we'll move on to chapter three. Chapter three is the Holy Spirit is our director. Okay. We're just going to go through this chapter very quickly because you've learned about, you've learned about the Holy Spirit, the person work of the Holy Spirit, I think in the first year of the Holy Spirit, you also looked at, you know, few aspects of the work of the Holy Spirit when you looked at minister's foundation, basically the publication that you studied, receiving God's guidance. And also you'll continue to be studying about the person and work of the Holy Spirit. Okay, so we'll go for a break and after our break, we'll come back and look at chapter three, the Holy Spirit, our director. Okay.


What is your most memorable catering event since you started the company with your husband? Or maybe like top two. I'm sure you've done some amazing ones. We've done some really cool ones, but probably my favorite was the Breaking Bad finale. Diego mentioned that, that you did that. Yeah. I don't know if you saw the show before you watched it. Oh, my good smile. Every single episode. We don't have a television as you may have seen. Yeah. But Breaking Bad goes down as like my favorite television show of all time. I read Brian Cranston's book that he wrote. Yeah, what was the event like? So it was like the end of the day. Was there a theme? Yeah, I was breaking that. And Aaron Paul drove the Winnebago to. Stop. Yeah. It was that Hollywood Forever, the graveyard. Oh, my gosh. It's perfect. Drove it to the graveyard. And then they had a screening of the final episode. And then we had 900 people in the Masonic Lodge. And like plus full was to be today. Oh, my god. What an event. What did you serve? So how did you serve math? No. We did. We took isomalt, which is like a kind of stable sugar sort of glass. And we made tons and tons of sheets of it in blue. And so wow. Racks and racks of the blue crystal math. We have those tiles that are on the elements chart. Yeah. So we have the periodic table like cookies. And it was sitting on a pile of the crystal math. And then our waiters all dressed like either El Pollo Loco or not El Pollo. What was it? Yeah, that was at the Pollo Loco. Yeah. No? No. That's the chicken place. Los Pueos Hermanos. Yeah, that's it. Los Pueos Hermanos employees. So they all have those uniforms with like the red visors.  And they were wearing tidy whiteies and a green shirt. No. And like brown socks. And they were the cocktails for us. That they were everybody catching. Yeah, yeah, yeah. And then we had a pizza from the show that landed on the roof. Denizios, we had like pizza boxes made that we were serving that, you know, Denizios pizza, which we made ourselves. And we had all of our other trays for the servers that bells on it. Like, it was the same Tito. Yeah, you guys wanted to detail with those. How did you? We were like super fans. Yeah. We were really excited. So this is just you and your team thinking of this whole thing? That's awesome. That's amazing. And then the marketing and the designer that we worked with, Lacey Maxwell, she's from the Bay Area, everything we threw at her, she was like, yes, yes. From May to the end, she loved the show too. And everything was edible. Yeah, so all of the bars were labs. And there was beakers. And she designed these amazing bars with, you know, like bubbling beakers and steaming things behind our bartenders.  And we had the industry chefs wearing like respirators and math suits. And we were making liquid nitrogen ice cream. Wow. And yeah, so it was really fun. Do you have pictures of this on your website? Oh, I think so. OK. That's amazing. That's amazing.  But that was like super fun because we were such fans. Yeah.