So, today what we will do is to look at this in a more general context and write on multidimensional Fokker-Planck equations and apply it to the case of particles moving in phase space itself. So to recall to you what the equations were, we had an equation that said x dot minus v was equal to 0 that just says the velocity is the derivative of the position and then v dot plus gamma v this is the friction part the systematic part of the random force on it. This was equal to square root of gamma over m eta of t where this was Gaussian white noise. In other words it is a delta correlated Gaussian process stationary Markov process which is a delta correlated with unit strength. So, eta of t equal to 0 and eta of t, eta of t prime equal to delta function of t minus t prime. Now the question is what is the corresponding Fokker-Planck equation satisfied by the probability conditional probability density in both x and v jointly. So, we are asking for the equation satisfied by p of x v t given that it started from x naught and v naught at time 0. It is important to remember that earlier we had equations only in v and of course the equation satisfied by v was the Fokker-Planck by v was the Langevin equation. We got a Fokker-Planck equation for it and the solution turned out to be the Onstein-Ulenbeck distribution and we discovered also that the velocity process was exponentially correlated and that it was a stationary random process. So, that much we have already seen. Now the question is what does this look like? What equation is satisfied by this quantity here corresponding to the set of equations here. Now it turns out that if you have a general process, a multidimensional process of this kind, if you put them together in some vector form and you call that a vector xi say xi dot plus if you got a linear drift or xi, this is some constant matrix here acting on this column vector here. If this is equal to on the right hand side you have the usual noise of some kind. So, we have gamma over m and then it has a vector of 8 of t and this stands this quantity xi stands for x and e and this 8 of t stands for 0 8 of t. There is no noise here in this equation but there is a noise driving noise on this side due to the collisions with the other particles in the fluid. If you have a Langevin equation a stochastic differential equation of this kind corresponding to it this quantity xi its conditional probability density satisfies a certain Fokker-Planck equation and that Fokker-Planck equation is given by the following it is given by delta over delta t and let me call it rho. Let me call this rho so that you know that it is a phase space density this is equal to on the right hand side you have r i j delta over delta xi i xi j rho that is the drift term the first term plus d i j d to rho over delta xi i delta xi j where this matrix d i j is in this case a 2 by 2 matrix which is 0 0 0 and then whatever was the matrix which came out from one half the square of this guy here but remember that consistency required that gamma was equal to 2 m gamma k Boltzmann t that was the fluctuation dissipation theorem this is what kept the velocity in equilibrium in the Maxwellian was retained in time then one half the square of this guy the 2 goes away there is an m on top comes and that m cancels out here with this m squared and give you a factor m in the denominator so you are going to get gamma k Boltzmann t over m this is what this guy is and that is the Fokker-Planck equation okay all we have to do and there is a summation over repeated indices i and j run over 1 and 2 okay now in this problem what is r well from these two equations it follows that r in our problem here is equal to 0 minus 1 0 and gamma so r 1 2 is minus 1 r 2 1 is 0 there is no x here and then r 2 2 is gamma and that is it so that is the Fokker-Planck equation all we have to do is to write it out explicitly and you immediately see that delta rho this thing here is equal to in this particular problem r 1 2 is a minus sign so this is minus delta over delta i is equal to 1 so there is an delta over delta psi i r 1 2 is what we need so this is equal to 1 that gives me a minus delta over delta x and this is 2 so that is a v times rho out here and that is it there is one more term which is r 2 2 which is delta over delta v gamma v rho that is the 2 2 part plus the only term that survives in this term here is d 2 2 that is the only nonzero element here and then it becomes delta v 2 so plus gamma k Boltzmann p over m d 2 rho over delta v 2 now of course v is independent of x because it is a dynamical variable which is independent of x so this is equal to minus v delta over delta x rho plus gamma times delta over delta v v rho plus gamma k Boltzmann t over m this is the Fokker-Planck equation satisfied by rho it is a fairly complicated partial differential equation this portion was already there in the Fokker-Planck equation satisfied by the conditional density of the velocity itself so was this but there is an extra term here out here and this is like a convective derivative because it is v delta over delta it is like v dot del three dimensions it is a one dimensional analog of that so if you bring it to the left hand side you have d by dt d over dt of rho the full derivative equal to whatever is on the right hand side with this linear drift and this extra diffusive term here now the question is what is the solution well depends on the initial condition and we are talking about this conditional density so at t equal to 0 this satisfies rho of x v 0 at x naught v naught is delta of x minus x naught delta of v minus v naught so with that initial condition if you solve that partial differential equation with the boundary condition that these things vanish as x and v tend to infinity you get Gaussian's as a solution finally we are not going to do that here but you end up with some complicated Gaussian solution a bivariate Gaussian both in x and v so it will have terms like e to the minus x squared minus e to the minus v squared and e to the minus plus x v there be cross terms the interesting question to ask is what happens to this as t tends to infinity in this problem t tending to infinity means t much much greater than the velocity correlation time gamma inverse so it turns out that rho of x v t given x naught v naught tends as t becomes much much bigger than the correlation time it tends to the Maxwellian in v and the Maxwellian in v is of course m over 2 pi k Boltzmann t to the power half e to the minus v squared over 2 k Boltzmann t so the velocity thermalizes and then you have the positional part and that of course is 1 over 4 pi d t to the power half e to the minus x minus x naught whole squared over 4 d t you recognize this to be the limit of the Antstein-Ohlenberg distribution as t tends to infinity that is just the equilibrium Maxwellian distribution and that is the solution to the ordinary diffusion equation the question is what is this capital D and we saw by comparison of the Langevin equation with the diffusion equation in the long time limit we also saw that the other the other consistency condition was d is k t over m gamma so is there an equilibrium distribution no because as t tends to infinity this vanishes out here so it says that there is no stationary distribution for ordinary diffusion at all everything goes to 0 the whole Gaussian goes down flattens out to 0 the velocity however thermalizes the velocity is a stationary random process it thermalizes one could ask alright this is the phase space density what happens if I integrate over one of the variables I should get the distribution in the other variable right so if you read this if you took the exact solution if you take integral over dx rho of x v t x naught v naught over the full space what should this give you what would you expect it to give it should give you well it should be still a function of time because all you have done is to take the joint distribution in x and v and you were integrated over the position so it should give you the conditional density in the velocity alone and what is that distribution it is not the Maxwellian that is the limit distribution the on-strain ohlinbeck distribution so it will end up with exactly the on-strain ohlinbeck distribution so this thing here will become m over 2 pi k Boltzmann t 1 e to the minus 2 gamma t that is the variance to the power half exponential of minus v minus the mean value which is e to the minus gamma t remember this is conditional on this initial condition always so it is this squared divide m times that over 2 k Boltzmann t 1 minus e to the 2 gamma t this is the on-strain ohlinbeck distribution so indeed it will turn out after you integrate over the position you end up with this okay what would you expect if you integrate over the velocity instead what do you think you will get if you did minus infinity to infinity dv rho of x v t x naught what should you get what would you expect would it be this would it be this what does your intuition tell you would it be that distribution I got rid of the velocity altogether so do you think it would be that distribution that is only valid in the diffusion limit that is only valid for t much much greater than gamma inverse we are not saying anything like that here we just integrating over v so what would you expect it is the positional distribution no doubt about it it is the positional conditional distribution density in the position it will be a Gaussian because you have got a bivariate Gaussian you are integrating over one variable you are going to get a Gaussian in the other variable no doubt but much more complicated than that simple diffusion equation solution it is some Gaussian you need to know the exact solution of this equation in order to be able to do that I am not going to write it down here I will give it to you in writing elsewhere it is a complicated formula but that distribution will tend as gamma t tends to infinity will tend to that diffusion equation solution okay and interestingly enough there is no simple formula here whatever distribution you get here that does not obey any simple master equation okay so although the phase space density itself obeys this master equation the Fokker-Planck equation and the velocity process itself obeys a Fokker-Planck equation the position does not obey such an equation simple equation at all except in the diffusion limit and you go to this long distance and in fact several ways of seeing this that the position is not stationary random process we will talk about the Wiener process very shortly and then you will see how complicated the position actually can become okay and what Brownian motion means and so on and so forth okay so much for this that this distribution has an asymptotic limit which does this and when you integrate it over the position you end up with the Onstein-Ohlenbeck distribution now we can generalize this a little bit go a little further we can ask what happens if I had an external force on the problem what would then be the effect of an external force one way to approach this is to ask is to look at a slightly different model instead of a free particle let us look at the problem of an oscillator a simple harmonic oscillator now what we are talking about is a particle on the x axis here is x equal to 0 and this is a simple harmonic oscillator in the absence of any external force but we now imagine it is in a fluid with viscosity and this particle is being hit randomly by the other particles in the fluid okay and we want a Langevan type model for its motion here so there is a white noise exactly as in this case but there is also a systematic part of the force which corresponds to putting in the harmonically bound particle namely this thing is in a potential now this potential is going to do something interesting the actual harmonic oscillator potential is half m omega naught squared x squared as you know now the first effect of this potential is that the translation symmetry on this x axis is lost because this point becomes a special point is a potential here the other thing is your it is as if this particle is connected by a spring harmonic spring to the origin and therefore there is a strong restoring force if the displacement is very far away from the origin so you might expect okay there is going to be two effects here one of them is the random force due to the other particles which is tending to diffuse move this particle away and make its mean square displacement diverge as a function of time the other effect is this restoring force which is always acting the question is what is going to happen to this particle will it diffuse or will it not will the variance of the displacement diverge as t tends to infinity if it does so linearly you know that it is diffusive if it goes to finite constant then you know it is not no long range diffusion at all so the question is what is going to happen so that is the physical question we can answer it in the following way I can write this down by adding to this the restoring force and that of course is minus m omega squared x and I have divided out by m so there is also a term v dot plus gamma v plus omega naught squared x let me call it omega naught the unperturbed frequency of this oscillator so this is the case of a harmonically bound and that is the equation everything else is unchanged exactly as it is now of course the velocity once again thermalizes and there is an equilibrium velocity distribution which is again the Maxwellian distribution before but there is also this potential energy term coming from the potential in which the particle is okay so now what happens exactly as before I can write down the same problem I have rho of x v t and we could start it from zero for example just to make sure that it is a simplest condition but whatever it is given some x naught given some v naught at zero the question is what does this particle do what is the density satisfy again the same Fokker Planck equation as before same as this exactly but whereas in the previous case this fellow was equal to zero r was equal to zero minus one zero gamma what is it now remember there is an extra term here omega naught squared and this is appearing with a coefficient x in the equation for v so it means there is a term here which is omega naught squared that is the only difference between the two everything else goes through exactly as before so there is going to be here another term which is equal to plus there is a term which is equal to delta over delta and what would this be it is in the v equation okay so r 2 1 is what contributes so this is delta over delta v omega naught squared x rho the term of that kind because it is going to contribute to a delta over delta v because it is r 2 and this is a 1 so there is an x inside omega naught squared x that is the only change and we have an exact answer for the Fokker Planck equation satisfied by a harmonically bound particle in phase space and of course omega naught squared is constant so let us write that out so let us write this as equal to minus v delta rho over delta x plus omega naught squared times x times delta over delta v delta rho over delta because this time x is independent of v so that comes out just as out here v was independent of x so it came out and that is the Fokker Planck equation okay now the solution to this is very different from the solution in the absence of omega naught altogether what do you think will happen physically what do you think will happen think back to equilibrium statistical mechanics if you have a free particle free gas the position is irrelevant in the position is equally spaced distributed entirely in the container then it is only the velocity that is relevant okay so gas and thermal equilibrium classical gas you only talk about velocity distribution at a given temperature the position is taken to be uniform throughout but now there is a harmonically bound particle so it is clear that the energy of this particle there is a contribution which is half m omega squared x squared to the potential energy of this particle and we know that the relative probability of any value of the energies e to the minus epsilon over kt right so you would expect to get in equilibrium the distribution would be biased towards x equal to 0 obviously most probable value of x would be 0 in this case and what is the actual distribution it is the Boltzmann distribution right so what is the actual equilibrium distribution row equilibrium of x and v or the stationary distribution from equilibrium statistical mechanics what would this be it is e to the minus the energy over kt and that is it normalized appropriately normalized so this guy would obviously be e to the minus mv squared over 2k Boltzmann t that is the kinetic energy part minus m omega naught squared x squared over 2k Boltzmann t that is the potential energy part right times normalization factors e to the minus ax squared where a is a positive constant if you integrate you end up with square root of pi over a so it is square root of a over pi is a normalization right this thing here equal to square root of m over 2 pi k Boltzmann t square root of m omega naught squared over 2k Boltzmann t this is the normalized distribution such that if you integrate over x and b you are going to get unity so the distribution factors into something in x Gaussian in x and a Gaussian in v velocity you expect this on physical grounds this is going to happen and that is it so this exact solution will reflect that will reflect precisely this now tell me does the variance of the mean displacement diverges a function of time do you think or no it cannot it cannot because the limit value limiting distribution in position is a Gaussian and Gaussian has finite variance completely so there is no long range diffusion the behavior is not diffusive standing to a Gaussian it is going to do exactly what the velocity does except for change of constant here and so on notice dimensionally everything is okay because the Maxwellian distribution in velocity must have physical dimensions 1 over the velocity so that p of v dv is equal to 1 when you integrate right so there is an m here and there is an m k t which is m l square t to the minus 2 and the m cancels and you got a l square t to the minus 2 square root which is l t inverse so this whole thing the velocity is in fact 1 over velocity 1 over velocity in physical dimensions and similarly this is going to be 1 over length because the t to the minus 2 cancels here okay so this is perfectly correct dimensionally and you end up with distributions which are stationary distributions in this case so the mean square displacement will not diverge what will it be actually the mean square displacement what do you think this will be what is what do you think v square will be in equilibrium what should this be k t k t over m yeah because half m v square will have half k t so this is k t this is k b p over m x squared in equilibrium what will this be no long range diffusion and that is reflected by the solution to this guy again this is a bivariate Gaussian this fellow here is a bivariate Gaussian which will now tend in as t tends to infinity it will tend to this equilibrium distribution here but there is an interesting wrinkle here what do you think is the relaxation time for the position and the velocity you can tell what this is by looking at these equations here and in a minute you will see what the difference is in a minute you see what happens is that the velocity thermalizes over a time scale gamma inverse that is the correlation time of the velocity but the position does not thermalize with that well for instance if it is an under damped oscillator it is clear that it is going to reach equilibrium values in an oscillatory fashion always if it is over damped it is going to go monotonically and so on right so the omega not and gamma together will provide two time scales in this problem and what is the criterion for over damping of an oscillator given given this current a natural frequency omega not and a damping constant gamma both of which have time scale dimensions of frequency so you know that in this general expression you end up with gamma over 2 whether that is bigger than omega not or smaller than omega not etc this is over damped and of course less than omega not is under damped we do not care what it is this equation is valid in general whatever the value of omega not and gamma b this equation is true in general and there is a complicated solution in bivariate Gaussian solution but the relaxation times of the two are different from each other we will see in a minute show you what the relaxation effective relaxation time is for the case of the position because you can also ask in this problem in which I am telling you I am asserting even without writing the general Gaussian solution down that x does not diffuse the behavior of the there is no diffusion long range diffusion here and that there is a stationary distribution both in x and v this case both x and v are stationary Gaussian processes processes in this case so here if you integrate over the velocity you will end up with the distribution for the position whose solution is some kind of Gaussian and describes a stationary random process it is the distribution conditional density is a stationary that of a stationary Markov process similarly for the velocity as well okay. Now we could ask can we quickly see what the position itself does how do how do we get at it assuming that the velocity has thermalized what would one do in this case well this is what one would write down at start with the second equation and write it in this form x double dot plus gamma x dot plus omega not squared x equal to square root of gamma over m 8 of t okay and then I look at the over damped oscillator case in other words very high friction so this is called the high friction limit there is a systematic way of doing this high friction I have divided by m but imagine putting the m here and saying this term dominates very high friction gamma is very large then the inertia term can be neglected okay the effect of the mass is supposed to be like the friction is supposed to be very high and then m can be neglect this original term and then it becomes a Langevin equation just like the velocity Langevin equation for a free particle right except a slight difference here so now you got an equation which says x dot plus omega not squared over gamma x let us put it on the right hand side so x dot equal to minus this guy plus square root of gamma over m gamma 8 of t that is the Langevin equation for the position in this limit okay so let us simplify it a little bit and what does this give you this is equal to minus omega not squared over gamma x plus remember this was 2m gamma whatever it was right and there is an m gamma out here so this is square root of 2k Boltzmann t over m gamma 8 of t in this limit in this high friction limit so that is a Langevin equation compare this with the Langevin equation for the velocity for a free particle what did we get we found that v dot compare with v dot equal to minus gamma v plus square root of gamma over m that was equal to square root of 2 gamma k Boltzmann t over m compare with that so apart from this thing gamma here being replaced by omega not squared over gamma and then this constant changing to this other constant it was exactly the same in structure right so what is the Fokker Planck equation for this p here so what is this going to be p of x t assuming that you start from some origin at p equal to 0 x not we do not care satisfies delta p over delta t equal to and what is the first term on the right hand side this is a simple Langevin equation for which you can write the Fokker Planck equation immediately right so this is equal to omega not squared over gamma delta over delta x because everything is in x x p that is the drift part plus one half the square of this guy whatever this was plus k Boltzmann t over m gamma d 2 p over d x that is the Fokker Planck equation for p of x comma t right do you recognize k t over m gamma what is that that is the diffusion constant that is the diffusion constant so what we have got here is what happens in the high friction limit to harmonically bound particle as it diffuses so there is a correction if you like due to the potential to the diffusion term this is d capital D without this omega not it would be delta p over delta t is d times d 2 p over d x so that is the plane diffusion equation and you have long range diffusion but now you have got this extra term here as soon as you have that what does that imply it says something very interesting it says immediately that a stationary distribution would exist p of x as t tends to infinity and it should satisfy an equation in which this term is 0 delta p over delta t is 0 right it must satisfy stationary distribution p stationary of x must satisfy d over d x so let us write it down of k Boltzmann t over m gamma d p stationary over d x plus omega not squared over gamma p stationary equal to 0 what I have done is to say this term is a function of x alone now so this goes away and I have just put that in and this is it or there is an x d over d x x times p stationary which implies of course that this comes out and then if I take this down there it says m omega not squared over k Boltzmann t therefore this is a constant independent of x okay that is the current and if you say the current is 0 at infinity p stationary vanishes and so on and derivatives vanish then the constant is 0 everywhere so we can get rid of this in infinite medium and is equal to 0 implies p stationary equal to e to the power minus m omega not squared x squared over 2 k Boltzmann t all I have to do is integrate x and that is x squared over 2 with a minus sign and then a normalization constant which is m omega not squared over 2 k Boltzmann t square root 2 pi and that is the solution which is exactly what you get from equilibrium statistical mechanics. So you have got the Gaussian stationary Gaussian there is no long range diffusion etc but the general equation is this this equation is the equation satisfied by a diffusing particle in the presence of an external force a potential in this case a harmonic potential in this case it is called a Smolukowski equation this this equation here is an example that term is generally applied to the diffusion equation in the presence of an arbitrary external force we are going to apply to other situations as well for instance if you put a magnetic field the particle is charged and you put a magnetic field it moves in 3 dimensions and the question is what kind of probability density to the position and velocity and the phase space density have that is a question we are going to answer but this is an the simplest example of a Smolukowski equation okay what can generalize this a little bit and say it does not have to be a harmonic potential it could be any function any force you like we need that because we are soon going to do the case of a magnetic field so what do these equations look like so particle let us write in a potential let us say v of x here let us write the phase space equation down and then see what happens so again let us do it in one dimension first and then we will generalize this to higher dimensions so you have x dot minus v equal to 0 but now let us put the v on the right hand side just dot equal to v and m v dot equal to minus m gamma v that is the systematic part and then there is a term which is minus v prime of x that is the force on the particle plus whatever is the random force plus so let us divide through by this guy as always 1 over m plus square root of gamma over m and now this v prime of x may be nonlinear in x that makes the equation extremely hard to solve because you no longer have a linear drift now we got a really hard problem on our hands so the question is what does this solution look like what does this thing look like in general what is the phase space density Fokker Planck equation look like okay so for this we need a slight generalization of the linear drift case where the drift matrix was given by R now it is some complicated nonlinear function in general so let us say as usual that we have an equation of the form psi dot equal to some vector valued function F of psi I have in mind putting this psi as the x's and v's depending on how many dimensions there are all the dynamical variables plus a multiplicative noise which is let us say g of psi on this a dot and let us be completely general we do not know what the dimensionality of the psi is if it is a particle moving in one dimension then there is an x and a v if it is moving in three dimensions then there are three x's and three v's and so on and so forth and we have so far been looking at the case where the noise is just in that single component of the velocity but there could be different noises on different components they could be completely uncorrelated noises for instance if the particle moves in three dimensions you resolve it into three Cartesian coordinates there is no reason to expect that all the forces in all the directions are identical they are completely uncorrelated in fact what we will do and end up doing is to say that this force eta i of t eta j of t prime expectation where i and j are Cartesian components would be uncorrelated with each other if i is not equal to j so I am going to say this is equal to delta i j delta of t minus t prime for instance we are going to do things like that so we need to keep that in mind so let us be completely general and say that this is an n dimensional object so I write it as a column vector in other words n by 1 object column vector of this kind so is this and this noise could be in some of the components it may not have any noise at all in some of the others for example there is no noise here no explicit noise okay so let us say this fellow here is nu by 1 so it is nu dimensional white noise there are nu of these guys eta 1 eta 2 up to eta sub nu then this g in general this would be an n times nu matrix so that when it acts on the nu times 1 it is going to give you an n by 1 okay that is the general situation then corresponding to this equation this implies by this correspondence we have between Markov process the Langevin type equation for a Markov process and the corresponding Fokker-Planck equation for a for a diffusion process and the line Fokker-Planck equation we have delta rho over delta t equal to in this case delta over delta psi i psi i times rho okay and summed over the components i sorry f i times rho plus delta 2 over delta psi i delta psi j d i j which could in general be a function of this psi because this guy is times rho where d i j equal to one half as you would expect g g transpose i is a transpose that is the generalization okay remember that g is an n by nu matrix so g transpose is a nu by n so this whole thing turns out to have the right dimensions and that is the general Fokker-Planck equation and what we need to do is to apply to this case out here the reason I need this generalization is because this is not linear in the oscillator case this was linear and I just identified that our matrix in the matter was over but now this is not linear so the solutions in general would not be Gaussian's or anything like that fairly complicated things at all and at the moment we do not know there is a stationary distribution or not at all so what does this give us it says in this case delta rho over delta t equal to and first we got to do this term here so exactly as before you are going to have F in this problem so let us write down the F as I in this problem is just X and V F in this problem is equal to there is a V and there is a minus V prime of X over M minus V d ij mercifully is easy enough it is exactly as before because again this is just constant out here this is 0 out here what is new in this problem n is 2 little n is 2 because it is a 2 dimensional object what is little new just 1 just 1 so it is a very trivial problem in this case so d ij once again this matrix in this problem is again 0 0 0 gamma k volts may be half g g transpose etc so we can write down what this equation is in one dimension delta rho over delta t equal to minus V delta over delta X rho that is one portion that comes from here because there is a minus sorry minus I took F on the right hand side so there is a minus sign okay and then plus gamma delta over delta V V rho that term will always remain and then there is a term which is delta over delta V so this is equal to plus V prime of X over M delta over delta V rho that is this term plus the usual plus gamma k volts many over M d to rho over delta this is the one dimensional phase space density conditional density for particles moving on the X axis alone one just one component and it is got this extra term here due to the external force okay or the potential applied potential everything else is familiar as it stands this equation is called the Kramer's equation we will write down generalizations of this to 3 dimensional problems etc there it is fairly very straightforward to write it down in 3 dimensions in fact I urge you to do this as an exercise do this in general when you write this as a vector and this equation too as a vector now this gamma arises from a fluid and it is related to the viscosity of the fluid which will take to be isotropic so it is the same in all the directions does not matter for all the Cartesian components and this of course will be replaced by the gradient minus the gradient of the potential and then for the noise take this to be eta sub whatever I corresponding to V I and eta I is delta correlated such that different Cartesian components are not correlated to each other okay and write the general Kramer's equation down in 3 dimensions for the phase space density as a function of capital vector R vector V and T so that is a fairly straightforward generalization what would happen if you put this particle in a magnetic field what kind of force does it see that is an interesting case I mean we are going to do this problem but just to anticipate what is going to happen what do you think will happen so I put a magnetic field in some direction like this and you have this particle moving in 3 dimensions inside a container fluid and it is diffusing and there is also this magnetic field what do you think will happen well the particle will tend to do cyclotron motion around the direction of this field right on the other hand it is also diffusing what is the force applied by this magnetic field on the particle it is the Lorentz force right so there is a Q times V cross B so there is going to be an extra term on the right hand side which is Q over M V cross B that force is not derivable from a potential from a scalar potential which is a function of position it is a velocity dependent force but the problem is still very tractable and solvable why is that what is it about this force that makes the problem solvable completely it is linear it is linear in V so as soon as we have a linear problem this drift matrix becomes R which is a constant matrix and then I have a problem of exponentiating this constant matrix the green functions can be found etc etc may be complicated but in principle but in practice it is doable completely so the fact that it is linear is very very crucial so this problem can be solved it is a velocity dependent force of course it is going to look like this term it is going to have a V but the difference of course is that V 1 dot will involve V 2 and V 3 whereas this term is just V 1 in EB 1 dot etc so it will mix it up and secondly this gamma is represents the effect of dissipation as you know if you put a particle in a magnetic field a free particle its energy does not change kinetic energy does not change so there is no loss of energy at all so that portion of the drift term will be reversible in time but this portion will not be this is what leads to things at tending to an equilibrium etc now tell me suppose I have a gas of charge particles and there is overall neutrality maintained by some background and then these particles are diffusing at some finite temperature clearly if I put a magnetic field that field does no work on these particles whatsoever so will it change the Maxwellian distribution at all if it is in equilibrium it should not change this Maxwellian distribution the temperature will remain exactly the same nothing is going to happen but will the diffusion constant get affected what will happen what do you think is going to happen well we know that otherwise if it is one dimensional motion we know that this d was equal to k Boltzmann t over m gamma that by the way remains true no matter how many dimensions the particle moves in because each Cartesian component the variance goes like 2 dt and if you have 3 components in the r squared goes like 6 dt that is all that happens so the d is exactly the same k t over m gamma but now I put a magnetic field what do you think is going to happen this isotropy is broken there is a specific direction single doubt by the magnetic field will the diffusion along this direction be affected at all it would not be affected at all because there is no force in this direction at all on the other hand in the perpendicular plane there are 2 other directions in this plane the diffusion is inhibited because when it tries to diffuse there is a cyclotron motion kicking in trying to make it curve its path back again so we are going to discover that this diffusion tensor d i j is not going to be a constant times a unit matrix it is going to be such that the x and y components are not as large as the z z component so d 3 3 if you put the magnetic field in the 3 direction is going to be bigger and in fact we expected to be just k t over m gamma but in the other 2 directions the diffusion constant is going to be inhibited we will see explicitly how that comes about but that is just a physical argument we will write it down explicitly and see what happens here. So this equation here this basic equation the Kramers equation is a starting point for discovering whether the system has a stationary distribution or not and so on what we are going to do now next is to first take care of the problem of the magnetic field and after that I am going to go back and say let us look at the diffusion process itself the simple diffusion equation itself a little more carefully and ask what sort of process is the process x where the density of x the probability density of x obeys delta p over delta t equal to d d 2 p over dx 2 the original diffusion equation in one dimension and the corresponding Langevin equation which in this case was just x dot equal to square root of 2 d times 8 over t in the diffusion limit this means that you have gone to time scales much bigger than gamma inverse and then the particles is essentially as if the velocity is uncorrelated it is delta correlated it is a noise in this case in this approximation you have x dot is just white noise on the other side and corresponding density obeys the ordinary diffusion equation this is called Brownian motion mathematical Brownian motion and this process x this x process is called a Wiener process and it is the integral of white noise because formally what is happening this will imply that formally x of t minus x of 0 equal to integral 0 square root of 2 d times integral 0 to t dt prime 8 of t prime so what we have is a process that is the integral of white noise and it is much less singular than the white noise itself which has this delta function kind of correlation and we are going to ask what is this x of t it is called a Wiener process and we are going to study it and its sample parts in some detail so that will be the next program because this thing is what acts as a paradigm the very model for random process as random as you can get in some sense and it has a lot of interesting properties once we do that I will come back and make a connection between this and the Onstein-Ulenbeck process which as I told you is a unique continuous stationary Gaussian Markov process it is a unique process and we will see that there is a theorem which will tell you that in some sense if you studied this process the Wiener process you studied all Gauss-Markov processes we will see how this is done very mapping okay so that will be the next article thank you.


Let's do this thing man 30 upgrades. Give me some numbers boys. We can go all the way up to 30 We can pick any any one we want in between give me some numbers Give me some numbers. Let's start this thing off with a bang number seven two three four five six seven We want to see foot birthdays and foot birthdays only from these all right first one All right, not bad Not bad Amad says 15 so this is 12 13 14 15 Pack 15 oh All right, not too bad. Yo, thank you for all the fouls boys rock it in Dijerry Louie young Ron and fields I appreciate it boys pack four Pack four. All right. Here we go. Oh Look at that one. Let's go third one in and we have an 88 PK Actually are any of these duplicates. Oh, they're not it'll say already owned if it is 88 PK. Let's go. All right. This is number 23 Come on. Hey, that's a team of the week moments. Let's go 84 pulsing boys. These are doing all right These are doing all right Nine for bail. Oh 12 11 10 9. Let's do it. Come on Gareth Bale. No Gareth Bale Pack 10. All right pack 10. Let's go These have been decent so far already owned I'll take him Do it. I'll throw it to the end. We'll mess with the duplicates at the end boys. All right, JD This one's for you In minutes, that's a dub. That's a dub. Hey boys. These are doing all right. All right. Let's go pack 30. Come on. Oh I Think we have to take him again boys. I Think we have to take him again, right? And then we just rinse him into an SBC I think we take him again take him again. Yeah. All right. Take it. All right. I think so too, man How many do we have left? There's like nine. We have like 10 player picks left You said pick it up. Yeah, let's pick it up. Yeah, sports. I've seen a way too many 83s recently, man I mean, they'll be good for the SBC tomorrow, but we want something better. Come on foot. Oh That's what I'm talking about. Yes, sir. Both of those are nice ones, bro. We'll take the levela We'll take the levela. All right, so we've got him in as pulse and twice we've gotten pk a a lot of 82s and 83s and A couple 84s. All right, that helps that helps. Let's keep going That's that is a dub. That was a dub right there. Oh Get in Another dub back-to-back Sadio money another 88 rated. Let's go All right, we're turning up now boys. Let's do it Let's do it. Can we can I'm gonna I'm gonna open the pack right next to it, man Maybe they're waiting if they're next to each other All right, here we go. Come on somebody else for birthday. I'm taking that one We don't want the dupes the one before pulse in is Ramos. Okay, bro. I believe it when I see it Well, no, not terrible. All right. How many do we have two more? Are these the last two? These are the last two boys. All right, so out of 30 upgrades.  All right, last two. Come on for birthday. Let's go full send it. Come on. Oh My gosh, dude. Are you kidding me? Another one of these is it? I Think we're gonna do we have we have to don't we? We kind of have to Hey CJ, have you used your quick sell recovery yet? Because if we if you haven't used your quick sell recovery I could rinse one of them send one to the club and quick sell this one and then you'll have it to recover That's always a possibility to there's our third pulse and last one. Come on. Let's go be big and then a big one All right, I'll take Lopez. I'll take Lopez So this is what 30 upgrades just got us We got three team of weak pulsins some dupe 84s and 85s 83s him in as These two back-to-back were dubs. Those are like 60,000 coins a piece almost right now. I don't know how much money is And then we got pk as well in the front page with a lot of 83s So look at this page with all the 81s and 82s that page is kind of bad like that set of 12 Then this page was a lot better 85 84 with some 88 89 I'd say that was just about that was all right. That was all right That was all right All right boys It's time for 1180 plus play your picks on my account big pulls only you're calling a terstegon in a Lamar I already have terstegon bro, but I would love a foot birthday. Let's get it. Come on. Yay sports Boom first pack. Let's go Let's go get in First 80 plus upgrade that I open on this account. Get the frick in baby. I don't care that it's eggsteen It's an untradable. Let's go You'll love to see it man. Look at this the whole gang eggsteen. Oh Jean of the G's I'll get excited about that get in So bro such things to the follow bro appreciate it appreciate it man First one man literally the first player that we get tonight 11 of these built up and the first one is a big dog Unfollowed but is it renaldo it's not renaldo man, but it's a start. All right M. Duke. Thank you the follow appreciate it Let's go back to back. What are we saying? All right. All right back to earth back to earth boys Back to earth Come on. Hey this this set is up My gosh, oh my goodness. Oh my gosh, dude We have Rashford and now Pogba. What? What? Let's go Let's go Get in Chris is filming out here boys. What are we saying? Let's go Let's go Boom, let's go, baby. Oh Oh The Red List pack look is in Welcome to live stream boys. You have now entered the pack king dumb Let's go. Let's freaking go I'm next more excited for this than I was for Rashford dude. Let's go Let's go Yo, I'm telling boys. What is up with my pack luck dudes? What is up with my pack luck? I have no idea. I am on the Red List. I am on the Red List. Oh My gosh, oh Dude, yo My heart is beating so fast Pogba's in baby. Let's go Let's go pick arteries. There's no way give me Paul Pogba boys Three picks in and we have Pogba and Agstein. I don't deserve this, bro. I do not deserve this We move let's go Can we get another one? Oh, we'll take a team of the week This is incredible by the way boys Whenever you're opening packs, all right Take this knowledge put it in your brain and remember Whenever you're opening whenever you're opening team of the week packs or whenever you're opening packs in general One guy in the chat taught me this and it's helped me this since day one Open your packs from right to left. Look how I started opening these player picks, man And look what we've gotten out of this. All right, right to left Right to left when you're opening packs, whatever packet is always go right to left We'll take it a sell-so for the badge. I come back to celebrate. Ah, that's probably a no from me chief All right, we'll take a Felipe. Yep Dude, what are these player picks, bro?  I get that a lot, man These are the 80 plus player pick packs, bro 80 plus Player pick packs why start right to left? It's a superstition, bro, but it works It's a superstition, bro, but it seems to work Latara Martinez get in. What's up con? Welcome to stream, bro Hey, we'll take that as well All right, last one boys. I don't even care what this is Yeah, sure, borai Glacius get in What a set of player picks 86 rated Muller Pogba Untradable first owner in the squad Wow and x thing as well. This is this is my account and this is just ridiculous, man unbelievable, I think I can Officially qualify myself as red listed boys. I'm not one to say that I should be red listed But I think I I think I'm red listed boys. I think I am


Welcome back. Now to the next major conversation. We have a human rights lawyer joining us. He is Justice Uhuebu, who will do justice, of course, no pun intended, to the issue up next. A legal practitioner, Anigoro Mathias, has falsely the judgment of the National Industrial Court, ordering the immediate upward review of the judge's salaries on the grounds that it was delivered out of the jurisdiction. And should be voided forthwith. Justice Obasaki Osage had in her judgment held that the minimum monthly salary of 10 million should be paid to the Chief Justice of Nigeria and 9 million to each justice of the Supreme Court. The judgment followed suit filed by senior advocate of Nigeria Chief Sebastian Hon, who argued that since 2008, when judges' salaries and allowances were last increased, inflationary trends and socio-economic realities have rendered such pay inadequate. However, picking holes in the judgment, the lawyer said the plaintiff, Mr. Sebastian Hon, lacked the requisite local standee to institute the case and the court was duty bound to have declined its jurisdiction and dismiss the case now. To help us unpack this and understand what's going on in Analyze, we have, once again, Justice Uhuebu, who is a human rights lawyer. Mr. Uhuebu, thank you very much for your time. Thanks for joining us. Thank you very much. Now, for those who do not understand what's going on here, we have, we hear that the lawyer was gone to court, who's what I commented, commented on the decision of the National Industrial Court regarding the salaries of the members of the bench. What exactly is this lawyer's problem, as far as you can tell? When issues come up like this, you look at the interest. Why, first of all, the person that went to court with the court, although they want to achieve and follow the law. The lawyer actually talked about it. And actually, when a lawyer comes up in court, the court in the court looks at the issue of local standee. That is the court of action. Talk about local standee, talk about local standee. Then, local standee in the court, what is your interest? What is your interest? First, secondly, how would that case or judgment of the court affect you personally, for you to be able to go to court in the court? I think the lawyer is just the people that are supposed to go to court to challenge or to request for agreements, sorry, or whatever, because people that are directly affected by the court or just as we're talking about. So, the law is no respect for the court to challenge any situation they are at. All right. Audio hasn't been too clear, but are you agreeing with the view of this legal practitioner, Mathias and Igoro, that the lawyer who filed this suit lacks the local standee? Is that what you're saying, or are you disagreeing with him? No, no, the lawyer, remember that the person that filed the suit, the lawyer, because he's the member of the bar, that's the major ambassador. And all the judges, all the magistrates, they are also members of the bar, of the major ambassador. That's part of the fact that they are in the bench, which they are also members of the major ambassador. So, for me, a lawyer, any lawyer can come up with a name, any issue that assails the judiciary, because we're talking about the judiciary here. So, for me, I believe he has the local standee, because if the court is not filing, you have the local standee to go to court for court matter, the court don't arrive as in the district. So, in the wisdom of the court, the court department has the issue of the decision, because I believe the person that filed, for me, he needs to be another person who is not a lawyer, who is not an employee, who is not a candidate. Alright, so you've talked about the fact that this lawyer who filed the suit, the National Industrial Court, is a member of the bar indeed. He's a very senior member of the INA bar. And of course, when you talk about local standee, you've broken it down for us. You have to have something at stake for you to file a suit. Being a member of the bar, you've said that all members of the bench are members of the bar, if I heard you correctly. But are members of the bar members of the bench? The answer is no. Yeah, the answer is no. But remember, you don't meet a lawyer's type. You don't meet a member of the bar before you're called to the bench. In fact, let me say to you, when the puta, the puta, the JSC of Bezell Memorial, was asked a question, the difference between the bar and the bench. He simply answered me and said that a lawyer who stands in the bar to argue matters, while a judge is someone who after long standing at the bar, is supposed to sit on the bench. So the implication there is that the judge also is a member of the bar. You don't meet a lawyer before you're called to the bench. But Justice Huaybu, a member of the bench, this is a judge who is sacked or who retires. Can this judge go back to practice law? Of course. If you're a member of the bench and you have retired, you can set up a law firm, but still a lawyer. I've heard that once you leave the bench, you are not allowed to go back to practice law. The truth about it is that when you set up a law firm, you can involve people in the law firm. But can you work as a practicing lawyer after graduating and retiring or leaving the bench? I'm sure it's no because I've spoken to lawyers who tell me it's no. You can practice. Hello sir. Okay, so you're saying as a retired or fired lawyer, maybe you were sacked. You cannot practice once you leave the bench. You're talking about sacked. There's a difference between sacked and retired. Okay, please allow me to rephrase my question. Once a lawyer who becomes a judge leaves the position of being a judge, leaves the bench, can he or she practice again as a lawyer in Nigeria? Well, you are still a member of the firm. So the issue, the standard in firing rules, once you have retired as a judge. Because if the purpose of your effort is to go and start practicing as a lawyer, you can start standing in the back. Okay, so if that is said that you can go back to practice as a lawyer, then where in lies the connection between you, for instance, you as an example as a lawyer and a judge? It therefore means that there's no connection, wouldn't you say?  You cannot be appointed as a judge without being a lawyer. No, but you're saying that when you leave the bar and graduate to the bench, you are saying goodbye, bye bye, au revoir, to the bar? That does not mean that you're not a member of the firm.  That does not mean that you're not a member of the Nigerian Bar Association. Because you just don't know who's going from you. Okay, I did some research and you were able to give us a sort of definition and explanation as to why this lawyer who filed this suit before the National Industrial Court raised the issue of local standing. And a little bit of what I was able to fish out from the material I had access to is to say local standing is a condition, just like you've said, that a party seeking a legal remedy must show they have, by demonstrating the court sufficient connection to and harm from the law action challenged to support that party's participation in the case. So is there sufficient connection between the lawyer who filed this suit and how much the Chief Justice ends? Is there sufficient harm from this increase in the salaries of the Chief Justice of Nigeria and the judges, justices of the Supreme Court to this particular lawyer from the increase in these salaries? Is there any sufficient connection or harm? Does this lawyer suffer as a result of this national industrial court decision? Yeah, it's it for me personally. That's why the court has been living above the Supreme Court judges. If I were to go to court for the Supreme Court, I would even be converted for all the advocates of all the judges in Nigeria and even sent down to the magistrate. Why? Because, because if their standard of living and so far away is something to write from above, we continue to be having judgments in the court. And remember, judgments will affect everybody in the society because they want to come and respect laws when judgments are given in court. It becomes a law on to set aside. So everybody becomes a particular of him. So naturally, if the judges or the agitator are paid well, the issue of giving judgments by a plant or one thing or the other will be a thing that will not be successful. Because we want issues where some people put on an argument or an issue that exhausts policies right and other aspects and let some petitions are not good. But if the judges are not well taken care of, all these allegations will not be coming up. But it will affect the society. It will affect the judiciary. Okay, so what you're saying basically is that the harm or the adverse effect to this particular law or any law in the country is that if the judges are not paid well, it will obviously affect the way they carry out their work and maybe make them susceptible to influence. What they are doing in the chamber, their judgments are not right. Alright, so if they paid better, the lawyers will also have better, fairer judgments, therefore. Definitely, you don't need to know the court before you put your hand up. Because there's a saying that some people always say, not the law and not the court. For me, it's an affirmation. What you need to do is to not the law and not the court. What about the position or the issue of jurisdiction of the National Industrial Court, which we're all aware of the position or the responsibility and the duties of the National Industrial Court in Nigeria, you know, labor disputes and all that. The lawyer is also fighting this whole situation in respect to the jurisdiction of the court that are entertained this suit. When you say that you're getting told, the National Industrial Court is talking about the law and vis-à-vis employer-employee relationships are common. So the judge is also creating an employee in and out of the matter. Because each side must have taken all this into consideration before the decision. Okay, alright. What do you think will inform a lawyer having such a view? You know, it's hard to see all the lawyers in the country or in the state having the same view on a particular matter. So what do you think will inform a lawyer having such a view? Is it that he doesn't want to see the lawyer, the judge is better paid or he just simply feels that the right thing should... Can you read the mind a bit into the motives and the reasoning behind this lawyer raising questions over the case? Yeah, like I said in the assessment, I would be going to court of harvesting for a uniform, there are some states, for example, in Nigeria today, I mean the members of the bank, that are not being paid well, their worker is not being cadastic and visually in some states, you know, it becomes more like a larger stock. So how do you not treat everyone differently? I keep on saying this on and on again. Every country that judiciary has failed, that country is finished. The judiciary is the only armed that can revive a country, where the country is. And that is why we say that the judiciary is the last hope of the common man. That person that you are hoping to be the last hope of the common man, what about the work rate of that person and all the way? All right, all right. Back to the issue of local study, permit me to ask and bring that up again. Are you supposed to just maybe talk about, in general terms, to prove how a case affects you, how it harms you? Are you supposed to give, as a lawyer, or a plaintiff before a court, a case in point, particular cases in point to prove that injury came to you or harm came to you as a result of a situation for which you want to see change? So for instance, this lawyer is supposed to show, prove to the court, the national industrial court, that give them instances, specific instances where he suffered harm, injury, you know, adverse situation as a result of a low payment, perceived low payment, and salaries of judges in the country. Is he meant to show that? To me, he may or may not because, for example, they are neither driver, you always say that injury to one is injury to all. Let me also give you an example. You, you are the journalist. That's not the fact that you are in one media app. But you can also translate an issue that is happening in another media app. Why because you are a journalist? Yeah, but if I, if I, if I go to court, there's something I hear lawyers say or judges say, sufficient reason. Am I not supposed to present sufficient reason or proof or evidence that the issue at hand, the matter, the facts affect me personally and give them instances to say, on this date that this happened and affected me and that's why I'm coming to the court to say, please make this, you know, change and rule in my favor. So, sufficient reason is different from those standing. Now, let me, let me give you an example. For example, in the issue of fundamental human rights acquisition, because the law is that you can, it goes to court, it goes to your right, if your right has been free done, about to be free done or likely to be free done. So, you will go to your right, if it's free done, it's going to go to court. The price of the reason is not right. Okay, all right. In your view, do you feel that if this is implemented, if there's no challenge to this decision by the Honorable Justice of the National Industrial Court, Milord Justice Obasekyo Sage, who's directed that the National Assembly, the Revenue Mobilization, Allocation and Fiscal Commission, the Attorney General of the Federation in the midst of justice, and the National Judicial Commission comply with the directive of the court to initiate immediate compliance with its order. Do you expect that after such an order is complied with by these apartheid states? Do you suspect that the issues of bribery or alleged bribery of judges in the country will cease automatically? Miss Ohui, why are you there, sir? All right, we seem to have a natural connection and we've lost the guest, but I think he's been able to do justice to the topic at hand. Quite interesting one, a very interesting one indeed. I even had to catch a break, a sweat or two on my face. Right, thanks for joining us on this episode of Legal Education 101. Let's call it that. The breakfast will be back tomorrow with more interesting analysis of the relevant issues in national life. My name is Kofi Patels. Please follow us on social media at Plus TV Africa on YouTube. You can also follow us. We have two accounts there, Plus TV Africa and Plus TV Africa Lifestyle. We're also on the internet at PlusTVAfrica.com. Thanks for joining us. We'll see you tomorrow in the morning.


the infantrymen who dropped from the skies and fight on land. With allied successes on all fronts, the training of paratroopers continues with increasing intensity. These men are all volunteers, and the army is now accepting men from every branch of the service who can pass the rigid mental and physical tests to become paratroopers. The daredevils who ride down on silk, paratroopers were the first to land in Normandy. There they proved their worth beyond question. Paratroopers will continue to play a leading part in the allies' global fight to victory. In Fifth Army in Italy at attention, awaiting the arrival of the distinguished French leader, General Charles de Gaulle. Fifth Army commander greets General de Gaulle, and the two high officers talk together again for the first time since North Africa. The Medal of the Legion of Honor for General Clark, presented by General de Gaulle. Allied brothers in arms and a tribute to all of the officers and men of the Fifth Army, liberators of Naples and Rome, now moving forward again in northern Italy, operations off the Atlantic coast. A huge naval floating salvage crane goes into action. Navy experts working in heavy water battle to make fast a cable to the submerged hulk of a sunken ship. The divers and trainees gain practical instruction in the tough and invaluable science of naval salvage. Second birthday of the waves, America's women of the Navy. In Cedar Falls, Iowa, they celebrate by setting out for nearby farms to help relieve farm labor shortages. Detasseling corn. To produce the proper kind of seed, the tassels must be stripped from each stalk, and these Iowa waves are experts at the job. On a moving platform, they detassel the tall ones. Fresh corn on the cob. For the champion detasseler, a helping of the best. Far from the cornfields at Hunter College, New York City, 3,500 waves in a formal review to mark their anniversary. LaGuardia on the reviewing stand. In two years' time, the waves have made a brilliant record. Soon, the first contingents may go overseas. Only one saipan in the Marianas come Admiral Ernest King, United States Navy Commander-in-Chief, and Admiral Chester Nimitz, Pacific Fleet Commander. They view some of the wrecked Jap equipment that litters the island, and confer with Marine Lieutenant General Holland Smith, whose amphibious core, invader of Tarawa, Makan, and Pogelain, has now added saipan to its list. The capture of saipan places Japan and the Philippines within striking distance of Allied Pacific fleets. Paracutin, Mexico. A new volcano pours millions of tons of white-hot lava along a 25-mile front less than 200 miles from Mexico City. A church is destroyed by the fury of the invincible avalanche, and the organ is hurriedly salvaged. Communications are destroyed, five whole towns engulfed. Tarascan Indians take flight to escape one of nature's most terrifying displays. Steaming and hissing, the lava moves on at the rate of 200 yards a day. Its temperature, 1200 to 2000 degrees. For the first time only a little over a year ago, now earth-shaking explosions, higher and higher waves of melted stone, and destruction everywhere. Paracutin, newest of major volcanoes.


Hi everybody and welcome to today's barn's takeout. Today we're going to go upstairs into the corner room number 17 and look at this teeny little picture right here. Souvence somewhat entitled um light focus done prior to 1949 by a German artist who went by the name Vols and you can see on this wall it's um displayed well immediately next to this painting that Arnie Matisse had done in the 1940s in Los and I do love this wonderful like tangerine color near another work by Vols um a lot of other watercolors by um by Chagall over here we can see many by um Albert Prens his very good friend Charles Demith I particularly like this one of a woman in a punching bag as well as other artists like say George Rowe over here and so many work many little works on paper that we can see and before we start to zoom in closer to light focus I do want to pivot in this room 180 degrees to look at this is the north wall we'll look at the south wall here we go because um let's go in more closely two more works over here by Vols but here we've got many done by the artist Paul Clay like here and here and here and here and here here um and I think that's important because Clay was a major inspiration for Vols actually and um Vols give you a sense of who he was called Clay like an angel recreating the wonders of the world and art whereas he um Vols was a poor devil he said okay so with that said let's let's take a closer look at at light focus here it is um it's done in a style that we call um art and formel meaning basically a post-war style of painting that's abstract but not geometric it's got this kind of quivering central motif zoom in a teeny bit um it's mostly gouache watercolor but you can see there's pen um ink pen we've got some kind of scratching some dabs we can even see fingerprints from Vols like here over here up here um it looks the central motif looks almost like it's being compressed but at the same time expanding maybe kind of moving if we look at these lines here or perhaps it's even almost like a flower on the stem um and to me it looks like it could be cosmic as much as it could be um like almost something depicted on like an atomic or on an atomic level and then you can also see of course the on the wallpaper emerging through the canvas and I mentioned a minute ago that it was done in the late 1940s and down here we've got the name Vols in the year 1950 which seems like it would be very helpful but it's not but we know it's done prior to that because Barnes had purchased his set of Vols watercolors in during a trip to Paris in 1949 and we know that on a basis of a letter that I'm going to read to you that he wrote to his friend and um and colleague named Mel Mullen at the Barnes and so what he wrote what Barnes wrote was I didn't get any old masters but I got a fine redont oil um one that you can today see in room 14 two early rule of watercolors one that um one of which I showed you on on the wall of 17 four clays one oil by a new man Van Halvevelde one by a Spanish woman De Silva six gouache watercolors by a German Vols living in Paris the last three artists named are the last three artists named are the only painters whose work I saw that are worthwhile very modern but good in individual they help our collection and I think this is just a wonderful statement from Dr. Barnes toward the very end of his life and collecting career where he always is interested in what's modern what's new saying that he he thinks it's going to help his collection um and and anyway I just think that's kind of a fascinating glimpse into his mindset to Barnes's mindset so with this Vols for a bit of perspective on it um one one thing that like I kind of mentioned Vols was looking to were earlier images by Paul Clay one of which was Clay's twittering machine painted back in the year 1922 and to decipher this a bit we've got this kind of platform um with this vertical element that's crank that seems to operate these birds which themselves seem to be alive and so for Vols um what fascinated him was the kind of synthesis that clay got brought together between the mechanical and the organic and maybe if we look back for a sec to light focus we can see the the kind of um organic but very abstracted energy that we're getting from it and something else that um that's important to engage with with Vols partly vis-à-vis Clay was his engagement with different intellectual trends and what I'm showing right here is um actually a a photograph taken in Paris of the existentialist philosophers Simone de Beaufort over here on the right um and Jean-Paul Sartre I'm sharing a cup of coffee at a cafe I think this wonderful encapsulation of post-World War II French existentialist philosophy now Sartre was somebody who after World War II um was kind of a benefactor of Vols even wrote about him and so to an extent the works of Vols have gotten to be associated with what Sartre said about him and what Sartre was writing was essentially here let's go back to the Vols that images like light focus showed this kind of existential like horror rejection of the quiddity or like the thinness of the world and were kind of revolts and for Vols's life at the point at that point it would I suppose in its way make a certain amount of sense in as much as um Vols at the beginning of World War II have been imprisoned in France because of his German citizenship um after that fell deep into alcoholism a problem that was major by the time um he knew Sartre and ended up basically leading to the end of his life prematurely in the year 1951 but we know from Vols himself who wrote often that the philosophies that did intrigue him were manifold he was really interested in the fusion of art and music like Paul Clay and so perhaps we can see a sort of um I don't know almost stringed instrument like vibrancy to this he was interested in Chinese mysticism and how images could um could promote that he was interested in the ideas of Freud and how art might be able to manifest something interior and he was also really interested in um in the philosophy of Friedrich Nietzsche which was not really nihilistic but asked in a world that's kind of devoid of meaning that's been kind of blasted apart how do we make our own and so in a lot of ways I think that an image like light focus might there's a torment to it for sure um both individually as well as culturally at the time this was painted in 40 49 probably on the other side there's a kind of hope and I think part of that can do with an individual engagement with the image and there's one last thing I'd like to show which is actually this um early medieval manuscript a carpet page that opens the book of John from the book of Kells and I want to put forth the caveat that as to the best of my knowledge Clay was um the Vols was not religious I do not believe that he made specific reference in his own work to the art of the middle ages and yet something that he wrote um let me find this quote for you um Vols wrote that a tiny sheet of paper can contain the world and I think that the kind of looking he was going for was something like this where it um begins the book of John here's it spells a big I N and then P you might be able to see over here the Latin is in Principio or a Rot Verbum in the beginning was the word and the word was God and the word was about God and the word was light and in the world and zooming in um you can see the way in which the word and the world wrap together in a way that you just have to immerse yourself in the transcendental and the abstract um to see how it all does come together and that indeed a tiny sheet of paper can contain the world and so I hope that you'll take a chance to see this work let's go back this fascinating I think very stimulating work by Vols when you come back to the Barnes Foundation and I look forward to seeing you there and that's it for today's takeout and thanks so much for joining I'm Tom Collins new Bauer family executive director of the Barnes Foundation I hope you enjoyed Barnes takeout subscribe and make sure your post notifications are on to get daily servings of art 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