We have given a lot of pieces of evidence that things are made of atoms. And in fact at the beginning of the 20th century, there are so many pieces of evidences, but still there is a debate that a lot of people consider the kinetic theory of gases, consider that things are made of atoms. It's just out of our imagination, its just an efficient way to do the calculations. But still at that moment, a lot of people did not believe that things are made of atoms. For example, very famous scientist Mach. He didn't believe that things are made of atoms, how, eventually that every working scientist believes that things are made of atoms. That comes to the story of the Brownian motion. What is the Brownian motion? If you take small pollen from flower and then you take tinny particles from the pollen and you put the tiny particle from the pollen to water. And then you put the water under microscope. What do you see? Brown noticed that, once you do that, the tinny particle from the pollen just dance. Okay, moving randomly under water. Why? Why this tinny particles dance in the water? It is properly scientifically explained after quite a long time, after decades. And by Einstein, how, Einstein explained this Brownian motion. He explains the Brownian motion as the strongest evidence, the direct evidence showing that things are made of atoms. How? Let us use a slightly different way to derive that, now known as the Langevin equation, which is a stochastic differential equation. Don't be scared by the name. Let me explain it, first what is the equation of motion for that, tinny particle from the pollen? let me remind you, that when I'm saying tinny particle, it is a tinny particle in the sense, compared to the flower. To that pollen, however is huge compared to the atom. it is much greater than 10 to the minus 10 meter. Okay, it's huge compared to the atoms. But nevertheless, from observing the dancing of the tinny particle, we can have direct evidence of the atom. And this is the equation of motion for that small particle, the tinny particle and here the capital M is the mass of this particle and v is the speed of motion. And this lambda is a friction force which is dissipation, that if you are not kicking this particle, the particle is slowing down in water, which is friction, dissipation because of water. And this is dissipation and if there is only dissipation the particle will stop. However, there is not only dissipation in water but also fluctuation that the molecules in the water are randomly kicking the particle. The huge particle in the molecule sense, It's kicking the particle and once kick the particle, the particle moves a little bit kicks the particle the particle moves another little bit randomly. so this is how the random motion of the particle arises. And mathematically, how do you derive that? Mathematically in this equation of motion, you add a stochastic term. Okay, you add a random source of fluctuations. What is the random source? It is random, so we don't know for sure, how large is the force at a particular moment, but we know on the one hand, it's time dependent; on the other hand, the average of the force, average among a lot of possible systems. possible systems of the same type. The average is zero, and then the average of the two point correlation of the force. In other words, if t is not equal to t prime, then there is no correlation between two kicks. one kick and then the other kick is totally random. but if t equals to t prime, there is a nine trivial correlation is just a variance. f(t) squared is non-vanishing, and the mathematical tool to describe it, is the Dirac delta function that we introduced. And the coefficient in front of this Dirac delta function is the strength of kicking, which is the strength of fluctuation. All right. And this is the stochastic equation. The Langevin equation way to describe the Brownian motion. But, the connection is really an Einstein style discovery, why? Because Einstein is the first person who noticed the relation between this fluctuation and this dissipation. Okay fluctuation, and dissipation has the same origin. What do I mean? Imagine, if I am in the rain, the rain drops and I'm in the rain, I feel random kicks by the rain forward and backward. The rain maybe not exactly dropping downwards. Okay, there may be small motion, parallel motion to the ground. So I am kicking randomly forward and backward by the rain. I'm kicking by the rain and this is fluctuation. Because of the rain drops, I encounter some fluctuation everywhere. On average is zero. However, what if I'm running in the rain, if I'm running in the rain, then I feel dissipation from the rain drops. And what is the dissipation that I feel from the rain drops? If I'm running in this direction, the fluctuation in the front direction, hit me stronger. Compared to my back, so that is dissipation that I feel, that is more difficult to run in the rain. That is friction force, that is dissipation. And the dissipation, because of the rain drops and the fluctuation because of the rain drops. They are of the same origin, just up to a factor of kinetic energy. And this is the relation between fluctuation and dissipation. This is very profound, not only in the case of Brownian motion, but also for example the noise in a circuit, it is related to the resistance of the circuit because resistance is dissipation and noise it is fluctuation. But nevertheless, let's pay attention, focus to this particular case and in this case the equation, the Langevin equation to relate dissipation and fluctuation is as follows, and how to solve this equation? if you know how to solve this equation, you know the constant variation method. If you don't know how to solve this equation, doesn't matter. Just insert this solution into the equation you find, yes, it is the solution of that equation. where v0 is the initial velocity and this exponentially dumping term is telling you the effect of dissipation, that once you put this drop of particle into water. In case if there is no fluctuation, the velocity is decreasing exponentially by the coefficient of dissipation. All right. now that we have this fluctuation term, there is a remaining part of the random motion and how to describe random motion. Again, you can take the two point correlation function of this v, of the variance of velocity. The average of velocity in random motion zero. But you can take the average of the velocity squared, which is the variance. And you will find the variance is non-zero because, take the two point correlation, the variance of the fluctuation. The variance of the random force is non-zero weighted by dissipation and eventually at the limit t equals to infinity, it's a good exercise for you, that you get the v squared expectation value, I mean, Imagine if there are so many different systems, with the same initial condition but with different random force. In that case the expectation value of v squared can be calculated as the ratio of capital Lambda and small lambda, which is fluctuation and dissipation. And what this means, is we can further write this equation into, that the kinetic energy of this particle can be written as the ratio between fluctuation and dissipation. Just as the case that I'm running in water. I'm running in the rain. The kinetic energy is fluctuation divided by dissipation and both fluctuation dissipation can be measured okay? Dissipation can be measured by a moving particle put into water, how fast it slows down. And fluctuation can be measured by putting a drop of ink into water and see how fast it is spread. And by this ratio we know the kinetic energy. To be more precise. The average kinetic energy of this little particle. And where this kinetic energy comes from? this kinetic energy comes from the kick of the little particle. That means on average that kinetic energy of the big particle, which is the particle from the pollen and the little particles which are the water molecules. They should be the same order. And that means, this ratio is not only the kinetic energy of the big particle, the particle from the pollen, but also the average kinetic energy of water molecules. And from here, you're not only directly see, that the dancing of the little particle is because of the kick of the water molecules. Because of the atoms in general sense, but also you have measured the kinetic energy of an atom.