 Right, so yesterday we went through very quickly, we recapitulated some aspects of equilibrium statistical mechanics, what equilibrium was all about. Today we start asking time dependent questions and this will be the beginning of non-equilibrium statistical mechanics because you now do not have any longer the safe comfort of the fundamental postulate of statistical mechanics and the equilibrium level which is that for a given system in isolation all accessible microstates are equally probable. We have no such guarantee now, we are going to push and pull on a system, we are going to apply perturbations to it, we are going to move it away from equilibrium and watch what happens as a system is left alone. Now we need a model to start with and we will start with the simplest of models perhaps, we will start with the model where you have a fluid and it is in thermal equilibrium at a fixed temperature, the fixed number of particles, fixed volume and it is in thermal equilibrium. If it is a gas for instance, if it is an ideal gas then we know that the Maxwellian distribution of velocities applies and you also know that the particles are uniformly distributed throughout the volume, we write down in equilibrium what the phase space density is for a given particle. So you can easily write down the fact that if you have this container with a whole lot of particles inside n of them for instance and if it is an ideal gas then we can write down the phase space density immediately, it is clear that let me use a symbol for it, rho equilibrium and now I want the single particle phase space density, this is the probability that the probability density that the particle is at any point r inside the container and its velocity is some value v and it is in equilibrium and this is an unconditional density in the sense that I do not say what happens, what the initial value of this velocity and position variables are or anything like that, I simply say I pick a particle at random and I ask what is its equilibrium phase space density, okay. Now this is intuitively clear that equilibrium statistical mechanics of course provides an answer but it is intuitively clear what the answer is, as far as the position variable is concerned the density is absolutely uniform, it is got an equal probability of being anywhere in this entire volume, so this is just a 1 over the volume and then there is a portion which tells you how the velocity is distributed and we know then that since the energy of this particle, if it is ideal it is not interacting with anything else, the energy of this particle is just its kinetic energy and the system is in equilibrium at some temperature, inverse temperature beta, so this is equal to e to the minus m v square over 2 K Boltzmann times T because 1 half m v square is its energy, just the kinetic energy and the rules of equilibrium statistical mechanics tell us that this is the density of particles apart from a normalization factor and of course we can write the normalization factor as well, so this is equal to 1 over v m over 2 pi K Boltzmann T and there are 3 of these degrees of freedom, so this is 3 halves times this quantity here where v stands for modulus of the velocity vector, so that is the phase space density in equilibrium, but right away we can ask a more complicated question, we can ask what happens if I take a single particle and I somehow manage to tag this particle, I am going to keep track of it and we are working in the level of classical mechanics, so I tag this particle and I look at it by saying look at the instant of time when I take a look at it, it is got some position r naught, some velocity v naught and then I given that I ask what happens to it as a function of time, so now I ask for the conditional probability density function for which I will continually, let me write it as capital conditional probability density function, I will use the abbreviation pdf for it throughout and I ask for the conditional probability density function rho such that its position is r, its velocity is v at some time t given that it was some r naught v naught and at time t naught, so this is the conditional density and I ask what is this quantity equal to, so r naught and v naught could be any arbitrary values which are allowed and I say given that condition what is it going to do as a function of time, well what we do know intuitively is that as time elapses as t minus t naught tends to infinity, so I expect that if t minus t naught tends to infinity, this thing will actually tend to the equilibrium density because the system is not going anywhere, it is in equilibrium and I expect physically that intuitively that if enough time elapses since the time I measured the initial conditions, this particle will relax to equilibrium and attain the equilibrium density, it will lose memory of what its initial values were of these variables and it will tend to this quantity here but the hard question is now what is it as a function of t for arbitrary values of t and not necessarily t minus t naught into infinity, well it is again sort of clear and we have not established this but it is again clear that if the system is in thermal equilibrium then it does not matter when I start this t naught and that this whole quantity will be a function only of t minus t naught, only the elapsed time because when I start the clock is completely irrelevant since the system is already in equilibrium and the process is stationary, so I do not really have to ask specify any particular t naught, all I have to worry about is the difference between t and t naught, the elapsed time, so I might as well set t naught equal to 0, right, I will often do that, we will often do that assuming the system if it is in thermal equilibrium has no time dependence of any kind, so the time when you start the clock and measure the initial condition is irrelevant but the hard question is what is this function as a function of t, how does it reach this because I know what its value is initially, I know what its value is not as t minus t naught tends to infinity but as t minus t naught tends to 0, in other words at the instant t equal to t naught what is this density, it is obviously equal to delta 3 of r minus r naught delta 3 of v minus v naught because I am telling you that the values at t naught time t naught are these values, deterministic values, so it starts off in this very spiked way, this delta function distribution and broadens out or does something crazy, it becomes uniform in space and in velocity it becomes Maxwellian with a mean value of the velocity which is equal to 0, so I know the limiting cases, I know the initial condition on it and I know the limiting distribution but I do not know what it is in between, okay, that is a very hard problem, it turns out this is a very very hard problem, if you look at a realistic dense fluid it is an extremely difficult problem to answer because these particles are an interaction with each other to start with, number 1 they are correlated with each other, so there are all kinds of correlations between the particles in such a way that you cannot write a single equation, closed equation for this density, it gets coupled to 2 particle distributions, 3 particle distributions and so on in a well defined hierarchy and this hierarchy is a never ending, it is not an ending hierarchy, it does not for a finite number of particles n but of course if you have an infinite number of particles in the thermodynamic limit, this hierarchy is not a finite set of equations, so already you begin to see that this is a very very hard problem to start with, taking into account all the possible correlations and if these are quantum mechanical objects and you have to impose for example symmetric or anti-symmetry in the wave functions, if you say that these are fermions or bosons which obey certain kinds of quantum statistics due to indistinguishability then it becomes even more hard, it is a very hard problem in many body theory and we are certainly not going to get towards solving this problem. So let us look for a simpler problem to solve, a simpler model to solve where we get some insight into how time dependence appears and we will use physical guidelines to guide us through this problem here. So let us suppose that in this fluid, in this dense fluid you have a few much bigger particles, a few tagged particles, colloidal sized particles. So these atoms, these particles are of atom size, atomic or molecular sizes which are of the order of fractions of nanometers but now let us put into it some objects which are my, are a little smaller than that but which are already several thousand times the size of atoms, individual atoms, right. So let us call those the tagged particles, I mean let us call those the larger particles, I will use little circles for that and let us put a dilute concentration of those things. So these are much bigger particles with individual masses which are much, much greater than M atom, they undergo Brownian motion in this fluid, they are moved about from side to side, they are jolted about, they have very jerky motion because they are being buffeted constantly by these little tiny particles, the atoms or molecules, okay. We will also assume that the concentration of these objects is sufficiently small that they do not interfere with each other. So essentially we can treat each one of them in isolation. So that is the second assumptions, the mass assumption will appear in a very subtle way and we will see exactly where it comes about. So this is like for instance in realistic cases it would be like having pumpkins being bombarded by mustard seeds even worse because these things colloidal particles would be micron sized and the other fellows are angstrom sized, okay. So many orders of magnitude separate them and the question is will these particles get anywhere? Obviously they are going to stay there, get bounced around, etc., etc. And let us see what happens to the motion of these particles here, okay. Now what we can do is one of several things. We could try and find out, since we want to know what the distribution and velocity in particular of these particles is, we could try and find out what the actual probability density function of the velocity does as a function of time by writing a little model for it, making some assumptions about the way in which the atoms bombard this big object. Alternatively, we could start with the equation of motion. We could guess an equation of motion, use Newton's equation for it, for instance, write down this equation of motion but take into account the random forces due to all these other particles which we cannot compute which are completely random for all practical purposes by using a stochastic equation for this particle and making suitable assumptions about the nature of the noise, okay. So let us use that. We will try to do both but let us write this down to start with the equation of motion and this is called the Langevin model. It will not be apparent in the beginning where this assumption has gone in. That is a little subtle but we will see how this figure is a little later. We will run various consistency checks and we will see that you need this assumption otherwise it is not true for the reasons I mentioned earlier. Namely, if you are looking at the atoms themselves and one of these particles is one of the atoms themselves, then it has got all kinds of correlations with the other particles and you cannot write a simple one body phase space distribution function, okay. So we keep that in our mind and we now write down something for this equation of motion for this. And let us make life very simple to start with. We will assume that this system is again in thermal equilibrium. So these particles are also in equilibrium at the same temperature except that they are also uniformly dispersed. We will neglect the effect of gravity for the moment and the velocity distribution is Maxwellian with this M here where M represents the mass of one of these particles here, one of these large particles, okay. So the heat bath is provided by the fluid which ultimately is in contact with the external world at some fixed temperature and these particles are in thermal equilibrium in this fluid and their distribution function and velocity in equilibrium is the Maxwellian distribution. It is precisely this and in equilibrium the position is also uniform so it is precisely this with M now standing not for the atomic mass but the mass of one of these particles here, okay. Now I ask this question. What if I start with some point and I go and I ask at t equal to 0, I start somewhere and I ask what happens as a function of t? How does it attain the Maxwellian distribution eventually? So it is exactly the same question by little sleight of hands hand instead of writing M atom, I wrote this M for the large particle and this is exactly the problem we have to solve, yeah. The degree of the larger particles are very small compared to the bath, yeah. How do we define a temperature for such a low density, a dilute gas? I can still define a temperature for a dilute gas. This is a good question. How do I define a temperature for one object, right? I am going to assume that its average kinetic energy defines its temperature which it does in an ideal gas that is precisely how temperature is defined, right. Temperature is a sort of wrong, is a sort of irrelevant notion that we unfortunately introduced for historical reasons. Before the nature of what heat is was known, they did not know that heat was random molecular motion, the energy of random molecular motion. Had we known that we would not have used another unit called temperature at all and then introduced a Boltzmann constant to convert from one to the other, they just called it average energy and that would have been the end of it, right. So in that sense I am going to say that the average energy of this particle, each of these particles is 3 halves kT, okay, right. Now I asked this question here and for that purpose we are going to aim at that but to start with let us do the following. Let us start with an equation of motion for this particle. So what is its equation of motion? It says m and for notational simplicity for the moment, we will put this in complication in a little later. For notational simplicity let us look at one Cartesian component of the velocity, x component for instance and let me just use v to simplify the notation. mv dot, this is a given Cartesian component. It is a little abuse of notation because I call v the magnitude but forget about this now, I will just use this. Otherwise I should write v1, v2 or v3 which is a nuisance. And what is this mv dot? It is equal to the force on this particle, instantaneous force on this particle which is varying as a function of time because the force is caused by the collisions of the molecules around and it is varying very rapidly and very randomly. In fact in the gas in this room for instance on the average there are about 10 collisions per picosecond also for the molecules. So it is really happening very, very fast. And this is being buffeted by these particles very randomly in a fluctuating manner. So this is the total force and it is supposed to be random. This fellow is supposed to be random. It is random in the sense that it is random for all practical purposes because if I were able to actually write the equation of motion down for all the particles in the system then of course I know where they are going, who is going to hit which other particle and so on and then there is nothing random about it in that sense. But of course once you have a very large number of particles for all practical purposes this force is totally random here, right. I want to use a symbol for this random force. So let us write it as equal to some eta of t. I want to use proper notation. So let me write it as a quantity gamma times eta of t where this is a constant. Well I want to simplify this notation as much as possible. So let me just write it as eta of t and we will come back to what this means. The randomness that I am going to put into it is to say that this eta of t is what is technically called a Gaussian white noise. So I will explain these terms what is meant by that as we go along. But it is some random force in this kind. It has physical dimensions of force as it stands here, okay. So I have this simple equation and the matter is over in the sense that I can integrate it formally. So once I integrate it I have v of t equal to an initial value of the velocity because I want to find out this quantity here. So I am going to start by saying to solve a differential equation you need to specify an initial condition. So I am going to say at t equal to 0 the value of the velocity was of this component was some number v naught, okay. So v of t equal to v naught that is the initial value plus 1 over m integral. I might as well choose the initial time t naught to be equal to 0, set t naught equal to 0 because as we have said as I have argued this whole thing is in equilibrium. So it does not matter when I start the clock. So this integral runs from 0 up to time t dt prime eta of t prime. That is the formal solution to this first order differential equation, this triviality itself and it satisfies the condition that at t equal to 0 v is v naught. But this is a random force and what I want is averages because once I have a random variable what I am interested in are its averages of various kinds, averaged over all realizations of this force. So all the force histories that are possible and each time there is a different history you get a different equation. So really this equation looks very trivial but it is actually an infinite number of equations because each time the noise is a different kind of fluctuation or realization you have a fresh equation really because it is a different function eta of t but we would like to average over all realizations of this noise but now we got to be little cautious. What do I mean by averaging over realizations of the noise? What is the input variable here? Well we will do this averaging in 2 steps. When I want any quantity which is an average of referring to this particle's dynamical variables such as its velocity or its mean square velocity and so on I have to tell you what am I averaging over? Not only should I average over all realizations of this noise but I should also average over all possible initial conditions because there is no reason why I should have chosen v0 as the initial value. I could have chosen something else as the initial value. So there is a double average involved here. First an average for a given v0 you average over all realizations of the noise for a given initial condition and then you average over all possible initial conditions as well. So I am going to use overhead bar for conditional averages whatever is inside here for conditional averages for a given v0 and I am going to use this for a complete average for a given v0 over over realizations of eta of t complete average over all v0 as well. So we will do this average in 2 steps. First we average over start with a given v0 so we look at the ensemble of these particles such that the v0 is given and we repeat this experiment with different eta of t each time take the average and after that we will average over the initial velocity as well. Now implicit in this is the fact that these averages should commute with each other. Which ever order I do it in I should get the same final answer and that is more or less obvious with little subtle assumptions involved here. First of all averaging is an arithmetic averaging is just adding numbers and addition is always commutative but more than that there is a deep assumption here that whatever is the fate of this particle does not affect the bath the heat path. So the assumption is the heat parts properties do not change as a consequence of the motion of these particles. That system remains in thermal equilibrium at temperature t okay. So this noise its statistical properties are not affected by what an individual Brownian particle is doing. So that is a physical assumption and you have to make sure that that is really satisfied before this model becomes realistic. So given that let us take the average over this. So what is v of t equal to? There is nothing to average here because this bar is for a given v naught. So this is a deterministic number and you get just a v naught plus 1 over m the average of this integral okay but an integration is a summation and the average is also an additive operation right arithmetic average. So this is equal to integral 0 to t dt prime 8 of t prime. So notice how carefully we are proceeding step by step so we know what the assumptions are at every stage. So every one of these operations has to be justified. So you agree that this is indeed the average value of the velocity but now I argue saying that this particle is not going anywhere it is staying inside the container and it is hit as much from the front as from the back and the average value of this force it is very reasonable to assume is 0. The average force exerted on this Brownian particle by all the other particles is on the average 0 because its average velocity is in fact 0 okay. So we will make the assumption that 8 of t average which indeed is equal to 8 of t as well because even if you change all the initial conditions it is still true this is equal to 0. So as far as the properties of 8 are concerned it does not care what the Brownian particles are doing and its average value is 0. That is the first thing that it is a noise with 0 mean okay. So this term vanishes and this is equal to v naught. No, no, no realization is if I plot this 8 of t which is the x component of the force if I plot this 8 of t versus t this might be one realization of the noise but it could equally well have been this it could have been this so I am going to average over all these realizations pardon me average, average on the average because if it were not 0 this particle would move systematically in some direction. So it is clear on the average it is 0 this these are not these fellows are not going anywhere they will have a fluctuation about their mean position but the mean itself is 0 to start with. We have already said that these Brownian particles are in thermal equilibrium and the mean value is 0 completely. We are not going out of thermal equilibrium we are just saying I focus on a particular particle and look at its initial condition and ask for conditional averages subject to the condition that the system is still in thermal equilibrium right. So we are asking a time dependent question because we chose an initial condition that breaks the fact that this velocity is 0 on the average it starts with some v naught and I ask what is it doing or to put it yet another way in pictures let us do that if I plot the velocity distribution because the position is irrelevant here that remains uniformly distributed throughout but if I look at this row of v t given v naught this is the quantity I really want for this tagged particle and I plot v here then at t equal to 0 it is starting with a delta function at v naught. So it is clear that this is the distribution where this is v naught it is starting with a normalized delta function as the distribution and as t becomes infinite the value of this distribution here becomes independent of the initial condition it loses memory and it gets into the Maxwellian distribution. So it is finally like this and the question we are asking is how does a distribution manage to start with a delta function there and end up with a Maxwellian here as t goes to infinity. So this is the distribution this is at t equal to 0 and this is at t turning to infinity. So that is the question we are really trying to answer how does this distribution do this what does it do in between not very surprisingly you will discover that this mean value starts drifting to the left until at infinity it becomes 0 here but in the meantime it is also broadening because this is 0 width and this is a width dependent on the temperature k t over m square root of so that is the kind of question we are trying to answer but we are doing it through the equation of motion. So we will first get information on the mean the mean square the correlation function and so on and then work backwards using that information to try to get the functional form for this quantity here and perhaps even write an equation down for it and maybe even solve it. So this was the first step here and now the next thing to do is to ask what happens if I average over initial conditions as well. So we can do that right away we have t pardon me yeah no no it will it will work you will see this will work because now I am going to ask what is the average value if I now average over all initial conditions but remember this particle was an equilibrium. So we know the following is true we know that rho equilibrium of v the equilibrium distribution is the Maxwellian distribution. So we know that this is equal to m over 2 pi k Boltzmann t to the power of half this is one dimension one component only so it is just a power of e to the minus m v square over 2 k Boltzmann t one component v is denotes so that is the reason for the half here instead of the three halves but that is a Gaussian symmetric about the point 0. So if I now do this this v of t this is equal to an integral from minus infinity to infinity d v not times v not which is this quantity here multiplied by rho equilibrium of v not and what is the value of that integral it is 0 and why is that so yeah because this is an odd function and that is an even function this distribution so it vanishes which is completely consistent we know that the average value has to be 0 in equilibrium and it remains 0 at all times so we are completely okay because at no time is it actually moving off anywhere or anything like that the full average the conditional average of course will start at v not t equal to 0 and perhaps drift towards this but the full average since the system is in thermal equilibrium the velocity is a stationary random process it the mean value cannot depend on time because then it is not a stationary process okay the mean the mean square mean cube none of them can depend on time no moments and we must check that. So you agree so far that this has been perfectly alright in other words what this has tested is that this assumption is not leading to any contradiction so far it is a physical assumption and seems to have worked now let us find the mean square and see what happens so the next step is to ask what happens to v square of t this is equal to I take this quantity and I square it so I get v not squared plus 2 over m times v not integral 0 to t vt prime eta of t prime so I take this integral and multiplied by v not twice that plus term which is 1 over m squared an integral now let us get rid of this prime notation and write something sensible 0 to t dt 1 integral 0 to t dt 2 eta of t 1 eta of t 2 that is the value of the square because it is a definite integral I really have to use 2 symbols of integration when I want to square this term right now let us take average conditional average so this implies that this quantity this thing here there is nothing to average over here because it is given v not squared is given and then I average over this nothing to average here that average is over eta but that average is 0 because this average is to 0 and then inside I have this average here so now we need a model I need to know what is eta of t 1 eta of t 2 average we know by our assumption that the bath is not affected by this particle that this is really the same as eta of t 1 eta of t 2 equal to what is the question now here is where the physics has to come in okay and this is why I said this is applicable to a particle whose mass is much much larger than the mass of the molecules so the statement is that this noise which is individual molecules hitting this particle okay is completely uncorrelated in other words what hits it at time t 1 is very different from what hits it at t 2 and then nothing to do with each other at all so this thing here which is telling you how much of memory sticks on here is essentially 0 if t 1 is not equal to t 2 this memory would in fact drop exponentially and it would drop like a characteristic time there would be of the order of the mean time between collisions that is of the order of a picosecond but we are looking at a much longer time scales so this is effectively a white noise that is the whole point about delta correlated white noise it is true noise which is delta correlated so it is proportional to delta of t 1 minus t 2 which has the wrong dimensions because this is the square of a force so you need to fix the dimensionality what is the dimensionality of this quantity one over time so put in a constant gamma here this is a constant which tells you in some sense the strength of this force the larger gamma is the stronger the force so this is why I called it a white noise we will come to a technical definition of white noise a little later but it is something which has a flat power spectrum essentially a delta correlated noise we need to make additional technical assumptions about what sort of this is very nice for the correlation function but if you give me a noise a stochastic process not only should I tell you all its mean values I should also tell you it is all its correlations an infinite number of correlations at different time and I should tell you the shape of the probability density functions for this random process not just one time but two time three time etc all the joint probabilities I have to tell you completely and we will make the simple assumption that they are all Gaussians because we invoke at this stage the central limit theorem it says if you have a large number of effects which are independent of each other and each of them has a finite variance and a finite mean then when you add them all up in some linear combination the resultant distribution is a Gaussian that is a very roughly speaking the central limit theorem of statistics we will invoke it when the time comes but right now we need just this assumption about the correlation notice that this is a function only of T1 minus T2 what I expect station it is also a symmetric function of T1 and T2 does not matter which came first so this is the assumption that I have been talking about delta correlated stationary noise stationary because it is statistical properties do not changes a function of time so all these called the average value 0 this is a function only of the time difference the three point function will be a function only of two time differences and so on the absolute origin of time does not matter so let us put this into that and see what happens so I put that in this implies that V squared of T average is V0 squared plus 1 over M squared and now I got to do this integral the gamma comes out and I have to do a delta function integral inside here but the delta function just says replace T2 by T1 right you have to be careful here again because if I plot T1 here and T2 here the integral runs from 0 to T in each of the variables and that is the line the 45 degree line is a line on which T1 is equal to T2 and now if I do the T1 T2 integration first it means I fix a value of T1 and I integrate in T2 from 0 to T and the delta function fires here I go to the next value of T1 and the delta function fires here so it is clear that through the range of T1 from 0 to T there is the delta function fires for every value so I can close my eyes and remove T2 and put T1 wherever it appears right and of course there is nothing to put there is no T1 here it is just a delta function which goes to 1 and I get an integral 0 to T which is V0 squared plus gamma over M squared T very bad news because this is now telling you that for this initial condition this fellow is actually increasing with time linearly and now if I average and do this V squared of T this is equal to an average over V0 squared which is just the mean square value of the velocity over the Maxwellian and that is k T over M plus gamma T over M squared there is nothing there is no V0 dependence here so the average value remains where it is and it is this thing here and it is completely unphysical because it says if you leave this beaker of water alone put a colloidal particle in it then its energy the particles energy will spontaneously increase till it becomes infinite as T tends to infinity so it is actually taking energy from all the molecules and reaching infinite temperature or energy if you like okay which is wrong it is just plain wrong so this is completely wrong what do you think has gone wrong so this is practically incorrect it is clear that some assumption there made there is wrong one or more assumptions so what assumption do you think is wrong we will eliminate it one by one what assumption do you think is wrong one possibility is that this is not correct a very strong possibility is that this is not right this is really not doing this at all but it is got some exponential correlation it is a finite time it cannot be there is no such thing in real life as a delta correlative noise because there is some correlation time no matter how small it can never be mathematically 0 right so that is the first step so let us do that let us say that this quantity is not that at all but it is some some other constant K times e to the minus T over tau e to the minus mod T 1 minus T 2 over tau that is a symmetric function it is stationary and so on so we could say good we will put this in so this integral is not so easy instead of a delta function you got to put this I leave you to check that you can do this integral once again now it is a very reasonable assumption to make it is an exponentially decaying function instead of just a delta function and one can say alright let me do that integral now and see what happens a little more hard work because you have to do both integrals T 1 and T 2 but I leave you to check that you will again run into exactly the same problem this it will not quite be there will be a little more complicated expression but there is no doubt that eventually you will end up with something that increases with T and does not stay in equilibrium okay so check that the incorrectness of this model is not due to the assumption that this noise is delta correlated even a finite correlation time will still produce the same thing so that is an exercise for you to check out but it still does it so that that is out that explanation is ruled out what is the other possibility what are the possibilities there would you say Newton's equation is wrong classical particle has to satisfy Newton's equation would you say the fact that the mean value of the velocity the noise is 0 is wrong that is not because that will just remove the mean value of the velocity itself will disappear will not be correct if so we ruled out all the other possibilities so what do you think is wrong the particle is moving in the fluid it will probably have a drag yeah so there are subtle effects like that there is a there is a back flow there is all kinds of complications like that there are many many such effects there is a reaction on the particles definitely there is certainly a back flow and a reaction on the bath particles that is not taken into account here but that is not the reason in this case one can take that into account in some fashion but it is not but the other statement you made that there is a drag on this is certainly a place to pause and give some thought to it how does this drag show up in this model how will it show up well imagine this particle and the particle and one dimensional motion so I move this way I am constantly being buffeted on both sides by molecules what happens if I move relative to these particles in one particular direction what happens when you walk in the rain and the rain is coming straight down but you walk where do you get wetter in the front or at the back the front certainly what happens is when you move in a crowd there is more particles hitting you per unit time from the front than from the back right and they impede your motion you turn around and walk in the other direction there are more particles hitting you from this side and the impede your motion that is what you call viscous drag that is precisely it so there is a part of this random force which is actually dependent on the motion of the particle itself that force disappears of the particle had zero velocity instantaneously but the moment it starts having a velocity there is a drag on it a viscous drag right so it means that this model is incomplete this force this random force has really got two components so there is a systematic eta systematic of p plus eta truly random only random just put it in quotation marks this is acting like noise but this depends on the state of motion of the particle itself and what is the simplest model of viscosity a linear model proportional to its velocity so it is clear that what we ought to do is to leave this as the white noise out here but this quantity here is in the direction opposite to the motion of the particle multiplied by some constant gamma and then v of t itself I want this gamma to have dimensions of time inverse so let's put an m here yeah this is a much bigger particle right what you call viscosity is the internal friction in a fluid which any given fluid layer sees due to the other particles in the fluid right so we are now adopting an extremely simple nigh view of viscosity as being simply proportional to the velocity to start with but we will see if this can be refined it should be refined but that's the 0th order guess eventually I have this vague feeling somewhere that this quantity this force this gamma this constant gamma will get related in some mysterious fashion to the viscosity of the fluid that's what I would expect because that's what measures the level of viscous drag drag forces on objects in the fluid right but this object is much bigger than a given molecule no it's much bigger than the molecules like it when you calculate Reynolds number to tell you when turbulence sets in etc the viscosity plays a role because you put a big circular obstacle or spherical obstacle or whatever inside a fluid and then you measure what the drag on it is so that's one of the reasons I said this particle we are talking about is tens of thousands of times bigger than the molecules so it's not at the molecular level but the manifestation of that will come from a property of the fluid which is its viscosity and yes it's a hard problem to calculate what the coefficient of viscosity is from the molecular level but that's a standard problem in many body theory there are computations for it I might even mention some of them so this is the model they're going to put in here I'll stop here today since we've run out of time but once you put this in and make the same assumptions about eta of t we go through exactly the same calculations let me just write down just the first step in it the solution of course is v0 e to the minus gamma t here because there's an integrating factor which is e to the gamma t so I put it to the right hand side you get this plus and you no longer have this simpler solution so you have plus 1 over m integral 0 to t dt prime e to the minus gamma t minus t prime so that's the solution and I request you to find out now what the mean value is first the conditional mean and after that the complete mean and then similarly what the mean square value is first the conditional mean square and then the full mean square value and we take it from that point