 Yeah, we have so far concentrated on dynamical systems which are generally non-linear with special reference to systems which are deterministic in the sense that there were specific rules of evolution and then we looked at the nature of the solution, the dependence on the initial conditions and sensitive dependence in the case of chaotic systems. We looked at both differential dynamics where systems were described by sets of differential equations of the first order coupled first order differential equations and discrete time systems versus the dynamical system was described by a map of some kind of recursion relation which gave the rule for evolution and time. But you see in real life most systems that we deal with would also have influences from outside or even internal sources of noise which can't be given detailed descriptions. You can only give statistical descriptions for such influences and then the resulting equations that you have no longer are deterministic equations but they would involve components which are random some kind of noise which is imposed on the system. So the general dynamical system would have portions which are deterministic and portions which are pure noise with some specified statistical properties and this would lead us to the area of stochastic differential equations and stochastic differential equations and so on. We haven't considered that very much in this course but I just like to introduce you to the very elements of stochastic so that you can make some contact with what happens in real systems some of the typical noises that one deals with. Now of course we also took a little excursion into probability and a little bit about Markov processes and so on. So we will build upon that knowledge and try to go on from there to write down the most elementary stochastic differential equations. Here is one instance of where it can happen in a practical situation. You know that if I have a set of chemicals reacting with each other in general you would have the concentrations of these chemical species the different molecular species as your dynamical variables and you perhaps have a set of differential equations of the form if the concentrations are of the form C1 C2 etc at some given instant of time then you would perhaps have a description for the way the reaction proceeds in terms of differential equations which say C1 dot equal to some function of C1 C2 etc and so on for the remaining concentrations. So this would give you a very gross description of what happens in the system and one of the ways you would write down these functions is to write the individual reactions down and then use perhaps the law of mass action along with the rate constants to tell you the rate at which the concentration of each species changes. Now that is a dynamical system in the sense we have understood it. A set of coupled first order differential equations with possibly some nonlinear terms on the right hand side and then the question would be is there an equilibrium is there an attractor etc etc what kind of steady state or equilibrium solutions you can have for such systems that would be one brand of questions but of course you could also look at this at a more local level and you could say well perhaps the concentration C i of the ith species changes from place to place if you course grain the system and it is a spatially extended system in which case this becomes a function of r and t then you could ask how does this change as a function of time and typically you would get equations of the form delta over delta t of this would be equal to again perhaps some function of all the concentrations C1 etc some deterministic portion which would be governed by maybe the law of mass action or whatever the set of reactions concern gives you plus there could be diffusion of the species and that would be described perhaps by writing a diffusion constant I for the species I times del squared C i as a function of r and t notice that we have not put in any noise here at all this is still a deterministic system but it is spatially extended so these are partial differential equations and now you have a much more complicated problem than before because you have a degree of freedom corresponding to each of the species at every point in space and so this certainly is much more complicated than the earlier kind of system we talked about these are called reaction diffusion equations their solution poses several challenges you now really have to deal with much more complicated system of partial differential equations of this kind but it is still not a stochastic differential equation it is still completely deterministic you could say okay I do not have something as regular as this but I have noises I have perturbations external forces and so on in which case you would have genuine stochastic differential equations let us look at the simplest of these so let us now change horses and go and look at the simplest kind of stochastic differential equations that one could possibly have I start with extremely simple examples of such equations and then we go on from there to look at more complicated things and let me in fact use an example which is familiar to all of you the example of a fluid say an ideal gas like the gas in this room and try to write really a microscopic equation for a single molecule we assume this to be completely classical and I try to write a differential equation or its equation of motion including the effects of collisions due to the other molecules and try to see where this gets us now what would you say obviously if I took a simple system of this kind like an ideal gas or a simple fluid and I write Newton's equation down for a given molecule we will assume these to be ideal point particles point masses then let us make life even simpler by focusing on one particular Cartesian component say the x component of the velocity and then the what you write is to say the mass times dv over dt this is the acceleration is equal to the force on this particle and the simplest assumption you could make is to say that I focus on one particular particle of a system like the gas in this room in equilibrium which is in thermal equilibrium so I know there is a Maxwellian distribution of velocities and then I try to write down a microscopic equation of motion for a single particle given tagged particle molecule this is equal to the force that acts on it so let us write this as the force and this force of course would depend on the time it would change from time to time this is some time dependent force here in this fashion and what would you say this force is equal to well it would be the vector sum we are talking about a single component so it would be the resultant of the external force on the system if any maybe you have got it in a gravitational potential or an electric field or anything like that some external force if any that could of course change with time plus the internal forces due to the system itself on this on the particle and this would come due to the interaction of this molecule with all the other molecules in the gas or in the fluid and we are going to make a very simple assumption and simply say for the moment you can always handle something with an external force let us do something in the absence of the external force and ask what kind of internal force you have here so now this internal force since it is due to all the other molecules and I am going to now simply assume there are just elastic collisions that happens at completely random instance of time and so a simple assumption would be to say mv dot is equal to without this external force an internal force there and it is a random force of some kind let me call that random force eta of t so this is a random but it is not enough to say that is random the moment I say something is a random variable I have to specify statistical properties of this random variable otherwise you cannot do anything more so we have to now start saying what kind of force is this what kind of random forces make the simplest possible assumptions this force is changing extremely rapidly because on the average the molecules in this room each molecule undergoes collisions such that maybe the time between collisions the mean time between collisions is of the order of picoseconds so certainly within any measurable time of even a microsecond it is undergone a large number of collisions completely and its motion is totally randomized so one possibility is to say that this force here very reasonable thing to say is that the average value of this force is 0 at all times because the gases whole is not going anywhere therefore the average force is 0 it is a reasonable assumption to make independent of time simply 0 you could also make the assumption that this force has no memory at all the different collisions is very unlikely the same two particles keep repeatedly colliding with each other recollisions are very very improbable so a very reasonable assumption is to say under these circumstances that this force is completely uncorrelated what happens from one instant to another if nothing to do with each other so a reasonable assumption would be to say that 8 of t1 8 of t2 two different times this thing here it factors into a product of 8 of t1 average of 8 of t1 times average of 8 of t2 because they are uncorrelated with each other when two random variables are uncorrelated then the average of the product is the product of averages since they have nothing to do with each other so this thing here is 0 for t1 not equal to t2 and of course when t1 is equal to t2 this thing here is a nonzero number it could even be unbounded because we have in mind something of this kind if I plot this correlation function as a function of the time difference between t2 and t1 I would expect that if t2 is bigger than t1 I would expect that this thing comes down goes off like that physically this is what I would expect such that the whole area under the curve is finite because that would specify a correlation time so a crude assumption a zero thought or assumption is to say if I look at this force on some time scales much larger than the characteristic time scale of the decay here this thing looks like a delta function to me and therefore a good assumption is to say this is equal to some gamma of delta of t1-t2 this is the a white noise approximation this simply says that the force eta is delta correlated there is no memory 0 memory time and gamma is some constant I have yet to determine this but it is some kind of constant here at the moment so it does satisfy the requirement that is 0 when t1 is not equal to t2 more over we have seen earlier that in the case of a stationary random variable in other words a random variable whose statistical properties do not change with time the average value is independent of time and that satisfied here it certainly 0 and the correlation function is a function not of t1 and t2 separately but a function of the time difference t1-t2 or t2-t1 in fact it is not hard to show that it is a symmetric function of the time difference that is because you can subtract from each of these time arguments the same amount of time if I subtract t2 then this becomes eta of t1-t2 eta of 0 here if I subtract t1 then it becomes eta of t2-t1 eta of 0 here but these are classical variables you can commute them in either order therefore it is easy to see that as a function of t1-t2 it is symmetric in other words it is a function of modulus t1-t2 and that is exactly what the delta function is this thing here is symmetric under the chain of sign of its argument so this is a good model as it stands we still have to see where it gets us but this is a good model now if I make these assumptions look at what is going to happen what consequence we have immediately it is an extremely simple model but we must try to see what it does for us so I start again with the equation of motion with these statistical properties and now I ask what is this imply for me what kind of average behavior will this imply for the velocity but remember that now I have to specify initial conditions to solve a differential equation I have to specify initial conditions that is easily done the solution of this equation the formal solution of this for a given initial condition so given the initial condition v of 0 equal to v0 so I focus on some particular particle whose velocity at equal to 0 happens to be v0 and that is a specified initial condition and I ask what is the fate of this particle of this molecule for this initial condition this equation implies that v of t is equal to v0 we have to integrate this equation out trivially it is v0 plus 1 over m 0 to t 8 of t1 t2 I simply integrate this equation that is the solution remember the system is in thermal equilibrium so I know that there is an equilibrium max valiant distribution of velocities so I already know that there exists an equilibrium distribution p equilibrium of v which is e to the minus m v squared over twice k Boltzmann t normalized appropriately the normalization constant in this case is that of the Gaussian so it is 2 pi k t to the power half so that is the probability density function of the velocity in equilibrium that is the Maxwell Maxwellian distribution it should really be called the Maxwellian probability density function but of course in physical applications we very often use the word distribution to mean the density function instead of the cumulative distribution function we just call it distribution so I know that but remember this is a conditional quantity here if the velocity is found to be v0 at t equal to 0 then the velocity at time t is given by this and now I average over the entire system over all those particles all those particles whose initial velocity is v0 so essentially I average over all realizations of this random noise and let me call that average with denoted by an overhead bar because it is an average conditioned upon the fact that the initial velocity of whatever system particle I am looking at is v0 some given v0 later I will relax that and average over all possible v0 as well with the Maxwellian distribution so what happens here I do this this implies that the average value of v of t for which I denote use an overhead bar is equal to v0 because this is a given number there is nothing to average over it has nothing to do with the noise plus the average of this quantity here this is an integration it is like a summation and averaging is also an arithmetic summation procedure so they commute with each other and the answer is this is 1 over m times the average value of v of t1 t1 but we saw that the average value is 0 we assume that it was 0 and therefore this is it as it stands but this is already extremely unphysical because it says if I start with a particle with velocity v0 then these particles that subset of particles which start with velocity v0 remain with the velocity v0 on the average for all time that is clearly not true it is clearly this is unphysical so something is wrong somewhere and we will see what is wrong but you see we still have to discover where we went wrong so it looks very reasonable that all we said was this thing here is due to all the other particles and they are completely randomized very very fast but then if I take this conditional average it leaves you with just v0 here so we already begin to see there is something wrong here but let us see what further it says we also find that if I square this yeah why should this velocity be the same I would actually expect why why should it be so why should it remember its initial velocity I start with the set of particles whose initial velocity is v0 and now I discover that their average velocity remains v0 at all times on the other hand if v0 is very far from 0 I expect that the average velocity should actually be 0 according to this yes yes so therefore I would expect it to go to 0 that is precisely the point I would expect things to go to 0 no if this v0 happens to be extremely large positive I would expect it to remain that I would expect it to transfer momentum to other particles and slow down I would expect that it remains as it is because in an elastic collision take it in one dimension in an elastic collision between two equal mass particles their velocities actually get exchanged that's the problem that's exactly the problem so it's going to tell us that this is it's going to get worse and let me let me do this and show you what's going to happen here let's find the square of t v squared of t what is this going to give us well first there's a v0 squared and then there are two cross terms where this multiplies that and vice versa but this is a constant and the average of it as 0 so those things go away plus 1 over m squared 0 to t dt 1 0 to t dt 2 8 of t 1 8 of t 2 averaged over all realizations of this noise subject to some initial the tag particle having a velocity v0 but there is nothing to do with the noise here this is a large heat path in which you have your tag particle sitting and therefore this bar here is exactly the same as far as eta is concerned as the full average over the system and that we saw was gamma of delta t 1 minus t 2 so let's put that in and this gamma comes out and you have a delta of t 1 minus t 2 but this integral is trivial to do just says replace t 1 by t 2 and that's the end of it so this integral goes away and you are left with this which gives you a t now if you go back here from this distribution we know the following we know that v in equilibrium is 0 because this is a component of the velocity it's an even function of v and therefore we know the average velocity is 0 every component of it is 0 we also know from this Gaussian it follows trivially that this is k B t over m which is just a verification of the fact that one half m v squared average is half k t which is an accord with the equipartition theorem every degree of freedom translational degree of freedom as an energy half k t on the average in non relativistic free particle situations but what is this telling us this is crazy because if you find now define one half m this to be the average energy multiplied by one half times m on top sorry this goes away and this gives me the effective temperature of the system then it simply says you leave a beaker of water alone in contact with the heat bar and its temperatures spontaneously increases to infinite values given enough time which is clearly absurd so not only was this bad this is disastrous where are we wrong where are we wrong of course you could immediately question this the assumption made on eta and say well eta of t cannot be delta correlated because that is unphysical it says that the correlation time is 0 and there is no physical force whose correlation time is 0 could be very very short but it cannot be 0 strictly this is true but on the other hand we have managed to separate time scales I tell you as a matter of hindsight that the actual correlation time of this random noise would be like the correlation between the time between collisions that is of the order of picoseconds or less on the other hand if I look at the system on the time scale of microseconds or more then this is certainly a good approximation it does not matter at all but the force is it is random but there is something missing in this random force and a moment's reflection will tell you that if a molecule has instantaneously a very high positive velocity it is clear it suffers more collisions per unit time retarding it than other than pushing it forward and therefore the right thing to do is to go back to the model and say that m v dot is not equal to eta of t alone but also has a systematic part which retards it which is proportional to its velocity instantaneous velocity but is in the opposite direction and that is best model by saying it is minus on this side something proportional to the instantaneous velocity with a constant gamma I would like to make this constant gamma have the dimensions of time inverse so let us put an m this is also random why because this is a random driving force that makes the velocity random therefore this is also random but it is a systematic part of the random force it is a specific function of the output variable and we made an assumption we have said that this retarding force is linearly proportional to the velocity instantaneous velocity itself and that of course immediately tells you that v dot plus gamma v is equal to 1 over m in the absence of an external force you could of course always add an external force to this but this is the solution in the absence of an external force when the system is in firmly equilibrium and what is the solution to this this equation and gamma is a positive constant that is crucial now what do you think is the solution to this equation well all we have to do is to solve this first order linear differential equation with that initial condition instead of this thing here all we need to do is to put in the integrating factor which is v0 e to the minus gamma t that side and since the integrating factor is e to the minus gamma t in this case this gets multiplied by e to the minus gamma t minus t1 that is the solution which obeys this boundary condition the initial condition you could put in other models but this is the simplest thing what made me decide on it was simply experience based on what happens in a fluid I know that the friction for small velocities is proportional to the velocity itself that is the assumption I made and it is directed opposite this is called the lines of my equation and where does it get us well for a start please notice what happens to the mean here it is this term here it is not that that goes to 0 because the average of eta 0 so indeed you have this what happens to this at t goes to infinity goes to 0 so it seem to be on the right track we seem to be getting back equilibrium but we have to still check this out so this goes as t tends to infinity 0 which happens to be this we have further corroboration for that because how would I compute the full average of v full average of v of t would be this average averaged over all possible v0 because this is a conditional average it is an average over that subset of particles whose initial velocity is v0 I take that subset and I use all realizations of the random force eta of t but now I would like to get a full average velocity does not matter what initial condition I choose so do you agree that this quantity here is equal to dv0 v of t bar over all possible values of v0 with what weight factor with what distribution with the Maxwellian because the system is in thermal equilibrium to start with at t equal to 0 so I put in p equilibrium of v0 and what do you get for this integral this is from minus infinity to infinity well notice this is v0 e to the minus gamma t e to the minus gamma t does not figure an integration this is an even function of v0 you put a v not so you get 0 so actually we seem to be on the right track as required it is actually restoring the equilibrium but there is another lesson we have learned from this is a very important one to find this v equilibrium I can do in two steps one of them is to find the partial average over a subset whose initial condition I prescribe and then I either average over all the initial conditions or I simply wait for long enough I do not average over all initial conditions I say I do not care about the initial condition I wait long enough and the system forgets its initial condition and indeed it goes to 0 my initial assumption is that a system is in thermal equilibrium with a Maxwell distribution of velocities that follows from equilibrium statistical mechanics independent of this time dependent stuff completely this thing here I would like to find the average of the power velocity of the particles in this room at any instant of time and I start by saying at t equal to 0 the system is in thermal equilibrium I do not do anything to it now I know the average value must remain unchanged with time now I am trying to build a model for what happens if I ask time dependent questions namely if I start with the particle whose initial velocity happens to be v0 and I watch it what can happen to this on the average well this model tells me its average velocity will decay from v0 to the value 0 given enough time with some time constant gamma inverse but I could be impatient I could say no no to find this average value at any instant of time given the fact that the distribution is known at equal to 0 all I have to do is to find the conditional average the partial average for a given initial condition and then average overall initial conditions with the equilibrium distribution so either I let t go to infinity or I let this happen and I must get the same answer and the reason is I expect I expect that if I if this process by which this whole thing is happening is a mixing process I expect that the conditional probability density that the velocity of a particle is v at time t given that it was v0 at time 0 this thing here I expect that at t tends to infinity this memory is lost and I would expect this to go to p equilibrium of v but if that happens then the averages should also follow the same rule I would expect that if I wait long enough I should get equilibrium average in this fashion so this is what we have to corroborate whether this happens or not well it seems to work at the level of the mean but the real test is at the level of the mean square because that is what involves physical quantities like the energy average energy of the particle so let us try to find this quantity now v squared of t and where does that get us in the Langevin model well this is equal to I have to take this solution square it and then take the average value the first term will be very trivial it is v0 squared e to the minus 2 gamma t so I square this guy plus a quantity which is the product of this times that average over eta that is 0 because average of eta 0 and similarly the other direction plus a 1 over m squared integral 0 to t dt 2 then e to the minus gamma t minus t 1 e to the minus gamma t minus t 2 times the average value of eta of t 1 times eta of t 2 but that we have a large one more we have a model for it in this white noise approximation which is nothing but gamma over this times a delta function of t 1 minus t 2 we have to do this slightly more complicated integral but that is not hard because it is the integration region is symmetric both of them run over the 0 to t range and so the region of integration is a square on which you have to integrate on the 45 degree line which is which corresponds to t 1 equal to t 2 so this means you simply replace t 2 by t 1 everywhere get rid of this and this becomes a t 1 which just gives you a factor 2 here this goes off and I pull out this integration is gone I pull out e to the minus 2 gamma t and I have e to the plus 2 gamma t 1 and that is e to the 2 gamma t 1 minus 1 divided by 2 gamma right so this goes away and I get a 2 m squared gamma and I get 1 minus e to the minus 2 gamma t so where does that get us it says v squared of t conditional average is gamma over 2 m squared gamma plus v naught squared minus gamma over 2 m squared gamma e to the minus 2 gamma t now we can do two things one of them is to let t go to infinity and see what happens so if you do that this is just gamma over 2 m squared just a constant of some kind on the other hand if I take this quantity and I do v squared of t that is equal to integral d v naught v squared of t bar p equilibrium of v naught I could also do that I could also average over all values of the initial velocity with the Maxwellian distribution which we assumed is valid at t equal to 0 what happens here if I do this this is just a constant so this part does not average to anything that is a constant does not average to anything what happens to this what is the average value of the initial square of the initial velocity over the Maxwellian distribution that is of course given by the equipartition theorem so half m v naught squared average is k t half k t so v naught squared average is k t over m so therefore this thing here would give you gamma over 2 m squared gamma plus k Boltzmann t over m minus gamma e to the minus 2 gamma t that is the result I would like this to be equal to this if the system is in thermal equilibrium I would like the two to be equal I am not using the equipartition theorem I am simply saying be very careful I am saying in equilibrium statistical mechanics the principle of equilibrium statistical mechanics tells me that when a system is in thermal equilibrium then for an ideal gas or a system of non-intracting particles of this kind the average value of kinetic energy in any component in the average value of half m times the square of the velocity any component of the velocity is half k t I am not I am not using any equipartition theorem I know that the distribution is Maxwellian that is an equilibrium distribution it is the canonical Gibbs distribution and I know that from equilibrium statistical mechanics so that is an input automatically now I am asking time dependent questions which are outside the purview of equilibrium statistical mechanics and therefore I need to have a specific microscopic model and the model that I have chosen is the large of our model but this has to be consistent with equilibrium statistical mechanics so I am not doing anything to it I am simply taking the system in thermal equilibrium focusing on a particular particle at t equal to 0 and asking time dependent questions of this without changing the thermal equilibrium situation and it is clear these two must be equal because the system has not been affected at all what is in thermal equilibrium should remain in thermal equilibrium this quantity should be independent of time and this should be equal to k t over m when is that going to happen if I equate this to this this has got to be 0 there is no other way the time dependence can be got rid of moreover the answer has to be k t over m and that magic is obtained simultaneously not only does this vanish if this is equal to that but this coefficient also becomes k t over m so you have complete consistency provided only that this be equal to that so what does that tell you finally this is possible it is consistent this large of our model is consistent with equilibrium statistical mechanics if and only if consistency if and only if gamma over 2 m squared little gamma is k t over m or if I take it across to the other side is equal to m in the large of our model so what have we achieved what is happened here finally this quantity was put in phenomenologically as a friction constant in the systematic part of the random force this quantity here was put in as a strength of the random part of the random force truly random part of the force the molecular collisions this measures the strength in some sense because it figures it is a coefficient of the delta function which is in the in the autocorrelation and it says the strength of the fluctuations as measured by this gamma the fluctuations in the random force due to molecular collisions must be related to the damping coefficient so you cannot have arbitrarily large fluctuations for a given damping coefficient and vice versa the stronger this is the stronger that is to to preserve thermal equilibrium and this is a relation it is the first example of a very deep relationship it is called the fluctuation dissipation or theorem it goes by many names in this version it is called a Nyquist relation the theory of thermal noise in a resistor due to Brownian motion of charge carriers this is precisely the sort of relation you get between the power spectrum of the voltage the random voltage and the resistance that you have here at any fixed temperature it is exactly the same relation once you put that in now that is input for consistency we are guaranteed now that the system is consistent with equilibrium statistical mechanics so once that is taken care of then we can actually go ahead and compute averages we can compute all kinds of averages now so this is the first crucial point you have to put this in as a condition on the Langevin equation having done that let us now see what p itself would be how would we get at what the probability distribution itself is so what is p of v t given a v 0 I will drop the 0 to show the instant of time and drop this and ask what is this guy equal to well I need an equation for it I need to make some assumptions it turns out that if I make the assumption that the velocity is given by the Langevin model the Langevin equation and eta is what is technically called a stationary delta correlated Gaussian Markov process for a Gaussian white noise then it turns out that this process V as described by the Langevin equation is a Markov process in itself it is not delta correlated it has a specific correlation time which will find out now but this conditional probability here satisfies a master equation characteristic of of characteristic of Markov processes in this case a very simple Fokker-Planck equation but we will write the solution down go back and look at the Fokker-Planck equation I am going to write the solution down by assuming that eta of t is a Gaussian white noise so eta of t a stationary Markov delta correlated with 0 mean that is the technical assumption so it is reasonable because it is reasonable to assume that the statistical properties of the molecular collision force due to molecular collisions does not change with time reasonable to assume that is delta correlated on the time scales we are looking at it is got 0 mean that is physically reasonable we are going to make the technical assumption that it is Gaussian namely its probability distribution function the density functions are all normal distribution they are all Gaussian distributions now that is a technical assumption but there are reasons to believe that that is also plausible assumption based on the fact that this eta is a resultant of a very large number of uncorrelated events so something called the central limit theorem comes to a raid and therefore this is not an implausible assumption it is probably is the most likely thing the fact that it is Markov that is an assumption we want to make an assumption that this is completely uncorrelated it is a very short time memory one step memory that is a technical assumption you could relax it in fact in real fluids you have to relax it later but that is the simplest assumption to start given this and given the Langevin equation which says v dot plus gamma v equal to 1 over m eta and eta has these statistical properties it turns out that v is also Markov it is also stationary it is not delta correlated it has a finite correlation time which in this case will turn out to be gamma inverse but it also turns out to be Gaussian so it will turn out that the Gaussian property remains it is very robust under this equation so the driving variable is Gaussian distributed so is the driven way if you grant me that then I could actually write down this thing here this conditional probability density function because you know that the Gaussian process a Gaussian random variable is determined its distribution is determined by two parameters the mean value and the variance those are the only two things you need so let us compute those quantities here we can actually write down what these quantities are because to start with I am going to require the following on this p I am going to write it down by hindsight so at t equal to 0 what is this density function it says this is the probability that the velocity is some v at time t given that it was v 0 at t equal to 0 so what is the initial condition on this guy what does that imply for this it is got to be 0 unless v equal to v 0 it is a Dirac delta function it is got to be normalized to unity so it is a Dirac delta function and what would you expect this to become as t tends to infinity no no no the probability density function the Maxwellian nothing is happening so I would expect this to become p equilibrium of v must check out that my solution in fact does so and then if you have knowledge of what the mean value is and what the variance says then we can actually solve the problem completely but we do have that knowledge because I know that v of t is v 0 t to the minus gamma t and I also know that v squared of t is equal to that thing there this quantity here with for consistency gamma over 2 m squared gamma replaced by kt over m so this is kb t over m plus v 0 squared minus kb t over m e to the minus 2 what is the variance of this process then therefore v squared of t minus v of t whole squared this is equal to the variance of v of t what is that equal to all I have to do is to subtract this out here and then you begin to see immediately that it is equal to kb t over m into 1 minus e to the minus 2 because this term cancels against this now if I make the assumption that it is Gaussian then we can actually write the solution down the normalize solution down what is the Gaussian what is the Gaussian probability density function look like remember that if the mean is mu and the variance is sigma and then the standard deviation is sigma the variance is sigma squared then the distribution density function is 2 pi sigma squared e to the minus the variable psi minus the mean whole squared divided by 2 sigma squared that is the Gaussian normalized Gaussian probability density function so we copy that out and therefore we can assert now that p of v t given v 0 must be equal to 1 over root 2 pi sigma squared but sigma squared is this so this is m over 2 pi kb t into 1 minus e to the minus 2 to the power of half times the exponential of minus psi minus mu whole squared so that becomes v minus v 0 e to the minus gamma t whole squared that is the mean value which is a function of time is an m there divided by 2 kb t now of course it is tricky to show that at t equal to 0 as t goes to 0 you end up with the delta function but it does so because as you can see as t goes to 0 drop this this is v minus v 0 whole squared but this guy goes to 0 out here and this fellow also so you have a 0 from this whole exponential and then you have something which blows up from here and the result is a delta function in the limit although it is not immediately obvious as t goes to infinity indeed it goes to the Maxwellian distribution because this part goes away you are left with this here and this part goes away that goes away and indeed it goes to this so this is certainly and what will this distribution look like if I draw picture it would do the following so as a function of v if I plot p of v t for a given v 0 at t equal to 0 it is a delta function at v 0 it is a spike and then as t increases this broadens out the average value this thing is a symmetric distribution about its mean which is at v 0 t to the minus gamma t that slowly drifts so you have something it looks like this and then slowly drifts and eventually at t equal to infinity it becomes the Maxwellian and that is a symmetric distribution so the peak broadens and moves to the left drifts to the left and gradually hits this and the variance changes with time like this so variance starts at 0 which is delta function and gradually reaches the value kt over m this distribution function this is called the on steam on steam olembeck we will take it from this point tomorrow what what happens next but what we want to emphasize here is that this stochastic differential equation that I wrote down the Langevin equation is equivalent to a Markov process it describes a Markov process whose probability density function conditional density function is the on steam olembeck distribution we will generalize that lesson to other stochastic equations but this is in the physical context of a particle diffusing we will make further consequence we will look at further consequences of this meanwhile I would like you to check out the following and that is not hard to do it is the same integral that we did earlier but now check out the fact that v of t v of t prime in equilibrium you could ask what this quantity is we looked at v squared of t in equilibrium we took the same v of t and squared it but now you should take two different instance of time of this kind and ask what happens now you could do this by finding first the average value of v of t v of t prime and then averaging over initial conditions or by letting t go to infinity t prime go to infinity such that t-t prime is finite either way to do this and the result at urge you to do it both ways write the integral down and then you have now an integration over t1 and t2 which would be unsymmetrical so if this is t1 and that's t2 perhaps t1 goes up to t but t2 goes up to t prime and your delta function constraint due to the white noise would be along this line so what would contribute you can replace get rid of one of the integrations but the other integral is constrained to be up to here and not up to here so you can easily see that the smaller of t and t prime is going to be it is going to be the region of integration for the second integral whichever is left the lesser of the two so it turns out to work this out this will be kt over m that's the value at equal to 0 times e to the power minus well you can almost guess what's going to happen this is an equilibrium correlation function for a stationary random process so it must be a symmetric function of the time difference and there's only one constant of time here in this problem absolutely so it's just e to the minus gamma modulus t minus so check that out so the on-steen rule and by process is exponentially correlated it's not delta correlated so by putting in the Lanjama model what you've done is started with a white noise which has zero correlation time but the driven or output variable has a finite correlation time it's built into the system so it says even though your noise may be have zero correlation it's really instantaneous correlations the output variable could slow down could actually have a finite memory and it does in this case so it acts like a paradigm for random processes of this kind and we take it from this point here and I like to for example look at the ordinary diffusion equation in the same language and ask what happens there we'll do that next we look at the displacement we haven't done that at all we know that when these particles move around they diffuse so I'd like to look at the position variable and see what happens to the position here from the same model and ask whether we can draw similar conclusions for this and connect it up with the diffusion equation and then more general stochastic equations this is what I'll do tomorrow so let's stop here