So today let us look at some of the consequences of the Boltzmann equation vis-a-vis relaxation. One of the things that primary points to be emphasized with regard to the Boltzmann equation is that it gives you an evolution equation for the phase space density in mu space for systems which are not necessarily in equilibrium, in general out of equilibrium and then the question arises as to how the system could relax to equilibrium under suitable conditions. For instance if we switch off the external force how it would eventually go to the Maxwellian distribution of velocities and the distribution in space would be uniform if you did not have any external potential acting on the system. So the question is how does this relaxation proceed and this equation should presumably give you an exact answer. So let us look at such phenomena, let us call it relaxation to equilibrium from the Boltzmann equation but of course the full Boltzmann equation is too difficult to solve. Even if you linearize it it is still too difficult to solve. You need a systematic approximation procedure for in powers of some correction to the distribution from the equilibrium distribution and that is a long-winded story. So the question is can we make a quick crude approximation and get some 0th order idea of how relaxation occurs in various situations. So this is what we are going to do today, talk about different situations where you have relaxation to equilibrium in a very simple, particularly simple mechanism. The first of these before I do that let us write the Boltzmann equation down just for you to refresh your memory. So it is Delta F over Delta T and that is in general a function of R, V and T plus V dot Del with respect to R of F plus if there is an external force it is F over M dot gradient with respect to V of F and that is equal to the collision integral Delta F over Delta T collision. So all the physics is contained here. This is basically kinematics but this is where all the physics is. Now of course we know that it is not horribly complicated but even after you make a linear approximation it is still as I said very messy so let us make the simplest possible approximation. We have seen already that if you switch off the external force there is no force at all then this thing here tends to an equilibrium distribution which is F, do not remember the notation I called it F naught but let me call it F equilibrium, F equilibrium of V was equal to well recall on first we should write our normalization. The normalization of this F was integral d 3 R integral d 3 V F of V T equal to N the total number of particles in the system. If you integrate it over V alone so if you integrate d 3 V F of R how should I write this R V T V alone then you get a function of R and T which is equal to well it is the number but it changes with the number density but it changes as a function of position and time so this is by definition equal to N of R and T right. If you integrate it over R alone d 3 V F of R sorry whatever I done over R V T how did I write the normalization down did I mess up the normalization so this is okay as it stands but how did I get the number density oh yes so we wrote integral d 3 V F equilibrium of V was equal to N over V equal to N that is the normalization right thank you. So let me define this non-uniform density by just integrating over the velocity alone this distribution function and in equilibrium there is no T dependence there is no R dependence if I integrate over d 3 V I should get N over V okay so that is those are the normalizations. Now what we would like to say is that remember remember that F I need a little more notation F equilibrium of V was equal to N this thing here in space for the spatial part and then the velocity part was the Maxwellian distribution so the normalized distribution was m over 2 pi k Boltzmann T to the 3 halves e to the minus m v square over 2 k Boltzmann T so that is the equilibrium distribution this is a constant now this function is going to keep appearing all the time the Maxwellian distribution in the velocity alone okay so let us give it a name let us call this equal to identically equal to N times let us call this W of V to show that this is the Maxwellian distribution because in equilibrium everything is in the absence of an external force the distribution in space is uniform so we have uniform density number density N to unit volume and then multiplied by a velocity distribution which is the Maxwellian distribution and this is normalized to unity so we know that integral d 3 V W of V equal to 1 that is the reason for this factor 2 pi k T whatever it is right now what the approximation does the simplest approximation to the right hand side is to say that collisions cause if you have a small departure from equilibrium collisions cause you to go towards equilibrium they help you to equilibrium so if your distribution F of R V, T is a little bit away from F equilibrium then if you make the single relaxation time approximation the so-called single relaxation time approximation so let us write that down single then if you start with the uniform density the system is completely in equilibrium and you ask if I have this perturb it a little bit away from equilibrium the velocities are not thermalized they are little bit away from equilibrium how would these velocities equilibrate yes 0 force case I will do the system with the force present but 0 force case so that is the reason whenever this happens whenever this is a function of R you know that there is probably a force present but even in the absence of a force instantaneously there could be density fluctuations and there are so in general I would call this N of R and T R, T but when you have reached equilibrium in space and time and in velocity and position then the whole thing is just a constant here okay now how does the velocity relax that is the first question you want to ask so let us call 1 within the single relaxation time approximation relaxation of the velocity that is given by the following now there is no space dependence because it is uniform throughout the velocity is supposed to relax to equilibrium now we already know there is a model for this this was a Langevin model when I wrote down the Langevin equation and then I argued that if there is white noise which causes collisions and so on or the effect of collisions is mimicked by white noise Gaussian white noise etc then you are a Fokker Planck equation and the solution to that was the Onstein-Ohlenbeck distribution and then that constant gamma the friction constant determined how things relax to equilibrium and the velocity correlation time was an exponential single decaying exponential with the relaxation time gamma inverse that was that model okay but now we do not have that model at all there is no stochastic force or anything like that this is a completely self consistent system and we are not saying that the mass of this particle is different from the masses of the rest of it they are not saying that at all no such assumption but we made a huge approximation to this in this single relaxation time approximation so now this F is going to satisfy Delta F of V t over Delta t no external force the distribution is uniform in space so this term is missing and you only have this and this is equal to the collision integral and what the single relaxation time approximation says is that this is equal to there is a relaxation time brought in a parameter tau just one of them so it is minus 1 over tau and inside is just F of V t minus F equilibrium of V t so that is the approximation it replace by saying the deviation at any instant of time of the velocity distribution from the Maxwellian distribution is negligible is very very small and then what you have is exactly like in radioactive decay very similar to that this difference divided by tau and it is got the right dimensions it is the dimensionality of Delta F over Delta t except this F is discretized and said that is the difference between the distribution instantaneous distribution and the equilibrium one divided by the time scale tau so this is an extremely simple model what does it predict now we can solve this is a trivial equation to solve the first order equation so it is trivial to solve and of course what one should do is to write set when we can either write down the integrating factor or else set F of V t equal to e to the minus t over tau obviously want to relax with that characteristic time scale tau multiplied by something else so let me call this psi of V then the equation on the left hand side is the first term is minus 1 over tau times this whole business which will cancel against that on the right hand side and it says e to the minus t over tau Delta psi over Delta t equal to the first term cancels so it is equal to 1 over tau F equilibrium of moving this to the right hand side but you have to tell me what the initial condition is you have to tell me what is the initial velocity distribution so let us put that in so initial condition F of V t equal to some initial distribution some prescribed initial distribution so that immediately implies psi implies psi of 0 is F initial because this term becomes 1 at t equal to 0 so what is the solution to this thing it says psi of V t minus F initial of V is solving this differential equation you have to integrate this from 0 to t put t prime and integrate 0 to t so that integral becomes the t tau cancels as you can see so it becomes e to the t over tau minus 1 times F equilibrium of V which implies that F of V t equal to I move this to the right hand side and multiply this by e to the minus t over tau so it gives me e to the minus t over tau F initial of t plus I multiply this by e to the minus t over tau so this is 1 minus e to the minus t over tau F equilibrium that is the solution which is exactly what you would expect at t equal to 0 this thing is equal to the initial value because that vanishes and as t tends to infinity this term goes away that term goes away and you are hitting B F equilibrium exactly so this will work as long as this is very close to equilibrium so that is the whole point that you have replaced this entire collision integral by saying that the system is very close to max value in distribution pardon me whatever you prescribed there is no reason why that should be max value for consistency of the approximation this too should be fairly close to the max value in distribution of course no that is not the point there is a characteristic time scale tau we do not know anything about it we have simply said this entire collision integral has been replaced by this discretized version we have not had do not have any handle here about what this tau is at all what we need to do is to find some way of finding out how good this approximation is is to find some way of measuring this tau if it exists now in practice that is not going to happen but you can see what is what we can do what we should let me indicate how one should go about it in the general problem in the general case we are looking at a function of V this quantity is a function of V and T now we know that the equilibrium distribution is a max value is this one of here okay now any arbitrary function of V where we run each component of V runs minus V infinity to infinity can be expanded as Gaussian Gaussian and V times Hermite polynomials so in principle you can say this quantity here is a superposition of all the Hermite polynomials times the max value distribution with coefficients which are possibly time dependent which are certainly time dependent and that is how the equilibrium will happen now this is the crudest approximation to that it says that in some sense it just says the polynomial talking about is 1 okay you have taken the constant there is no Hermite polynomials anywhere there is no function of V anywhere you just have this function of V and that is it this is already the max value okay so there is a systematic way of justifying I mean finding out what this approximation means but in physical terms it says that if you replace this entire collision integral by saying that we do not care what is happening in the inner mechanism there is some effective time term over which the system equilibrates then this is what happens as far as the algebra is concerned but now how far this is good how good a good an approximation this is can we get an handle on tau can we measure it and so on are not answered questions within this framework as yet we are going to find out we are going to find out of this tau appears anywhere else and then discover how ask how what tau could possibly be okay certainly yeah it will not be a single molecular collision time because there are many many time scales involved here so this is some effective time scale on which the velocity is thermalized we already know from the Lanjama model that the time scale on which the velocity thermalizes in that model seems to have something to do with the viscosity of the medium but that was dependent on the assumption that the Lanjama equation was valid which is itself true only if the mass of that particle is much much higher that is not obvious here it is not obvious here so it is clear there is a spectrum of relaxation times and we have taken the leading contribution is it really what has happened now the question is can this also could this result have been produced in any other argument any other way is there a simple model which will produce exactly the same result some stochastic model which will produce the same result it is clear this is not the on-steen Ohlenberg distribution at all so it is not certainly the Lanjama model at all it is very different from that the Lanjama model has this on-steen Ohlenberg distribution which is again a Gaussian where the system tends to the Maxwellian distribution exponentially with this time scale gamma inverse but it is very different looking from this here it is just either they this is and incidentally this if I start with the particle like we did in the Lanjama model with a given velocity v0 this of course would be delta of v minus v0 this would be replaced by delta function but I am allowing a little more general solution by saying that it could be a distribution in itself not all fixed at one velocity is there some way of producing this thing here by a simple model assumption on the random on the nature of the random velocity we do this at all not the Lanjama model here now the answer is yes but it must be a much simpler model than the Lanjama equation because that involves the Fokker-Planck equation which we solved this thing here looks like it is a much more trivial thing to solve right there is an equation for the distribution function which is first order in time and involves the same thing out here so it just looks like a Markov process we have just assumed it to be a Markov process but it looks like it is a discrete Markov process well it is continuous because v is continuous but it looks like v is changing through a jump process of some kind so let us see if that is validated or not suppose v underwent suppose v is given by a Markov process suppose so what I am trying to do is to argue that there is an effective model where v is just taken to a Markov process by which you will reproduce this exact result so that is another way of understanding what is the meaning of this single relaxation time approximation suppose v is a Markov collision will take v to some other value each component of v varies continuously minus infinity to infinity the moment you say it is a Markov process then its probability density delta I need to use another symbol for it since I am reserving F for the symbol we use in the Boltzmann equation the distribution in new space let us just call it p so delta p of v and t divided by delta t must be equal to on the right hand side we need to write a rate equation because it is a Markov process so in general you can write this since it is a continuous process d3 v prime on this side times a gain term and a loss term that is what you do for any Markov process for which you can define a transition probability per unit time right so times the probability that you reached the velocity v prime at time t multiplied by the transition rate per unit time that you hit v given v prime starting from v prime so this guy here is the v prime to v that is the gain term and the loss term in the rate equation is minus w you jump out that is the Markovian master equation now the question is can I produce this solution by writing a model for w that is the question being asked detail balance must obtain in equilibrium so what sort of function should I put in here such that in equilibrium this will become p equilibrium that becomes p equilibrium what sort of function should I put in here such that detail balance will obtain well let us look at the physical process you have an initial velocity v in this transition rate you have a velocity v and it is getting knocked out into any other velocity v prime now what is the effect of these collisions one possible approximation is to say the collisions are extremely weak so whatever you started with that changes very slightly in a given collision in the limit of no collisions it will not change at all but in the limit of weak collisions it will change very small by a very small amount delta v to start with v a collision will connect you only to those velocities which are within a range delta v of the initial that is the weak collision approximation it turns out that approximation leads to the Fokker-Planck equation assuming that there is no memory in these collisions and it is a Markov process I am not going to do that but there is a way of systematically deriving from this master equation the Fokker-Planck equation by making the so-called weak collision approximation there is another possibility and that is a very strong collision approximation it says look each collision is such that it thermalizes it at once in other words it says that this quantity lambda v prime the transition rate from any velocity v prime to any other velocity v does not depend on the initial value at all the collisions are so strong that in a given collision it is immediately thermalized that is one possibility that the transition rate is independent of the initial state it depends only on the final state so suppose that were true suppose but equal to we need this is a transition rate so you need a constant of dimensions time inverse let us call it lambda times some function let me I should not use the function psi I should go to use some other function what is a good what is a good psi of v independent of v prime suppose so the transition rate every time there is a collision and the average rate of collisions is lambda the velocity changes whatever be the initial velocity it goes to a final velocity depends only on the final velocity suppose this is true and now you impose detail balance in equilibrium then what will it imply this will imply detail balance implies then with this assumption if you let t tend to infinity this term is 0 the time derivative it goes to equilibrium this becomes p equilibrium of v prime and this becomes lambda times psi of only the final state of v p equilibrium of v prime must be equal to lambda times psi of v prime p equilibrium of v and there is only one solution to that which is that this is p equilibrium of v itself so it is obviously the strong collision approximation so in one shot yeah so what it is saying is the following you have an equilibrium distribution in every collision the velocity is reset from whatever its value to some value drawn from the equilibrium distribution with those probabilities so if your distribution is a Gaussian and you are at this point you have this initial velocity after collision the velocity new velocity is reset to be one of these velocities with this distribution with this probability it is chosen okay so that is the implication of this statement here random process in which a Markovian process continuous process in which the transition rate is independent of the initial state and is drawn from the final state alone and is a function of the final state value alone and moreover there is detail balance this is called a Kubo Anderson process so what we have shown so far is that if you assume a Markov process for this guy and the Kubo Anderson process with detail balance then this equation simplifies enormously and look at what happens to it exactly apart from that factor n there is a factor n for our phase space distribution this is only in the velocity so I have distinguished it by writing p equilibrium separately so look at what happens to this equation you have d3 v prime this is a function is i of v p equilibrium of v and then you have d3 v prime p of v prime t which is equal to 1 because it is got to be normalized so this immediately says delta p over delta t equal to lambda times this fellow here is p equilibrium of v in this side minus lambda times and now this is an integral d3 v prime of p equilibrium of v prime which is 1 in this side times p of v and t so it is clear that with the identification lambda equal to 1 over tau you get the single relaxation time approximation it is exactly the same so the solution is exactly the same so one way of interpreting the single relaxation time approximation in the weak in the Boltzmann equation with a linearized Boltzmann equation is to say it is equivalent to saying that the velocity v is just a Markov process a Kubo Anderson process nothing more than that this gives exactly the same solution well you can ask well can I tweak this a little bit can I try to improve this model by saying look each collision need not take you to the velocity drawn from the equilibrium distribution so there could be one limit in which you have a very weak effect of collisions this is a strong collision limit in which it immediately thermalizes in some sense so is there anything in between I leave it as an exercise to you to show the following rather simple exercise that if you say independent of this if you say that this W is not this but it is equal to lambda times let us say some gamma times I should gamma is a bad number alpha times well alpha times a function of Xi this is p equilibrium so let me write it like that plus 1 minus alpha times at delta of v minus v prime you could do that this interpolates when alpha is 1 you have the Kubo Anderson process when alpha is 0 this goes away and you have no collision at all it does not do anything this will not equilibrate at all it will remain where it is but this here with alpha equal to 1 equilibrates with the time constant lambda inverse so now figure out what happens you can easily satisfy yourself that this again satisfies the detail balance condition so it is a kind of interpolation model between the strong collision limit and the 0 no collision there alpha is any number between 0 and 1 it does not equal if that depends on lambda this depends on lambda right so it does not equilibrate in one collision it simply says that the transition rate is drawn from the equilibrium distribution independent of where you started yes you can but this is an arbitrary parameter this is an arbitrary yes there are models for collision broadening in gases and so on where you use a strong collision limit seems to be the correct limit work with yes there are physical systems which display this behavior so the point I am trying to make is if you did not have this you have the strong collision limit it is a rather trivial solution it is equivalent to the single relaxation time approximation in the Boltzmann equation in the linearized Boltzmann equation on the other hand if you did not have this but you had this alone there is no collision there is no physics it is no equilibration at all but if you tweak this and made this a weak collision limit then you get the Fokker-Plank equation okay so it is possible to derive the Fokker-Plank equation from the Boltzmann equation by making a single relaxation time approximation and a weak collision approximation okay I am not going to do that that is a little bit of machinery but I am not going to do that I just wanted to point this out okay but now you could choose this alpha to be between 0 and 1 what do you think will happen this model also satisfies the detail balance condition for any alpha between 0 and 1 so what do you think will happen to the solutions again it should be solvable completely because there is a delta function here all it does is to rescale time it does not it just buys you some time it does not do anything the second term does not do very much yeah not entirely but this one here so it just changes lambda alpha and then because it is lambda alpha inverse as you can see if alpha was 0 that means collisions do not change the velocity at all so there be no equilibration which is equivalent to saying the relaxation time must become infinite so it is not surprising that it appears here in this place so the reason purpose of introducing this was to show you that relaxation phenomena can be modeled from the in a systematic way from the Boltzmann equation now let us look at the other thing we did with the Langevin model which absolutely absolutely yeah it is not the collision time very emphatically no not the time between collisions no in fact we will see what this tau is in the single collision approximation this was just a model the lambda was completely arbitrary here but let us go back to the single collision thing and look at the other phenomenon which was we have a gas in which you have a non-uniform initial distribution the velocity has thermalized and now the system diffuses that was our famous diffusion regime where the mean square displacement went like linearly with time and the velocity correlation time had long died down and you are looking at longer time scales let us see how that comes out in this language in the Boltzmann equation so this is the smoothing out of a non-uniform initial distribution in other words diffusion self diffusion this time because the particles of the medium are themselves no external force as before and the Boltzmann equation is delta F over delta T plus V dot gradient with respect to R of F this term is very much present because this is a function of R V and T that is equal to the single relaxation time approximation and what would you write it as this time it is 1 over tau F of R V and T minus what would you write it this time as there is an R distribution there is definitely an R distribution right we are trying to okay but the velocity has thermalized so you write it as N of R, T times W of V and recall that this fellow here is integral V 3 V F of R V by definition non-uniform distribution the velocity has thermalized it is the maximum we want to know how this guy relaxes how do you solve an equation like this well it is got both space and time derivatives so clearly you are going to do the plus transform with respect to time derivative time and Fourier transform with respect to space so you are going to define an F tilde of K V and S to be integral 0 to infinity DT 0 to the minus ST integral D 3 R 0 to the minus I K dot R F of I do not want to put 2 tilde I mean it stands for okay this stands for Fourier transform with respect to R Laplace transform with respect to time of this we plug that in here then let us write the equation down directly oh by the way I need to tell you what is the initial value of this fellow what is the initial distribution so I have to tell you what the initial distribution is so let us put N F of 0 is equal to N initial of R whatever be that initial non-uniform distribution that you plugged in times W of V of course so what is the equation you get this let us call this F tilde when I do the Fourier Laplace transform so F tilde S times F tilde of K V S minus this guy here that is the formula for the Laplace transform of the time derivative but I have got to do a Fourier transform with respect to space let us call that N tilde of initial of K W of V just sits there as a spectator this plus V dot gradient what is this guy going to do gradient with respect to R pulls on minus I K so we got minus I K dot V times F tilde S that is the transform of this fellow here this is equal to on the right hand side equal to 1 over tau minus 1 over tau F tilde of K V and S it should be plus so I expand this in terms of I expand this fellow yeah I expand this fellow and then it is F tilde I leave it you to put in all the 2 pi factors and stuff like that okay yeah yeah when I go to the Fourier transform in every one of these terms there is a 1 over 2 pi in the inverse transform that cancels out but this is a plus because what I am doing is to write this fellow in terms of as 1 over 2 pi D 3 K this guy it was a plus I K dot R so I K dot is equal to this this term plus 1 over tau the Fourier transform of this plus 1 over tau N tilde of K and S W now this looks like a hopeless task because we do not know what this is and we do not know what this is what should I do this is presumably given to me this is known function here I integrate both sides with respect to V what happens then so what happens to integral D 3 notice that integral D 3 V F of R V T integrated over V is N of R if I take Laplace transform this becomes N of R S if I do Fourier transform it becomes N tilde of K, S so I integrate both sides with respect to V in which case you are going to get this thing here and this is going to be integrated over with respect to V on this side but you got to be a little careful in doing this this N initial will move to the right hand side and there is this guy so we should not yet integrate over V you need to pull various things and then do the integral over V let us write all the F guys together so there is S that is this term plus I K dot V plus I K dot V so this term is gone this term is gone plus 1 over tau plus 1 over tau times F tilde of K V and S is equal to so this term is gone one I have written single written one extra term I think I have an extra term somewhere this is equal to let us see where this takes us N initial tilde of K times W of V that is certainly there plus last term plus 1 over tau where is the problem no no what did I do how did was this quite right how did this come about this is equal to this is F of R V it is F tilde of K V and 0 is it not this term is F tilde of K V and 0 and which we wrote as the initial distribution in R times the equilibrium distribution in V and then the Fourier transform with respect to that which was this guy so this is okay this is okay plus N tilde of yeah so that part is alright 1 over tau N tilde yes K and S W of V so this is gone that is the equation right now let us move this to the right hand side so this is divided by S plus I K dot V plus 1 over tau S plus I K dot V plus 1 over tau I am going to leave the rest of the completion of this algebra to you so I move this there and I integrate over V then if I do this on both sides D 3 V of this whole thing of the entire thing both the left and right hand side then what is this quantity equal to I have integrated over this so this is N tilde of K comma S so I get an equation which says N tilde of K S equal to the integral of W of V alone is 1 but you have got this in the denominator so you cannot do anything with it it is equal to N initial tilde of K integral D 3 V W of V over S plus I K dot V plus 1 over tau this fellow sitting there that is some number it depends on S and it depends on K it is some function of S and K plus the same number once again because there is W of V over this guy 1 over tau N tilde of K comma S times the same integral D 3 V W of V blah blah blah call that integral something or the other and bring it to the left hand side and you have an equation now for N tilde of K comma S you invert with this there are plus transform you get N tilde of K comma T you invert the Fourier transform you get N of R comma in principle now doing it analytically is formidable but you have a closed equation for it completely and what we will do next time is to take this and see how we are going to get the diffusion coefficient in the so called hydrodynamic limit first of all you have a relaxation time tau here we have seen that the velocity is relaxing with the relaxation time tau we know the diffusion regime is when you are at time scales much bigger than the relaxation time right so in terms of the Laplace variable we need not even invert the Laplace variable the diffusion regime would mean S times tau this guy here would be very very small this is small compared to 1 over tau that corresponds to long times small s is large t okay so this guy here should be equal much much less than 1 and K should be very very small also because you are looking at hydrodynamic modes you are looking at long wavelength fluctuations not on very short length scales long time scales long length scales should give us then the diffusion coefficient okay we will see how that emerges from here systematically we get actually more information from this equation but the basic trick is the following the basic trick is you write the single relaxation approximation by saying the collision integral is some minus 1 over tau times the difference between the distribution function and the asymptotic form or whatever and then you solve that equation in a self consistent way in this case the trick was to integrate over v and then get a self consistent equation for n tilde of K comma s which is what we are trying to find n of r comma t and what is done is to find this Laplace Fourier transforms first into this okay so we will complete this we will do this meanwhile try this out and then we will see how a systematic approximation procedure will give us the diffusion coefficient okay then there also remains the case of what happens if you apply a uniform but time dependent force on the system how will it take it out of equilibrium so you start with an initial condition that spatially uniform and the velocity is thermalized and then say I am going to switch on a force which does not depend on the position in the simplest instance but on time how is the system going to go out of equilibrium again in the single relaxation time approximation so we will look at that as well and those are things which these are the things which will help you to find things like the viscosity the diffusion coefficient the thermal conductivity and so on as I said before okay so let me stop here.


Upload with Henry Morgan. Jeffrey Hunter, anxious to save Henry Morgan from sailing into a trap, goes with hero to the governor, Sir Thomas Mottford. Between them they piece together what really happened on the night that the Aztec necklace was taken. Sir Thomas is anxious to know how Jeffrey became a convict. Jeffrey tells him that it was the innocent dupe of a man named Blake who later arrived as a convict at the island and who denounced him to Kitty. Sir Thomas warns both Jeffrey and hero that he has no option but to retake them into custody and he puts them on their honor not to try to escape. The law is, together with Diets and Kitty, still a prisoner, arrives in Cuba and Diets is well rewarded for his work and plans are made to capture Morgan. Eventually Sir Thomas returns to Jeffrey to tell him that Blake has confessed that Jeffrey is innocent and so he is pardoned. Due to the good work hero has done, he too has given his freedom. A sloop is about to leave Port Royal to overtake Morgan and his fleet and Sir Thomas tells Jeffrey that he is to join her. I can't express the Thomas the feelings I have but I'll doubt this must be the happiest day of my life. You are prepared to join the sloop? Join her? Why a hundred fighting men couldn't keep me away? I too don't know what to say, boss. When I left my home way back in Africa, I never thought I'd see a free day again. Lord, I thank you for letting the sun shine so bright.  My tongue is just tripping over itself. I'd like to go down on my knees and thank you for all that you've done. That will not be necessary. I realize that if you've not stayed by Jeffrey Hunter and nursed him back to health, Morgan wouldn't have been warned. He's not warned yet and he has a good stuff. But the sloop is fast. With the very good wind behind you, you have every chance of overtaking him. The sloop sails in an hour's time. An hour? Good. We want to waste no time. I daresay there's quite a lot you have to do. You want some money to set yourself up, some new clothes. I haven't thought of that. I'll give you what you need. But you better hurry. You've not much time. Thank you. Thank you very much, the Thomas. There's nothing to thank me for. Your case was a miscarriage of justice. You'll be compensated for the humiliation to which you've been. Hearing my name is all the compensation I want. You ain't staying goodbye to me, are you, boss? I ain't got no place to go now. Freedom to an old negro like me means a lot, but it's like a new world I've been put into. I'm a bit bewildered. I don't want to say goodbye to you, Miss Jeffrey.   I don't want to say goodbye. You're going away under ship. Maybe I'll never see you again.  I'd never do, hero. And there's no reason why you shouldn't go on helping me. What do you mean by that? I had no thoughts of saying goodbye. True, I'm going away on a ship, but I took it for granted that you'd be coming with me. You want me to go with you, Miss Jeffrey? Oh, Lord, I thank you for being so kind to me. Thank you for keeping me by my friend. You want me to come with you. I'm telling you, Miss Jeffrey, that there's one person you ain't never gonna lose now. No, sir. You're taking me away with you on the ship. Hallelujah. I go down on my knees now. Thank you from the bottom of my heart. Now, come on, you blacks, Kelly. Wake up on your feet. None of that nonsense. Of course you're coming with me. And we'll have to move pretty fast. You want to catch that slope? I expect it when I return, Kitty, to find you dressed in the fine clothes that I've got here. What are you doing still in those filthy rags you're wearing in Jamaica?  So keep your distance, dear.  And I don't want it, Titt. I'll ask you to leave me be. Are you not forgetting, Kitty, that you're not in a position to make demands of me? In Jamaica, when you're at the Dolphin Tavern, you would have born this servant. A little better than a slave. But here in Cuba, you're not even that. You're my poverty. I deal with you as I will. Don't you come near me.  I'm glad to see your spirit is not gone. I was beginning to think you had become a milk and water woman. And I don't become that, Kitty, because if you did, then I would soon grow tired of you. And when I grow tired of you, I'll not want you above me. And then, of course, I'll have to get rid of you. For your own sake, you'd best remember, always make yourself interesting to me. And now I know of this nonsense. Dress yourself in the fine clothes that I brought you. Appreciate the fine home you're living in. I'm a man of wealth now. I'm going to be a man of position. Look what I'm giving you. Don't you appreciate it?  I'd rather be a slave in a dockside tavern than living with you.  It makes you glow with life. I've been a very patient man, but now my patience is at an end. Don't you come any closer to me yet. I'll have nothing to do with you. And don't you put your hands on me. And who is there to stop me, eh? This is not the dolphin tavern, Kitty, when you can scream and have the men come into your aid. There is no one in Cuba to help you. No, and no one at all. Open your soul, love it, try to scream. Scream loudly.  It will only bring to their faces there a knowing smile. And Kitty knows that what he says is true. There is no one. She is helpless, as helpless as a jungle-colored butterfly which is plotted unsuspectingly into the enmeshing sticky strands of a spider's web. But to give without a fight is against all her nature.  Her heart pounds within her.  Her small, white teeth close over her lower lip. She looks searchingly into his face. It grows large before her eyes. Horrible, vicious, layering. It fills the orb of her vision. Fascinated with horror, she lets input forth his hands to grasp her. The shock of the cruel, deep-digging fingers snaps the bond which paralyzed her, the forked lightning playing around the tree. She becomes a live, squirming fighting thing, reserved and hidden strength born of desperation surges through her. But yet, her strength is even greater. Knows it out around the headlands of Port Royal Bay as daintily as a dancer. It skirts the reef over which the ocean's swell breaks into a feathery, white surge. Quietly, the steady wind fills the canvas so that the ship seems like a white cloud being strangely guided over the deep blue sea. A long, lean, narrow bow dips down to cut through the rolling sea then rises proudly up to meet the next swell. And the coast of Jamaica steadily recedes behind the pathway of swirling wake the ship leaves. Up on deck, the trade wind plays chasing among the gold of Geoffrey's hair, gently caressing his cheek and his eyes search wide over the expanse of sea, looking out to that point where the two blues merge to make the distinct line which is the horizon. But his happiness is clouded, the memory of the faces before him and a trust which he didn't fulfill. It was not my fault. My failure was beyond my control. But the inner voice will not be still. Kitty has showed to him that she had complete trust that he would rescue her.  But I won't think of that. I must not have so go mad. I'll think of Henry Morgan in the morning I must give him. I pray this wind will hold and we'll be able to overtake him. And the wind does hold and day makes way for night. Stars pop their coals, censor his old light out from their cushion of blue, deep and almost to black. Like diamonds in the hair of a raven beauty, they winkingly look down. Then their brilliance fades before the white waxing moon which climbs from out of the sea like a Dutch cheese which has been bounced slightly out of form. Though in a new world of strange beauty, the slope cuts through the ocean, its nightlights bobbing like fireflies, the moon reflecting in the wake, making a dancing silver lane. While the wind, gentle yet strong, fills the curved spreading canvas. Over the arch of heaven moves the lady of night, chased by coming day. Gently she sinks to abdicate to her lord, the king of the kings of the heavens. Bright and red and full of life, he sweeps majestically into the sky. The wind still holds steady and firm. Circle of time repeats itself again and again for the days run into a week. Curious her, Windward Island, Trinidad, all are pussed. Faintly the coastline of South America lurks just on the edge of the horizon in distinct black pencil smudge. South and further south sails the slope. It passes into that stretch of ocean which turns from deftless blue into a strange muddy brown caused by the mighty Amazon spewing forfeit waters into the sea. Through the muddy waters, out into the blue once again. High up in the mast head, the watch is ever searching for the small fleet which is known to lie ahead. And behind the slope, miles and miles astern, all out of sight, sails another fleet, proudly bearing the flag of Spain. But the watch is looking ahead, not astern. And then, then is a shot. Fails ahead! Then is a rush for the sails, voices jabber, fingers point, hand shields, sunlight from eyes, and they're in a way in the distance of the ships, looking small like pocket handkerchiefs. And the coast of South America begins to loom large and take shape. Cleaving through the rollers, the slope draws near and Morgan, toward the flying gull, views it with suspicion. They're flying the English flag that were an enemy of waters. And all guns train on her. Nearer comes the slope, and Morgan, the glass to his eye, watches the long boat lowered and put out towards the flying gull. That man in the stern, can I believe my eyes? Vice and David, it is? Yes, it's Hunter. That's I. Have a guard ready to give my visitors a welcome. What game does Jeffery Hunter think he's playing? The moment he comes aboard the flying gull, I shall see that he's pleased in irons. And then, patch eye, you can lock him below. I'll deal with him when I'm ready. Jeffery Hunter had not realized that this would be his reception. It is imperative that he makes Captain Morgan listen to him and believe his story. Will they meet in the next episode of A Float with Henry Morgan?


Welcome to module 8 of NPTEL NOC course on point set apology part 2. So, we shall continue the study of local compactness. We have already observed that in our definition of local compactness regularity is built in under certain mild additional conditions like T1-ness, it will be a T3-space and then it will be a hostile space also ok. So, what we would like to see is that under this local compactness many stronger separation properties are possible. So, here is illustration let us begin with that. The first thing lemma here is take a compact set compact and close subset of a locally compact space that n be an open subset in x which contains k.  Moreover there is a stronger thing here that this kind of separation is actually coming from a function namely there is a continuous function f from x to 0 1 such that f of k is singleton 0 and f of the complement of u is singleton 1 ok. It started with the u is contained inside which contains sorry started with n actually n is a neighborhood ok. So, I could have stated is f of the complement of n is also singleton 1. So, this is also correct anyway. So, how do we how do you how do you perceive this one? Local compactness ensured for each point like this suppose if point in the neighborhood then there is a neighborhood u contained u bar contained inside n and u bar compact. From a point we are now extending it to a compact sector. So, that is the first thing and then not only that you can actually get functions like this like your results type results are true. So, proof is not all difficult let us go through the proof. For each point x inside k by local compactness we can find open sets u x such that x is inside u x u x bar contained inside n ok this is point wise.  Union of u n so, I will take it as u each u i is contained inside n.   So, first part is over now see u bar is compact and it is we are working inside a locally compact space ok.  So, this is the standard Uridon's characterization which we have seen in the part one of any normal space any disjoint open closed subsets of a normal space can be separated by function itself like this that is what we have used there. Now, you define f from x to 0 1 finally, you want a function from x to 0 1 by the formula fx is gx inside u bar so far outside u bar I will extend it by identically 1. In fact, on entire of x minus u I can put it as 1 see g bar minus u is already 1 therefore, it will agree on these two closed subsets on the intersection of these two closed subsets and on this subset its g is continuous on this subset is 1 that is also continuous therefore, f is continuous from x to 0 1 itself restricted to u bar it will be g and it g has this property g of k is 0 g of u bar minus u is 1 right. So, whole function we have got this way ok. As an immediate corollary we obtain several interesting results the first one is quite useful in measure theory ok. In a locally compact host of space given a compact set k contained inside an open subset v there exists a continuous function g from x to 0 1 such that g of k is 1 g of v complement is 0 by interchanging g to 1 minus g you can always interchange the role of this k being 0 and g of this one being 1 that is always possible ok that is not that should not confuse you. So, what it says is in a locally compact host of space given an open set v containing a compact set we have g k equal to singleton 1 and g of complement of v equal to singleton 0 ok. All that I do is we know that every compact set in a host of space is a closed set. So, therefore, we can apply this lemma that is all ok. So, this is just a real statement of that part of a lemma in a special case that is all all right. So, this is what we get next by the way we will later on prove the same thing same result in a different way also ok different proof also we will give a locally compact space is completely regular is another corollary why I am calling this is a corollary because they are easy consequences of our lemma. A locally compact space is completely regular regularity we have already seen ok it is built in in our definition ok. So, that is again complete regularity means what given any point in x and a closed subset f outside you know x must be outside f this joint from x v k we have to produce a map continuous function g from x to 0 on such that g x is 1 g of f is singleton 0 ok. So, apply the previous theorem with k equal to singleton x and f equal to complement of n equal to complement of f ok previous lemma you can apply that then you have what it over ok. So, locally compact spaces are completely regular. So, this is what we meant by saying that there is more separation in locally compactness than which is obvious from the definition namely the built in built in property was regularity here now we have got complete regularity right of very different nature compact set and open subset of that one can be separated and so on. All right let us go ahead with this one a locally compact half star space if you put half star space or just put t 1 s that is enough it is a tick now space. Why because we have already seen it is completely regular plus t 1 is by definition tick now space that is all ok. Now, we shall go to a different kind of result for local compact spaces. So, local compactness is sits in say you know in a juncture of so many different kind of concepts in topology. So, this is what we will try to see. So, next thing is the bare category type result for locally compactness ok. In other words the Bares theorem is true for locally compact spaces. So, every locally compact space is second category or you can say it is a bare space or you can say it is not first category and so on. Let me tell you what it is also. So, first of all the this this result was proved for complete metric spaces in part one. The proof is here is more or less similar in fact it is even simpler. Simplicity comes because of the local compactness you will see that. So, for proving this one in the in the case of metric spaces we have to do quite a bit of preparation and all that here everything is done now. So, we can immediately prove this one. So, what is the proof? Start with any countable family of nowhere dense subsets of x. We must prove that the union of all these f n's ok will not cover x actually we should prove stronger thing namely it is complement of union of all these f n's is itself dense in x. That is what we will prove and that is precisely the staying that the space x is bare space or it is second category ok. Start with a family countable family each of them must be nowhere dense even the complement will be dense. So, that much we will prove not only just it is non-empty is enough but we will show that it is dense. So, let u be any non-empty open set in x. We must show that this in the complement is dense none of these f n's we will intersect will take away a union of f n's will not take away this part is what we have shown right. So, choose any open set any point inside inside this open set ok. In fact, point is not necessary first of all we will see that take a neighborhood u naught of that only for that purpose I am taking a point here such that u naught bar is compact and u naught bar is contained inside u. So, what have we used here local compactness only for that part I want to start with a point inside inside this non-empty open set ok. Now u naught wise compact ok. Now there exists an open set non-empty open set u 1 bar contained inside u naught minus f 1 because f 1 is nowhere dense f 1 closure does not contain any open set non-empty open set that is the meaning of nowhere dense set. So, if closure does not contain f 1 also does not contain any non-empty open set that just means that complement of u naught minus f 1 is non-empty ok. So, there it is a non-empty open subset ok exactly same way take a point here then take u 1 and then closure of that and contain inside that. So, I am not writing all that here.   So, what we have we have a decreasing sequence of open sets each contained inside the closure of each contained inside the next one inside the previous one and each of these is compact take the intersection ok. This w is clearly is in the complement of all the f n's why because even one of them does not intersect f n. So, this does not intersect the union of f n's ok and of course, w is contained inside u all right. u i bars are decreasing sequence of non-empty closed subsets of the original u naught bar and u naught bar is compact that is all I need ok once u naught bar is compact all u n bars are automatically compact anyway ok. So, a decreasing sequence of non-empty closed subsets in a compact set is intersection is non-empty. De Morgan law applied to compactness union of you know if finite union does not cover the whole space then the entire union also will not cover the whole space that is the property of compactness. So, we have produced a non-empty open subset inside this u outside all of u f n's. So, that is all what we wanted to prove ok. So, you see in three or four lines here yes I elaborated it fully we have complete proof of Bayer's theorem for locally compact spaces. Now, I am going to discuss something much deeper, but here I want full participation from you people. If for some chance you feel that this is too much for you you can you can you know you can take a leave I mean you can take ok I will I will think about it and so on you can you know sit on a side, but make it since an attempt to try this one ok. This is the way one becomes a mathematician or one starts doing research work.  So, immediately you should ask what is common between them? What makes them both of them a bare space ok? Where is the hypothesis coming from? Apparent if you just look at local compactness of complete metric space they are nothing to do with each other. Indeed there are lots of complete metric spaces which are not locally compact. There are lot of locally compact spaces which are not matrixable either. So, what is this makes that both of them bare spaces? So, I have tried to explain this one in a set of exercises which are all doable. In any case you can try them and then your TAs will help you to solve them if you have difficulties only if you try otherwise you can take it easy. For example, question and so on we will not bother you with this one that is what I meant by you can take it easy ok. So, let us go through this exercise just I am not going to tell you what is it but they are all easily done. The first thing is every compact regular space is a bare space. See what we have done is locally compact space is bare space. But when I say compact a compact in number definition may not be locally compact remember that. So, I have to put regularity here also. But the proof is if you have understood locally compact space bare space just now you can prove this one also. The next thing is once you prove this one you can prove this one namely every compact horse door space is a bare space. This is the consequences of exercise 1 if you think properly. The third thing is every open subset of a compact space is a bare space. If you prove 3 you have already proved 2 but why I have given it like this first you prove this one 2 and then deduce that every open subset of a compact space is bare space that will take a little more effort ok. Now all these three is going to tell you that there is more bare spaces than just locally compact spaces or complete metric spacing only for that reason I have to prove this one. Now comes the crucial thing every g delta in a compact horse door space is a bare space ok. So, every g delta concept comes here g delta remember just means that intersection of a countably many open sets. Open set is a in a compact horse door space is exercise 3 from open set you have come to g delta here. So, this is set of the first 4 exercises ok. Once you have done that now we will turn our attention to complete metric spaces. Ok, indeed the metric space is not completely is all that necessary you can just work all these things in a pseudo metric space ok. Pseudo metric space is just a metric space all other properties are true but property 1 d x y equal to 0 need not imply x equal to y that is all I am just recalling that ok. So, take a pseudo metric space it will also give you a topology show that every closed subset in it is a g delta. So, now we have got one single property common to both these families pseudo metric spaces every closed subset g delta and here you know every g delta set in a compact horse door space is a bare space. So, that must be something that is the key for these two things to have bare space ok. So, let x d be a bounded and complete metric space the boundedness can be always attained because every metric space every metric is equivalent to a bounded metric by just taking d divided by 1 plus d that will be a bounded metric always it will give you same topology. So, this is a mild restriction but it is needed in this exercise start with a bounded and complete metric space ok. Now, wide tau is a compact horse door space ok let tau d denote the topology induced by this metric on x deem induce a metric ok. So, that I am taking here let e from x tau d to y tau be a topological embedding see here you have metric here you do not have metric but both of them have topology in the topological sense this is an embedding embedding means what continuous mapping such that onto the image that mapping is a homeomorphism. Suppose for every bounded continuous function f from x to r that is defined on x you have an extension of that to the whole of y by continuous extension. Now, why can I can talk about continuous action but this is embedded here. So, you can think of this as subspace here.  So, suppose this property is also true all these I am hypothesis here for each x belonging to x let fx from x to r denote the function the distance function fx of x prime is distance between x and x prime this distance is the metric coming from x here ok. With this function I want to define a map d hat from y cross y to r by the formula d hat of y 1 y 2 is the infimum of f fact of x on y 1 minus f fact of x on y 2 take the modulus. So, these are all non-negative real numbers take the infimum. So, there you will get a non-negative function here ok on y cross y what is this this is has all the characters of metric except it is not exactly metric it is pseudo metric on on e on x x thought of a subspace of y where e x and e x prime distance d hat the same thing has the original distance. So, what it has happened is d hat is an extension of d on x ok to the space y. So, d hat is not a metric perhaps it may fail to be a metric, but it is a pseudo metric d hat is continuous on y cross y remember y has already a topology on it right d hat will introduce a topology we have to wait for that yet. So, but this d hat is continuous on y cross y and hence identity map from the original y tau to y the metric topology tau d twiddle this is continuous ok. The fourth part is the image of x under e is a closed subset of y tau d twiddle in the new topology metric topology pseudo metric topology is a closed subset all that y I have done I have put an arbitrary metric space inside a y tau which is a compact metric space ok now e x becomes a g delta set in y tau ok. So, this will really complete the picture why a complete metric space and locally compact spaces have this beautifully one property namely being bare space they share this property ok. So, that is the conclusion of this last exercise which you can read by itself, but you will have to wait to see the full thing ok you will have to wait till chapter 5 when we will do compactifications of various things. Namely a locally compact half dog space can be compactified what is called as one point compactification whereas a pseudo metric space can be compacted where as metric space can be compactified what is called as stone check compactification and so on ok that will fully complete the picture, but for preparing for before you can take this as granted and then you would have understood the difference here ok. So, you have plenty of time to understand this one not necessarily by the by the time we come to chapter 5 or finish this chapter 5 you can again come back to these exercises part of these exercise you might have solved before by that time and then you will be able to see the full picture ok. So, that is it for today. So, thank you so we will meet again next time.


It was you who lit the match what you thought you didn't have a plan. I don't know you tell me Does the title of this family seem too long? Let me know your thoughts. My name is brandy Keith Avery and this is just my pain Hey, what's going on everyone? Thank you so much for tuning in to my opinion slash review for the girl in the spiders way up a new Dragon tattoo story. I really do appreciate it now That title is super duper long the original title was just a girl in the spiders web If you did not know this film is loosely based around the girl in the dragon tattoo They came out in 2011 With Rooney Marrow Daniel Craig in a David Fincher, but none of those people decided to return in this one We have Fedet Alvarez as the director and Claire Foy as the hacker at Lisbeth slander now the girl with the dragon tattoo. I did see it I thought it was a decent film. I did not see the original the Foreign version which are here is much better and like I said this one is loosely based on it The director is Fedet Alvarez if you don't know him He did a don't breathe in 2016 currently working on don't breathe to great movie and also evil did Which I haven't seen Now this family here seemed like it would interest me And it did for the most part. I really do love a vigilante plot story so where especially if the Protagonist is perceived to be weaker than her foes But they still come through and kick a whole bunch of button things like them And that's really what I did like about the trailers for this film and leading up to it You know towards the beginning the first act Claire Foy. She does she does a great job She was also in first man. She may get an award nomination for best Female supporting actress I believe for that film. I mean, I think she deserves it in my opinion She did better in that film than this one, but I mean she still did a great job here I liked her character. She wasn't playing in games. She didn't waste any time.  She didn't take pleasure out of it but you know, she went around and she's seek justice and That's what I thought about the most of this film was going to turn around But it goes into a little deeper is with the you know criminal organizations national security Was it the national security? What's the agency administration? I feel like a dummy. I'm gonna look that up real quick. I forgot Now you're a national security agency and just some some other things hacking and things like that That all that I really didn't find that interesting even though some of the surrounding characters I did like they had to do that were you know, they had to do with like the keep stand feel he was a Dre in What was a movie called? Get out and he was also in side of body early this year. He did a great job I like seeing him on screen But you know, there was some times like I think after the first act I was okay Where are we going with this? I thought that this was a vigilante chick that was going around kicking in people's bus That beat up women. Oh, but now we're trying to steal a program. Okay, what does the program do? Oh, wait a minute now We're going back to the original film that has to do with her little sister, you know kind of just what's going on So I like the very beginning of the film I like the very in kind of how I made everything circle back around but the motivation behind it all was just kind of annoying to Me, I don't like when bad guys are wanting to take their anger out on the whole world Just because they're mad of the sibling or somebody they knew person over here just hurt that person You don't have to hurt everybody or maybe I just didn't understand this film at all because I was ready to go I mean, I don't know. It was just okay I didn't enjoy it as much as I wanted to but at the same time It's not a horrible film if I have to rate the girl In the spiders web out of a one out of ten. I'm gonna give this a six out of ten Yes a six out of ten, but guys that is just my opinion Have you seen the girl in the spiders web or do you want to see it?  Do you agree with me or do you disagree with me? Let me know down in the comment section below. Let's get this conversation going to keep it fun If you like this video go ahead and give me the thumbs up and if you don't that's fine But you can still subscribe to my channel and also look me up on social media Facebook Instagram Twitter all that good stuff It's right there at the bottom of the screen And I made it very easy by providing a link to all that good stuff down in the description box below But guys, I just want to thank you again for tuning in to my opinion slash review The girl in the draw the girl in the spiders web and before you go don't forget that my name is Brennan He they free and that's just my opinion peace


Better to use a luffa, washcloth, or your hands to get clean. Sorry luffa lovers, but it's actually better to use your hands. Because luffas can harbor nasty bacteria, using your hands to get clean is the recommended method. It's also not advised to reuse a washcloth unless it's cleaned immediately after use and fully dried. If you simply can't cut ties with your luffa, make sure to sanitize it weekly either in a dishwasher or with a diluted bleach solution.