Langevin and Fokker Planck equations

welcome to this video on statistical mechanics I am your Stuyvesant and I teach statistical mechanics in the master program Applied Physics at the Delft University of Technology in this movie I want to address the launch vehicle which describes heavy particles that are surrounded by large light particles so it could be smoke particles in a gas or solute particles in a solvent and these heavy particles they experience collisions from the light particles and the effect of these collisions we want to capture that in a force which is the frictional force and so-called random force which is also called the large event force and the resulting equation of motion is called the launch of an equation of motion we make several assumptions for the random force for the knowledge of our force and from this we can calculate many properties and the most important one is that we can derive the evolution equation for the velocity distribution in such a system so that means that we look at the distribution of the velocities of the heavy particles and we can see how that evolves in time and as we will see it will evolve towards the Maxwell equation which is not really surprising but the equation which describes this evolution it's the fokker-planck equation also describes how a non-equilibrium system evolves towards the equilibrium Maxwell distribution I hope you enjoy this movie here you see a molecular dynamics simulation of a lennard-jones gas those are atoms which interact via learn the Jones potential attractive well and repulsive core and the particles are atoms they are rendered by blue dots but there is one particle which has been singled out that's this blue particle here and the blue particle is just one of the atoms it's an arbitrary atom and by giving it a different color it's easy to see how its moves through the gas and as you can see it's kind of random walk motion the particle receives kicks from the other particles and that moves the particle the blue particle around and we shall see that although the trace the motion for each individual particle is completely impossible to predict it's possible to make definite statements about the motion in general that the particle makes so when we can make statistical statements about that motion and that is the subject of this movie in fact we focus on the special case where we have a large particle that's this disc here which is moving in a dense gas or fluid consisting of small particles and they bounce off this large particle and that gives rise to the famous Brownian motion and its motion is determined by the light particles which collide many times with the heavy ones so the disk is moving slowly and the other particles are moving a lot faster because equally partition tells us that the kinetic energy is equally distributed over the particles but I have your particle then has a much smaller speed the effect of the collisions by the light particles with the heavy particle is twofold first of all the heavy particle will feel a drag you can imagine that if the heavy particle moves at sub speed through the liquid it will experience many more head-on collisions then it will experience kicks in the back and therefore its speed will slow down until it comes approximately to arrest so in the first force that the heavy particle feels is a drag force on top of the drag it will feel random kicks in all possible directions and so you have random kicks on top of a drag so first of all you have to drag and then there are random kicks that even if the particle has come to arrest there is no drag anymore but there will be random kicks in all kinds of directions we want to capture the effect of both the drag and the random kicks in an equation of motion the drag force is easy to incorporate because we know it's form it's minus gamma times V that's the usual form for the drag force then the random kicks we represent them just by a force R which is a vector and it must have some random character and that represents the random kicks and eventually there is a feel like refa T or if the particles are charged it may be an electric field which is a systemic force and we can include that and we call it F and depending on position and time so the F is a systemic force the drag force and the systemic force are rather familiar but the same cannot be said of the random force however although the random force itself is by nature unpredictable we can make some statistical statements about it so we can make some statements about the statistics of the random force the first property of the random force is that it's expectation value is equal to 0 for all times so imagine the fluid or the gas being at rest which means that it's centre of motion is at rest then obviously if I put a particle in the inequity article initially at rest it will get random kicks from all directions and these will average out that's an average in time so what does the average here with the angular brackets mean because this is evaluated at one particular time T well the average expressed by these angular brackets is an average over all the possible realizations of the random force so if we start a simulation or an experiment we get a certain random force then we do it again and again and again so we have a long series of random forces at subsequent times and if we average is average over that series then it turns out that this average is that it's random force is zero so the angular brackets is represents an average over all the possible realizations of the random force the second property of the random force is formulated as a statement about the correlations so imagine we take the random force at some time T some arbitrary time and then we take the random force at some time tau later than T so T plus tau in that case the value the expectation value averages out to zero whenever the Tau is greater than zero so when when we look really at a later time the random cakes have nothing to do with the random cakes that you saw before now in a liquid we know that that is not true because in a liquid you have hydrodynamic correlations so this is an approximation and in fact that approximation is valid for heavy particles because they will experience many collisions before they change their position appreciably so if the time scale at which the heavy particle moves is a lot slower than the correlation time and the fluid this is a good approximation however if I take just one atom feeling random kicks by the other atoms it turns out that this assumption is not very valid so why do we make that assumption it's for mathematical convenience because it enables us to infer all kinds of properties without too much mathematical effort this is therefore an approximation that is made very frequently so it says that the correlations between subsequent random forces vary of course if tau is zero then I just have the expectation value of the random force squared the final property of the random force is that it's subject to a Gaussian distribution so each force is drawn from a Gaussian probability distribution P R represents the probability density for finding a random force of size R this should in fact be a vector but you can read this as one component of our so each component of R is subject to this probability distribution which means that if we run for example a computer simulation we draw a random force at each time from a Gaussian random distribution and then we feed that into the equation of motion and we do that for each component so that's why our here is not affected but just a number when discussing random forces it's any how convenient to think in terms of computer program and in a computer program what we do in order to solve the equation of motion is to discretised a time so if time steps t1 t2 etc they are equidistance which means that the distance between time between two subsequent time steps is always delta T in the simulation we generate a random force at for each time step and the properties that we have formulated before translate into the following a recipe the probability density for having a random force r1 at time T 1 r2 at time T 2 etc is given by this expression which is a Gaussian expression and the fact that the forces are uncorrelated at subsequent time steps just causes this probability to be the product of the individual Gaussian probabilities we can then return to the continuum notation we have a sum over us at subsequent time steps we can write formulate that as an integral if we divide by an extra factor of delta T and so we have now formulated the probability distribution as a continuum distribution and we use a shorthand notation to queue for the denominator here and so this is our continuum prescription for the probability density of having a force between T 1 and T n if we then recall our prescription which was that the correlation function of the random force led to a delta function we can write in the for our n RM and this is now one-dimensional that it's equal to R squared Delta M and M that's now a discrete Delta function in the discrete time it is interesting also to consider the correlation property of the random force we can formulate that also in discrete time note that I have taken here a scalar R so this is one component of a vector random force and in discrete time this will give me an R and RM random force at times T N and T M and that is equal to the average value squared times well this Delta function in discrete time turns into a delta and M divided by delta T because the integral over the Delta function is 1 we need 1 over delta T here and due to this property here we see that the width of R squared is always this Q so we have a Q over delta T times Delta and M so this R squared is just Q based on the properties of Gaussian integrals and the second moment next we turn to the solution to the launch of our equation and in general that solution is difficult to find but in the case where the systemic force is equal to 0 we can make some statements about the solution so that's the case where we will be now and the launch very equation then reads as follows we have M V dot is minus gamma V that's the drag force plus R which depends on T and which is a random force this equation is called the launch of our equation the launch of our equation is a differential equation in V and it's a homogeneous equation for one dimension and therefore we can first try to solve the homogeneous solution so we take R equal to zero and then we are left with this equation and that gives us the homogeneous solution the wiggle is V wiggle 0 times e to the power minus gamma T over M so this is the homogeneous solution then we need to find a particular solution and we do that using the standard method we try a solution of the form VT is the homogeneous solution times some function that depends on time and if we plug that into the differential equation we get the following first of all we have two time dependencies one here and one there so we get two terms from the M times V dot those are those two terms and on the right hand side we have here the drag force and here we have two random force and the drag force is cancelled on the left and right hand side so here you see that this term is identical to that term and so we are left with a simple equation for F this equation can be directly written as f dot is some function on the right hand side which depends on T and this can easily be solved when I say solved it means writing this solution formally as an integral over a function involving the random force and because we don't know the random force there is no way we can integrate in this function analytically putting this F into the form for V that we have written here we obtain the following solution we have here the particular solution that is the one we just found and this is the Imagineers solution which we can add with an arbitrary p f-- pre-factor and because we take the integral from 0 this pre-factor should be the velocity at time T is 0 as mentioned before the second term is not amenable to analytic solution but we can definitely infer some statistical properties of the solution for V using the statistical properties for R we can for example calculate the expectation value of this V we put angular brackets which means is that we average over all the possible realizations of the random force and because the average value of this random force equals 0 for all times the solution will just be the first part because the second part vanishes so average we can find what the V in the course of time is obviously that's the average behavior and it's interesting to see how he would vary around this average solution in order to find that out we calculate the V squared of T so that we can later subtract the V expectation value squared that's this expression so the difference is this term and now we have here two hours because we have two integrals we have first an integral for the first fee and then for the second one we call the integrant the integration variables t1 and t2 and therefore we have an expectation value of our t1 and our t2 and we know from the statistical properties of our that we can write this as Q times Delta t1 minus t2 t2 and that simplifies the integral a lot working out the integral we obtain the following result so here we have a q which now enters in front of a single integral because we had a delta function the delta function tells us that t1 is equal to t2 so we have now two times gamma and now I've called the integration variable T primed we can carry out this integral it's a very simple integral and the result is that we have V squared v-0 square times minus 2 gamma t over m plus Q over 2 gamma M times 1 minus e to the power minus 2 gamma t over m so what do we have we have an average behavior and that average behavior is typically the solution to a drag equation exponentially damped velocity and we have also an expression for the variation so if we subtract this term from the left and right hand side this turns out to be the fluctuation the average value of V with respect to its average velocity and we see that for long times this term decay and we are left with just Q over to gamma M so for long times in the first term of course decays to zero also this term decays to zero and we are left with a constant for the expectation value of V squared so here I've written down that very same statement it's Q divided by 2 gamma M that's the expectation value of V squared but if the particle is in equilibrium in a liquid or in a gas we know from the equipartition theorem that the velocity squared expectation value should be KBT over 2 over m and therefore we find an expression for Q in terms of the temperature and the friction coefficient so Q is 2 gamma KBT we have seen that the launch of an equation is an equation of motion in which there is a frictional force and there is a random force we will study that equation from now on just for simplicity with M equal to 1 and what we want to do now is not to study individual trajectories but we want to see what the probability is to find a particle with a particular velocity V at a certain time T so we search an equation for this probability density which is time and velocity dependent so suppose we know where the particle is at t what its velocity is we can find the velocity at the next time step we discretize the time now with a time step delta-t and we find that a new velocity is the same as the old velocity plus in the force which is minus gamma v plus r delta t so let's make a cartoon of this equation we have the old velocity which is VT and this first minus gamma V times delta T which is here shown in green that reduces that velocity so all the velocities are reduced by this term and then we consider the next term that has determined with the random force off so we show the distribution of the random force which is Gaussian in blue and here we show our delta T so this is the probability distribution for the second step our delta T we want to find an equation for P at the next time step so the distribution for the velocities in the next time step provided we know the velocity distribution in the old time step and therefore we have to integrate over all the possible the previous velocities and we call these now V old so we integrate over DV old and in addition we integrate over all the possible steps D RT our delta T like this the probability that we were had a V old is given by P V old at time T at a previous time step the probability that we make a jump of our delta T we call that curly P so it is my curly P that's a Gaussian distribution so please note that I have a right and a simple be like this and I have curly P the curling P is a Gaussian distribution and we don't know what the P of V is but of course we cannot have just any field and any displacement fidelity because we want to restrict ourselves did those combinations which give us the Nuvi here and we realize that constraint using a delta function so here I have inserted a delta function which guarantees that this equation holds so this is the V in this equation so in this equation here we call this V this is the V old and this is also the V old and so if I want to satisfy this equation then I can realize that using this Delta function and because we have a delta function it's now easy to work out the integral over the V old and we see that the Delta function does not have field in its argument but effector times V old and then we need to use this rule for working out the Delta function and we obtain then the following result so here is the result first we have a term 1 over 1 minus gamma delta T which comes from working out is Delta function with the 1 minus gamma delta T in front of the V old and this field here has been replaced by the remaining arguments in the Delta function so that's V plus gamma V minus R delta T this is the new V and the old fee is related to the new V in this way and I still have the probability distribution the Gaussian probability distribution here and I'm left with only one integral because the Delta function has been used to do away with the integral over the V old now looking at this equation we immediately see that it's useful because delta T is taken to be small to do a Taylor expansion for the P so that is the next step and we do that Taylor expansion to second order in the delta T so here is to his this is the zeroth order term which I find by taking delta T equal to zero then here is the first order term and here the second order term all the peas and its derivatives are all evaluated at V and T except that I have not mentioned those I've not reproduced those arguments explicitly in the last term we are going to work out all the terms to order delta T so let's first look at the first term which is x PR delta T and then we integrator for the our delta T but because we make always one step our delta T this probability density is normalized so that means that if we integrate over Delta R over R delta T like we do here we always get a result 1 and because there's no art of the T dependence in this term of the integral the first term we obtain the zeroth order term is just PE V T because the integral of the remainder just gives us 0 it gives us 1 and then if we want to work out everything up to order delta T we see that we can multiply this by 1 plus gamma delta T the next two terms each give us contributions which themselves are at least of order delta T so this delta T correction will give at least an order delta T squared and because we work consistently up to order delta T this term will not have any effect on the next two term terms not to order delta T now we consider the second term and first we have a gamma V times delta T which I've copied here multiplied by DP e DV and then we have again the same integral over normalized probability distribution so that gives me just the one this term here with the are the probability distribution here is Gaussian and it's symmetric so if we multiply there are we get an anti symmetric integral so this after integration will lead to zero so this is the complete result that we obtained from the second or the second term which is the term to first order and delta T in the Taylor expansion so let us continue with the second-order term from which we first obtain a contribution gamma squared V squared delta T squared over two and then again we have an integral of the second derivative of P multiplied by a normalized Gaussian so that just gives me a 1 the term which is linear and R vanishes for the same reason that this one vanishes we have an anti-symmetric integrand and then we have an extra term and that term is equal to delta T squared over 2 times d squared P DV squared and then we have an integral of our delta T squared so in the fact that I've put a delta T squared here means that I can leave it at I leave it out here then we have a P R delta T and we integrate over that our delta T so this is the second moment of my Gaussian distribution now we have seen above that the fact that this in fact it can be written as the expectation value of R squared times delta T squared and we had seen that that was Q over delta T and the Q had the value two gamma K BT and therefore we see that the second term acquires the following form it is gamma times KB T times delta T times the second derivative of B with respect to V squared now there is one final step we need to perform we have the difference here of P V taken at t plus delta T and P V at T if we take only the first one and we bring it to the left hand side and that's obviously the first derivative of P and so we arrive at the following equation an equation which gives us a time derivative of this probability distribution and the depends on the first derivative here and the second derivative of P in fact this term of course when I work it out it generates two terms one is two term here with the gamma delta T that's the term when I take the derivative with respect to V and when I take the derivative with respect to V P I get the second term here this equation is a famous equation and it's known as the fokker-planck equation in order to check whether the fokker-planck equation makes sense we look at this stationary solution which we find by taking the time derivative of P equal to zero in that case the right hand side of this equation should be zero and it is solved by the following expression PV is C times minus V squared over 2 KBT if we had included the M in our derivation we would have also have had also an M in this equation and then it would be minus MV squared over 2 KBT so what we have obtained is the in fact the Maxwell distribution it's the Boltzmann distribution for the speed and we have obtained that for the special case M is equal to 1 for the case where m is not equal to 1 we find the following for Co Planck equation which includes an M in the first term in the denominator and an M Squared and the second term so finally let me summarize what we have seen we have tried to formulate an equation of motion for heavy particles that feel an influence of many light particles which collide with the heavy ones there will be an average effect to drag down the particles and that's captured by this frictional force of the heavy particle - gamma V and on top of the frictional force there are random kicks and they are represented by this random force R and the random force will be different each time we start a new experiment obviously it's a random force but it has some statistical properties and we can express those in terms of expectation values and the expectation value is always interpreted as an average over all the possible realizations of the random force so the first thing we can say is that the random force the average value of the random force many realizations is zero which tells you that there is no bias in the force in the random force any bias is present in the frictional force secondly there are no time correlations so the heavy particles move quite slowly and before they significantly change their direction or their position there are many collisions and therefore these collisions averaged over some time interval are not correlated and we expressed that by this delta function for the autocorrelation function of the random force and finally the distribution of the random force is assumed to be Gaussian so the P of R is given as e to the power of minus R squared divided by twice and the average value of R squared we can make a surface that further if we really solve this equation of motion that's possible and the solution is given here V T is V 0 times an exponential decay with time with time constant M over gamma and then there is a second term which involves the random force if we don't take the average of all the realizations of that force it vanishes due to this assumption and we are left with a velocity that a case to zero so this one this solution only sees the effect of the friction on the other hand if we take V squared then we see first of all the effect of the friction but then we can use the fact that we have a result for this R squared and that gives rise to the second term here and that's the effect of the fluctuations that's the noise that we see on top of this decay if we wait infinitely long then all the decay has gone all the turn decaying terms vanish and what we are left with is then Q over to gamma m and we know that we can relate V squared it's related to the average kinetic energy which is KBT per degree of freedom and we find this relation between gamma KBT and Q which is the amplitude of the force recall that it occurs here in this Delta function we can also find the distribution of all the velocities in a system that's called PV of T which is the probability density for finding a particular velocity V at time T in fact we have derived a time evaluation equation for this probability density and that's written here DP DT is equal to gamma D DV of V times P plus gamma KBT times the second derivative of P with respect to V and this equation is called a Fokker Planck equation if we look for the stationary solution we put left hand side to zero and it turns out that the solution is then given by the Maxwell distribution and therefore this equation describes how in a system a non equilibrium distribution of the velocities evolves towards the Maxwell distribution

Channel: Jos Thijssen

Views: 14,240

Rating: 4.9836736 out of 5

Keywords: statistical mechanics, fluctuations, langevin equation, fokker planck equation

Id: H9I0PmXwhdo

Length: 36min 26sec (2186 seconds)

Published: Wed Nov 23 2016