Good afternoon and welcome from the United Nations Headquarter in Vienna where we are sitting here at the Vienna International Center at the Commission on Narcotic Drugs Meeting with three wonderful activists from Latin America, Isabella from Colombia, Sara Snap from Mexico, and Luciana from Brazil and today we will discuss what are the trends in drug policy reform in Latin America and my first question to you is that how much Latin America is highlighted at this meeting? Do you see any any any chance that things will move here and then what can what can your region expect from the United Nations? Well I think the first thing is really start together. We have a lot of problems in common and being together thinking in solutions that works for all Latin America is something that really help us to find new ways to resist to all the the dangerous thing that are in our horizons. For example now in Brazil we are trying to resist the the perspective of leaving harm reduction behind and go to abstinence as a option as a governmental option to treat drug issues. So being together and try to understand how other countries are leading with this kind of thing help us to think in a organized way to to be here and to try to make it it's an international discuss. Well I think Latin America continues to play an important role here at the CND. Mexico has the chair of is the chair of the CND this year and so in one way that's good because it allows us to be very close to the chair and to what's going on around the modality's resolution and what's going to happen in 2019 which is kind of we're in view of what will happen in 2019 but also what that means is that Mexico has to play a moderate position and they have to ensure that they don't come off as biased or pushing for anything too too progressive and so they have to make sure that they reach a consensus and so they can't necessarily be there saying no we need to make change we need to make change and then I think so I think that there have been we've maybe lost them as as the strong strong voice that they were but I think it will be good for them to be moderating the negotiations in some of these resolutions that are coming up here in Vienna. On the other hand there's certain countries that in years past have been more progressive so in Guatemala they they used to have a discourse which was much more progressive and what we heard this morning was not quite in that same line. I think Uruguay had a great presence today and they are moving forward with their reforms and they are talking about human rights being at the center and the cannabis regulation that's happening in Uruguay so there's certain countries that are that are speaking about these issues but I think you know we have elections coming up in the region so in Mexico we have elections in Colombia there will be elections Costa Rica has elections so so we're going to see some shifts in in the dynamics and the Latin America of today is definitely not the Latin America of 2012 or 2015 where we saw more consensus in the region but whereas now we see a lot more fractures in in how the region moves forward. Yeah just to echo what what you both said and compliment the fact that not only Latin America is not as cohesive and as strong in terms of advocating for reform in terms of drug policy but that dispersion has led to also not condemn the horrific things that have happened ever since 2016 so I was for example surprised at the fact that the Philippines spoke today and there was apparently no voice from the region to say that what is happening is not acceptable and as for your second question about what do we expect from UNODC I think that for Colombia at least we need to have more vocal support not only from UNODC but from the whole international community to the peace process and the peace agreement it's a very dangerous conjuncture right now with the elections coming up and some of the political forces that could be the potential winners of this election this year those are political forces that want to go away and go without the peace agreement basically so we need a strong voice from the international community from civil society abroad from like-minded countries to support the peace process and support specifically peace peace processes that align with eradication of coca-crups that are voluntary that are aligned with development that are aligned with human rights and that are not based on repression and and forced eradication. So let's start with the peace process in Colombia can you explain to those viewers who are not very familiar with the situation in Colombia like where this process started and how do you evaluate this process in terms of like how it affects the people who are growing coca-crups in Colombia. So the process is now in its initial implementation stage the peace agreement was signed in August of 2016 then it was held to a referendum public referendum and then no one by very small margins but it won nonetheless and this means that the the process has very fragile legitimacy for the Colombian public nonetheless the government has carried on with the implementation of the peace agreement and what it means for coca-growing regions is that some voluntary programs have gone into those regions to make cocaleros like part of the regional economy the main challenge of this is whether or not the government will fulfill its part. So far cocaleros have actually eradicated their plans and have been very responsible in the fact of agreeing to be part of this voluntary process and the government has not completed its part of building roads making sure that their access to credits access to land titling which is a main component of how you can go into rural development. So it's it's a critical moment because the trust between the community stores is very fragile these are communities that have been sort of abandoned for 30 40 years even more so now the government is saying well you just have to eradicate your plan and then we will come and make sure that a livelihood will happen but when it doesn't happen then what is left is that there are other criminal groups making pressure for people to continue growing coca and this leaves them in a very dangerous situation into which they can be subject to violence from these groups or they can be subject to adhering themselves to the program but without a livelihood to support them. And this again is why I go back to the this big consensus and supporting the peace process and supporting the fact that it's going to be a long process is not going to have results in the very short time is going to have results in the long term and so we have to be patient with the fact that that's that's the the smart way to just a little bit on I mean Isabella is definitely the expert on this but there's as she said new criminal groups might be entering the market in Columbia and I think that this is something that we need to be aware of and and be concerned about in Columbia human rights defenders and particularly human rights defenders of the environmental human rights defenders environmental rights defenders are having are being murdered at a very high rate because of their work people who are signing up to the eradication programs are also in very vulnerable positions of losing their lives and so this is one of the consequences that we've seen because I agree we need to support the peace process the the next government that comes into place will need to to try and have continuation of these policies because it's the only way that in Columbia they're going to find peace but there is a there's a huge issue around that if we don't deal with the issue at hand which is cocaine and that the majority of the coca that's cultivated in Columbia is destined to be to be turned into cocaine and then exported to the United States and Europe we have to address that issue if we don't we're we're really ignoring the true market that that is existing and so I think you know the rural development that's happening and the programs that are that are being done to address access to opportunities for those communities is highly important and needs to be continued but at the end of the day the government will at some point whether it's this government or the next government or the one after will have to address what are we going to do around the issue of cocaine and believing that voluntary eradication or even forced eradication is going to change something means that we are continuing a discourse of that these plants are no longer going to be on this planet that people are no longer going to be consuming and that's something that needs to shift because we know people are going to continue to consume substances and cocaine is one of the substances that's consumed in a primarily recreational fashion not in a problematic use around the world and so we have to figure out how do we possibly create a market for regulated cocaine that is grown under fair trade criteria and where people are willing people in the global north are willing to pay a higher price knowing that the product the majority of that money is going to the producers rather than to the people who are trafficking and who often are are undertaking violent activities in order to protect the product as it moves towards the the market and if you want to add yeah I figured and to to add on that yeah I I guess it's it's still shocking to see a lot of hypocrisy around harm reduction because you see this very progressive country speaking about harm reduction and how they've improved the lives of people who use drugs respecting their human rights which is all good and well and we all continue to support that but it completely ignores the fact that the drugs are coming from somewhere where people are being marginalized stigmatized criminalized deprived of their human rights so as long as those progressive countries did not broaden their spectrum of how they consider harm reduction what it is to be and we open up the debate also beyond cannabis and also two other plans to coca to opium poppy and say that this markets will not disappear this markets have also a potential if they are regulated to be a source of livelihood for these communities from our countries and just to not continue this hypocrisy of saying that harm reduction is only for drug people who use drugs it's also for people who grow this plant and it's a matter of social justice that we take into account both both populations and not only those who are using drugs in the urban settings but also those who are producing the plants of which the drugs come out of so actually basically you're saying that there is an elephant in the room and that's the consumption of cocaine in in like the let's say in like in the north part of the world like in Europe and then America and then that you need that we need like a kind of regulated market for growing and distributing cocaine trafficking it actually to to the north but still we see that even there is a very strong opposition to cannabis legalization so do you think that there is like a real chance in in the near future that there will be a regulated market for for cocaine products how how do you see the chance for that I mean so for the last two years action technical social in Colombia who I collaborate with and the Universidad de external Colombia have been working on a study a perspective study on what a regulated market for cocaine would look like in 2034 understanding that this is not going to happen tomorrow it's not going to even happen next year or in the coming years but that we really need to be future focused on what could be possible and from that we've identified specific variables that we need to be keeping an eye on and how those might change and how that might affect a possible market but I do believe that and we believe that at some point politicians and someone who wants to put their will and their political capital behind this will look to the to the to the cannabis example and say the market is different and the market will be structured differently but that a regulation is the best way to repair some of the harms of prohibition and and that's having that social justice focus and the Minister of Justice today in in the side event spoke about that and he actually spoke about the cannabis market in Colombia becoming an economic opportunity for the country through the through exportation licenses which are included in their medical cannabis law and he the last comment he made you know everyone else had spoken in the last and he said and you know cocaine was used with medical purposes and we should look into what that could be those medical and scientific uses that are important that we might have to you know move towards I mean in our study we're not we're advocating for the legal regulation of a recreational or adult use understanding that the majority of people and we did a survey with cocaine with people who use cocaine who are using cocaine use it less than 10 times a year and they use so that's not even on a monthly basis and that they don't necessarily want to register with the government they don't see their use as a medical use they see their use as it's part of enjoying and pleasure you know we try not to ever in these wall in these halls speak about why do people use drugs they use drugs because if they feel it feels good you know not always but you know they're leaving pain in some way or they're having a good time and that's something that that I think we try not to talk about and so so while I don't think that it's around the corner cocaine regulation I think that our job as civil society is to put the proposals on the table to do the research to become experts in these issues so that when someday a politician turns around and says we should think about this then we have something ready because otherwise this is going to take generations and I think for Columbia after 52 years of a conflict they don't have time Mexico we don't have time we cannot keep at the same pace that we've been doing it's just too much on our communities and every day we think oh maybe this is the tipping point maybe this is too much you know Ayotzinapa maybe that's but it doesn't it's hard to know when we're at this breaking point and when you ask analysts you know in the example of Mexico where are we in the scope of war in the country they say we're not even halfway there and that's really really scary and I think that the other piece of the coca cocaine debate is around smokable coca in Latin America and the need to get harm reduction services to folks who are consuming in those spaces and what are also the circumstances around their consumption of that of those drugs not necessarily just their consumption but rather what are the opportunities they need and Brazil is one of the examples where that's really showing up and where they had great programs but now they're being taken away. Actually there is something that I think that put all the countries together in Latin America that's the violence and how people who is living for products or sale drugs people are dying in Brazil nowadays we have more than 60,000 people who is dying every year and we really can't stand on this line we need to change the things and we need to do this right now we can't stand like this people are in jail people are being killed and it won't change the fact that people are using drugs because they feel like so it's something that really we really need to be together to try to change all the violence that is involved with the drug use not the drug use but how the government's leads with the drug use there is something that I really think it's important for all the countries too it's how the judicial system is treating these issues for example how can the judiciary or other judiciary or other institutions that are law institution is continuous to put people in jail for crimes drug crimes is continuous to be okay with so many people dying how can we continue to do like this and we know who is dying in Brazil who is who is in the jail who is in who is dying is our black and poor people we all know this 70% of our women in jail are there because of drug crimes so we cannot continue like this we really need to change how the things are going and the responsibility needs to be divided not on not just on the government but also with the judicial sisters in my point of view so what's new in brazil in terms of the drug war especially with police and favelas and what is the position of the government now do they want any reform are they open for any reform no we don't have good perspectives nowadays actually we have a military intervention in Rio de Janeiro that makes things even more difficult our government put a military chief in front of our security policies and people are completely afraid of what does it means enough when we look we also have elections in October and we are seeing these as a political option to to obtain more sustained sustainability for the program but it's so dangerous that we are doing this again in Brazil we have the military assuming the responsibility that is civil and we have a lot of army in the streets in favelas actually it starts with Rio de Janeiro but nowadays we are looking forward and seeing is it going to be our reality in many states we still don't know but we think it's there is a chance that it happens and we are all worried about it we don't see a good perspective in a short period what about Mexico do you see the light in the end of the tunnel the terrible drug war yesterday we had a session where we had a we had a great activist speak Maricela was speaking about how losing her sons in the war and drugs and and that's how dangerous the life of human rights defenders in Mexico so so so do you see any hope it's very hard to see hope because we had militarized our public security in 2006 and that's where we saw this increase in people being killed and people being disappeared people being displaced and what happened this last December was that essentially the government had been operating under a state of exception in allowing that militarization to take place and in December we passed a law which formalizes and legalizes that militarization and so and with the elections coming up we we also see that as possibly being used as a way to intervene or repress social protests that might happen around the election so we have a lot of concerns around this interior internal security law which is what it's called and how that might impact human rights defenders and the situation on the ground and and how that intersects with the the war on drugs and so I think I think that's what we're seeing in Mexico is that is the need to really I see the light at the end of the tunnel as movement building that's really where we're going to see a change and where if we are able to show the intersectionality of how the drug war issue with the human rights is completely issues are completely connected with environmental defenders with access to to essential medicine so all of these issue feminist organizations you know how do we really bring together all of these groups people family members of people who've been disappeared how do we bring together these groups so that we can create pressure from below to the to the political level because what we saw is we had this medical cannabis law that was passed also in December and what really was passed was the capacity to import CBD products from foreign pharmaceutical companies and so that's not going to provide any sort of economic stability to people who are currently cultivating cannabis in Mexico it doesn't face the reality of that we are a country that cultivates cannabis and has for for centuries and instead it's it's it's what the government and legislators it was the minimum that they had to do to appease the media that we had you know rallied around the case of a young girl who has epilepsy grace and so and and because of the Supreme Court case that gave the right to cultivate to four individuals in 2015 based on the constitutional right of the free development of personality so we had these moments these historical moments that were very important in drug policy and I think we were hoping that we would get a little bit further with ungas and now that it has it has calmed down it's hard to see where to go next I mean proposals there are many proposals why are we not legalizing the poppy crops that we had currently have in order to make morphine which is something that is not readily accessible to the to the population as a pilot project as a as a means of helping create livelihoods for those rural campesinos in the state of Guerrero which is the poorest state in the country that we should regulate the cannabis market because all of the west coast of the United States has regulated and Canada is going to regulate and will be the only ones in North America that haven't regulated cannabis so so we have these reasons we have these things that we should be doing we should have a supervised consumption site in the north of the country for people who are injecting drugs but it's hard because there's the government is not going to take any steps as long as we're in an election period and so essentially we are right now trying to be close to the candidates and see whether any of them would take on some of these proposals but it's already a very divided issue and so it's hard to see how they might really take that on and so that's where I I see the most hope in how do we work together across movements how do we build something that that allows us to to understand that our issues are common and that we need to be working on them together a few weeks ago there was a sacred plants conference in Ajijic Mexico and that was also interesting because there were lots of shamans and indigenous groups that came and that's an issue that we need to be working closely with and we don't see it that much in these spaces but with drug policy reformers and them I mean we I don't need to be an expert in peyote to be able to advocate for their rights and to be able to advocate for you know expanded access to the plant and conservation of the plant but then vice versa they they can also take on our issues around well the ones I represent around you know people are using drugs on a recreational basis and so how do we regulate those markets how do we get harm reduction services to people who need them how do we get education to people who who who need it so it's really about how do we share issues and how do we see this within a social justice framework all of it because at the end of the day and I think California's legislation is the one that best embodies this regulation the regulation of markets really needs to be within the social justice perspective so that we can begin to reverse some of the harms that have been caused by prohibition in the end of our discussion along me a personal question because I think it's very sometimes it's very depressing to to work in this field you hear so many sad stories about you know loss of lives and and sometimes human rights defenders are in danger just because they are doing their work so what is it what keeps you motivated and keeps you you know still working in this field like still fighting for for human rights where where do you get your strength from I think it's precisely there with the people and their families and how can we work to try to stop all these horrible things that are our day by day in our countries I think that's it I would agree with Luciana I mean it's that it's knowing people who've been so impacted or being part of communities that have been impacted and this is I mean we're all living a trauma for many of those in Latin America of just the security situation or the insecurity situation and and recognizing that the way that the countries are moving is not the correct way we're we're we're warmongering instead of building peace because it's much harder to make peace and so I think that yeah it's it's the people and it's also just knowing that that we're on the right side of history and this might take generations but we're just gonna keep at it and we have to keep going and and because it's just rationally pragmatically on a justice level it is the correct thing to be doing and so this can all go on and this is a simulation of something and it's really important that we're all here but to have the community that we build both in these places on an international level and then also on a national and local level that's what gives me energy is knowing that there's so many smart people who are just really passionate about this issue and that's how I feel and I just want to keep putting proposals on the table really thinking about the future and making sure that together we are coming up with with solutions rather than just saying this isn't working but rather know this is how it could work I think that also additional to what you've said because the strength comes from seeing the resilience and resistance from the people who are enduring them the most harms but also acknowledging this the place of privilege from where we all stand we are all here in a place of a lot of privilege and the only way to serve that privilege just leads to name the harms that have been caused in the name of the war on drugs and those are mostly not harms to me personally to my country they are but but mostly to people growing in rural regions for people women who have incarcerated for very very minor offenses we're just trying to provide for their families those are not harms to me personally but I am I have the privilege to be here and speak on behalf of those I definitely feel privileged to have this very interesting discussion with you thank you very much for accepting our invitations and thank you for those who watch the show


All right, so we are making a fair amount of progress at understanding how to understand equilibrium constants for molecular reactions. We know that the equilibrium constant is a product of partition functions, and we've also known for quite a while now that the partition function for a diatomic molecule we can write as the product of a translational piece, a rotational piece, a vibrational piece. We've seen more recently that we have to include a correction for the zero of energy that involves the bond association energy. And lastly, this G term here is the electronic degeneracy of the molecule. So combine all those pieces together, that gives us a partition function for a molecule. So if we now have a reaction that involves diatomic molecules, then we have all the pieces we need to be able to write down what the partition function looks like. So let's go ahead and do that for an actual reaction. So let's take as our reaction this HBr formation reaction, where one hydrogen molecule and one bromine molecule recombine to form two HBr molecules. And we're going to need lots of constants, we're going to need masses and rotational constants and so on for these molecules. So let me go ahead and get those on the board. Masses in grams per mole. I'll just write a little chart of these values for H2 and Br2. Also the rotational constants, those will be in units of Kelvin. Vibrational constants, again in units of Kelvin, bond association energies for these molecules, which I'll write in kilojoules per mole. And lastly, the electronic degeneracies of the ground states of these molecules, those are all just one. Each of these molecules has an electronic ground state that's not degenerate, that just has a single degenerate ground state. So those are the values we're going to need to plug into this equation. The equation that we need to plug into is this expression for the equilibrium constant. So I'm going to take each of these partition functions raised to a stoichiometric coefficient. So my reaction is products or HBr with a stoichiometric coefficient of 2. The reactants are H2 and Br2, each of which has a stoichiometric coefficient of 1 on the reactant side. So that's H2 to the negative 1 and Br2 to the negative 1. So those show up in the denominator. So I'll write this on a new line. So I'm going to take this expression for the partition function for each of these diatomic molecules, plug it in for HBr and the numerator H2 and Br2 in the denominator. The numerator is going to look like this expression for HBr. The partition function, remember, is squared. So instead of taking this to the 3-house power, I'll square it and make it to the third power. Likewise, the volume is going to be V squared. Each of these terms is going to show up squared. I'll go ahead and insert the symmetry number. The symmetry number for HBr is just one symmetry number when I get to it for H2 and Br2 because they have a twofold symmetry. I can rotate them, exchange the two H's, exchange the two Br's, and I have the same molecule. That symmetry number is going to be 2 for the H2 and Br2 molecules. All right, so now we have an expression for the partition function of HBr squared. I'm going to now put in the denominator a very similar looking expression just for H2 and for Br2. And those are not squared, those just show up to the first power. That's the term for H2 in the denominator. I also need in the denominator an entirely similar term for Br2, which I'll run out of room for, so I'll write that. This is still in the denominator. So there's the third of my partition functions now for Br2. So that's a fairly ridiculous looking expression. We can simplify it a little bit. Some of these terms, because I have something squared in the numerator and something very similar times another one very similar in the denominator, terms like this v squared will cancel a v and a v in the denominator. Likewise, 2 pi kT over H squared to the third cancels 2 pi kT over H squared to the 3 halves and another 2 pi kT over H squared to the 3 halves. See, these temperatures will go away. I can't cancel any of the terms that are specific to each molecule, any of these terms that differ for the different molecules. So I can't cancel the masses because the mass of HBr is not the same as the mass of H2. I can't cancel the rotational temperatures and so on. So what I'm left with after that cancellation, so I'll collect the like terms together. So for example, I have mHBr, this is actually cubed, divided by mass of H2 to the 3 halves, mass of Br2 to the 3 halves. Likewise, if I collect these rotational temperatures in similar terms, I have a 1 over 4 from the symmetry numbers for H2 and for Br2. All right, so now I've included the ratios of all these terms in the numerator relative to the denominator segregated by the terms that came from the translational partition function, the terms that came from the rotational, the vibrational partition function, the bond association energies, and the electronic degeneracies. So now it's as simple as I can get it before we plug some numbers in. So if we do the still fairly complicated arithmetic of taking these numbers and plugging them into this expression, it's useful to do that step by step. For example, if I first just take these terms involving the masses, everything that survived from the translational bit of the partition function, this ratio of the mass of HBr squared divided by the mass of H2 and the mass of Br2, each to the 3 halves, that term to a few sig figs ends up being about 20. I'll leave this 1 over 4 term by itself. The rotational term, so 20 times 1 over 4, the rotational term ends up being 0.07. The vibrational term, this long collection of exponentials involving vibrational temperatures, that ends up being 0.23. The term involving the difference in bond association energies, e to the bond association energy of HBr twice in the numerator and then H2 and Br2 in the denominator, those difference of bond association energies in the exponential, that turns out to be roughly 1,400. And then lastly, the electronic degeneracies didn't affect things at all. So we'll do, I'll go ahead and tell you what the result is. If I multiply all these numbers together, the final result is 1,800. So if all we cared about was the partition, I'm sorry, if all we cared about was the equilibrium constant, that's the number, the equilibrium constant for this reaction. And I suppose I needed to have told you, since temperature shows up in this equation all the time, when I did this arithmetic, I had to choose a particular temperature. I did these calculations, these are arithmetic calculations, at a temperature of 1,750 Kelvin. So if the problem was determine the equilibrium constant at a temperature of 1,750 Kelvin for this reaction, we've got an answer. At 1,750 Kelvin, the reaction goes pretty strongly towards products because this equilibrium constant is quite large. But what's even more interesting is now that we know where that number comes from, we can see which of these terms is most responsible for that large equilibrium constant. Why does it go mostly towards product? The largest contribution is from this 1,400 term, which was largely responsible for driving the reaction towards products. That's because the products in this reaction HBr have a bond association energy that's deeper, more negative than the average of the H2 and Br2 reaction molecules. So every time we break an H2 and break a Br2 molecule, which costs energy, we get back more than enough energy to pay for that by forming two HBr molecules. So the decrease in energy pushes the reaction to the product side. Likewise, there's some other terms that push the reaction to the product side. The masses, for example, the reason the translational contribution to the partition function pushes the reaction to the product side is in this case because H2 is such a quantum mechanical molecule that the degeneracy of the translational states, that the contribution from the partition function from HBr is bigger than the product of the two partition functions for H2 and Br2. Because Br2 and HBr are both fairly classical molecules especially at this temperature, H2 is still fairly quantum mechanical. So there's more degeneracy for these classical molecules. The reaction is pushed more towards the product side to convert the quantum mechanical molecules into more classical molecules with higher degeneracy. So those factors are countered by some of these other terms that drive the reaction more towards products. In particular, the symmetry number, the less symmetric, I'm sorry, the more symmetric H2 and Br2 molecules are responsible for this factor of 4. So the reaction is driven toward the degree of higher symmetry of the H2 and Br2. For the rotational and vibrational temperatures, the reason they drive the reaction backwards is because in this case, if we look at these numbers, H2 is certainly a very quantum mechanical molecule with a relatively large rotational and vibrational temperature. But even HBr, because of the presence of the H in that vibrational motion or in the light-reduced mass of that molecule, means that it's also fairly quantum mechanical. So turning HBrs into Br2s and some H2s, which the Br2 is much more classical, that increases the contribution of the partition function at least from the Br2 contribution. So the net result, there's lots of complicated factors pushing the reaction slightly forward, slightly backward, strongly forward, and so on. The net result is this 1800 for our equilibrium constant. So the takeaway after seeing this long-involved calculation, there's a few things to notice. Number one, we can calculate equilibrium constants from scratch. Notice there's nothing here other than fundamental quantum mechanical properties of these molecules from which we can predict equilibrium constants describing how these molecules behave at the macroscopic scale. We can see where that value comes from. In particular, we can see how it depends on temperature. If I were to give you a different temperature than 1750 Kelvin, you could calculate the equilibrium constant and it would turn out different at different temperatures. So the equilibrium constant will depend on the temperature, and now we can see exactly in what way. It's a very complicated temperature dependence, but the temperature dependence is all encapsulated here. This prediction of the equilibrium constant can be quite accurate if we have accurate measurements of these constants, and if we want a really accurate answer if we were to include effects of anharmonicity and other corrections to these idealized harmonic oscillator and rigid rotor versions of the molecules, then we can get quite accurate calculations of the equilibrium constant and how it depends on temperature. And then lastly, some of this cancellation that happened here, this calculation was already bad enough. It would have been a little worse if I didn't have this amount of cancellation that I did have. The reason that cancellation happened, the reason that I had two volumes in the numerator and two of them in the denominator is because I have two molecules on the product side and two molecules on the reactant side. So these coefficients match some of these two coefficients. Not every reaction works that way, so some reactions are a little more complicated than this to deal with, and that brings in some additional complications that we'll talk about next.


Hello all, welcome to yet another session of our NPTEL on nonlinear and adaptive control. I am Srikanth Sukumar from Systems and Control IID Bombay.  And we are now well into studying mathematics which will help analyze the stability of algorithms driving systems such as these.  So last time we had started looking at the notion of uniform stability. So this was the week three lecture one and we saw sort of what is the difference between uniform stability and the notion of stability. So essentially uniformity is here with respect to the initial time. Yeah, we then started to look at an example from the book of Vidya Sagar. This is a very, very classical example by Maserain and Vidya Sagar's book works it out. Why this is such a nice example is because it is a scalar system which can be easily integrated to compute the solutions. And B it sort of gives us some very nice features of stability versus asymptotic stability and things like that. Yeah, so this is rather nice for us. So what we had seen until last time, what we had completed until last time was the proof that this system is in fact stable in the sense of Lyapunov. Now in order to prove the stability we had given an epsilon computed a delta. So this was the critical component, it computed a delta. So we will continue to work with this delta even today. So that was what was required for us to prove stability given an epsilon we wanted to be able to compute a delta. So once we did that we were able to conclude stability and the next endeavor for us was to talk about uniform stability. So we started discussing it but then you know we didn't have enough time and of course we were missing a few steps. All right. So today I'm going to start there and what is it? What are we going to try to do? We are going to try to prove uniform stability and we know from the definitions here that the only difference is the absence of dependence on T0 in delta. So we need to be able to find a common delta for all possible initial times. That is the whole idea. We need to be able to find a common delta for all possible initial times. That is what is the uniformity. So now we do see that in our case where this delta, this initial time dependence comes from and that is precisely from this gamma. Everything else is in fact independent of initial time. It's only gamma which depends on the initial time. And what is this gamma? This gamma was this quantity. This gamma was this quantity right here. It is just a quantity which is a function of the initial time. All right. Now what do we need? So let's carefully think what we need. So given any T0, I can find a gamma corresponding to this particular T0 and then using this gamma I can compute the delta. Now the question that arises is what to do if I want a common delta for all possible T0. So this is where we need to be careful. So let me, before I even try to prove for this specific case, even before I try to prove anything for this specific case, let me try to give some kind of intuition for how to choose such a common gamma in the genetic, such a common delta in the general case. So let's see. So suppose I have, so I'm going to make this a bit bigger. Suppose I have T01 and corresponding to it I get a delta 0, sorry, I'm going to call it a delta 1. Similarly, I get another initial time and corresponding to it I get a delta 2.  And I know that this is of course some kind of an infinite series because my initial time can vary from whatever initial value, say 0 to infinite. So I can have all sorts of initial times. Now which delta should I choose is the question. So I have so many deltas here. So which one should I choose, which one should I choose is the question. Now let's look at the stability definition. What does it say? For all epsilon positive that has to exist some delta such that if initial condition is delta away, then my solution is epsilon away. So notice that this entire, although I didn't mention it, but yeah, so what although I didn't mention it explicitly, all of these are corresponding to, correspond to same epsilon. Because I use the same epsilon and I get different delta values for it corresponding to, you know, so these are all different initial times, right, if I, okay, here notice that the subscript gives me dependence on the initial time. And this quantity here shows me dependence on the epsilon. So basically all of these are for the same epsilon. I want to indicate that here. So once I have this list, the question is which one should I choose? How do I know which delta to choose? And for that we need to refer to this definition here, okay, that I, if my initial conditions are within delta then I want my state trajectories to be within epsilon, okay. Now usually when you have an infinite collection like these, like this, you know, then the problem, then usually what are the choices for which one I should choose? The usual choice is the largest or the smallest of this, okay, usually the smallest or the largest of this. And because this is an infinite collection, I would say the supremum or the infimum of this, you know, sequence somehow, yeah, I will take the supremum or the infimum of this sequence, okay. Now let's see what happens if I take the supremum, okay. So say, so say delta equal to sup over k delta, okay, so k greater than equal to 1, okay. So suppose I take delta to be the supremum, then what happens? Then what happens? Okay. If I take delta to be the largest possible one, and then what happens? Then I'm the surely, okay, suppose, I mean, I'll show you what sort of problem we landed, yeah. So suppose delta 2 less than delta, just suppose, I mean, see, because it's obvious that delta is the largest value of all of these, therefore, there must exist some k for which delta k is less than delta, all right. So I'm just simply saying that let's say it is delta 2 and say delta 2 is less than delta because it is, after all, I chose the delta to be the largest one, yeah. So obviously there are some deltas which are less than this, yeah, excellent. So suppose delta 2 is smaller than delta, okay. Then what happens? So what I know for sure, right, if I choose t0 2 as my initial condition, as my initial time, yeah, at t0 2, what happens? I know that norm of x0 minus xc less than delta 2 implies norm of xt minus xc less than epsilon, all right, okay, I know this. Now what's the problem? Okay, so what is the problem? The problem is I have chosen the delta as larger than this. So the question is, is it true that x0 minus xc less than delta implies xt minus xc less than epsilon? What do you folks think? Yeah. Delta here is a bigger ball than delta 2, yeah. I'm guaranteeing that if I start in the small delta 2 ball, I remain, my solutions remain in the epsilon ball. But now I'm asking that if I start in the larger ball, will I still remain within epsilon? Okay. So the answer is obviously that this is not guaranteed. This is not guaranteed, okay. If I start in the, if at time t0 2, if I start in the smaller delta 2 ball, I'm guaranteed to remain within the epsilon ball, okay. But if I start in the larger delta ball, I'm not guaranteed to remain within the epsilon ball, okay, all right. So what? So the choice of choice that we made of the largest delta k is definitely not a viable choice. It's definitely not a viable choice. So what is the other choice available to us, yeah? Suppose now I choose my delta as the smallest one. Suppose I choose my delta as the smallest one. Then what happens? That again, yeah, so I know that this delta is the smallest one. Now obviously I cannot say that implies obviously that delta i greater than equal to delta for all n, yeah, because I chose the smallest value. So obviously all other values have to be greater than or equal to this value, okay. So great. So for t0i, arbitrary i, what can I say? I know that x0 minus xe less than delta i implies x txc less than epsilon. Now does this also imply that x0 minus xc less than delta implies x t minus xc less than epsilon and the answer is a emphatic yes, emphatic yes. Why? Because delta here is smaller than this. So if I start in a larger ball and my solutions remain in epsilon ball, it is guaranteed that if I start in a smaller ball, my solutions have to remain in that silent ball, okay, alright. So the picture would go something like this. So let me try to make these so-called smaller and larger balls, right. Here you go. So this is the say the smallest and the largest ball and just trying to adjust sets the center. Give me a moment, right, this is more of the center. So what am I saying? So this guy and then I have this guy and then I have this guy, right. So this is of course the epsilon ball. This small blue thing is the delta ball and then the small black thing, sorry the small blue thing is the delta I ball and the small black thing is the delta ball, okay. So what is evident to me that if I know that my solution starting in the delta I ball remain in epsilon ball, yeah, so if my solutions that start in the delta I ball, right, remain in the epsilon ball, okay, then it is guaranteed that if I start in the delta I, delta ball which is smaller than the delta I ball. So anything inside the delta ball is also inside the delta I ball, right. So anything starting here also remains within the epsilon ball, alright. So, alright, so let's, this is evident, right. So this, why even without this picture I can still claim that this implies that norm x0 minus xc is less than delta I, okay, excellent. So what do I know? So this is a valid choice of delta, okay, so what do I know? I must choose the, I must choose the smallest of the deltas possible, okay. So if I have a sequence of initial times and correspondingly I get a sequence of deltas I must choose the smallest delta to get uniformity and notice that this of course is independent, right, I mean it should be obvious to you that this is independent of initial time because I took infimum over k and k is what was our initial time dependence, right, k was the initial time dependence, alright, great. So that's the first thing, I need to know that in order to remove the initial time dependence I need to choose the smallest possible delta, okay and that's important. Now let's look at our specific case, okay, let's look at our specific case. We have epsilon over gamma em where gamma is what depends on initial time, okay, gamma is what depends on initial time. Now the idea is I need to choose the smallest of the deltas, okay, so what is it? Let me be careful. So what do I want to do? So I want to, let me be more formal, I want to find inf over t0, yeah, say greater than equal to 0 doesn't matter, inf over t0 greater than equal to 0 of delta t0 epsilon, okay, alright and this is same as if you may, okay, I hope you agree and because what was delta, the way we found it, the delta way we computed was epsilon over gamma em and gamma as was the only quantity that was bringing in the initial time dependence, okay. Now if my, I want my smallest delta, I would need my largest gamma, right, because gamma is in the denominator, so it's an inverse relationship, so therefore this makes sense. I need to find my largest gamma, okay, so then if I move forward what is the largest gamma, so let's see, soup over t0 greater than equal to 0, okay, so let's look at the expression, okay, so it is I believe, let's move into copy it, let's see if I can copy it, right, so, right, so I make it smaller and so this is what is my soup over, right, okay, so I need to find the supremum of this quantity, okay, now it should be obvious that exponential is not damaging the supremum, so this is actually then, I mean, that's fine, so I will essentially say that this is equal to exponent of the supremum over initial time of minus 6 sine t0 plus 6 t0 cosine t0 plus t0 square, now can some of you already guess where we are going with this? What is the largest possible value of this quantity for arbitrary t0? So we start to hit the problem, this is actually equal to infinity or I mean, if you may, if you want to be more precise, this is tending to infinity, right, as t0 goes to infinity, okay, so last time we spoke about this in terms of increasing functions and all that was not correct, let me reiterate, it was not correct, this function is not monotone or anything, but one thing is clear about this function is that the largest value that this function can take is infinity itself, okay, this function blows up, why? The same argument as before, this t0 square is going to dominate all of these things, okay, it doesn't matter what happens to this, this is just between plus minus 6, this is going to be between plus minus 6 t0, but then this t0 square which is a quadratic term is definitely going to dominate all these kinds, okay, so this entire quantity goes to infinity as t0 goes to infinity, right, so what happens to the supremum? It means that the supremum itself goes to infinity as t goes to infinity, okay, so what happens to the delta? Implies delta which is inversely proportional goes to 0 as t goes to infinity, okay, and this is not allowed, we need the ball to have a non-zero size, right, this ball has to have a non-zero size, but here what are we getting? We are getting that our delta actually goes to 0 as t goes to infinity, and this is certainly not allowed, okay, so therefore it is not possible to choose a delta independent of t0 in this case, okay, so what does it mean? It means that the system we have is stable but not uniformly stable, okay, so now of course I mean there is an additional statement here, yeah, but we will talk about this a little bit later because you have not defined asymptotic stability yet, okay, we have not defined asymptotic stability yet, so we will talk about this at a slightly later stage, okay, so what one of the very interesting systems is that of the Van der Paul oscillator is a very, very, you know, commonly used system to design pacemakers for the heart and other oscillatory dynamical systems and very, very interesting system and with the dynamics given by equation 1.6 and for different values of mu the behavior of this oscillator changes, okay, so the question is, you know, the question is what happens, I mean, what can you say about the stability of the origin for different values of mu, okay, for different values of mu what can you say about the stability of the origin for the Van der Paul oscillator, yeah, of course in state space form, it can be written in this way, yeah, in the state space format it can be written in this equation 1.7, yeah, so what I want you to do is to actually try this out, so remember one thing, you will not be able to solve this equation for solutions just like we for this very special problem, it is very difficult to actually solve this equation, so I don't recommend that, what I recommend is for you to make these phase plane plots like this, you know, plot between the x and y, yeah, that is the two states x1 and x2 or x and y whatever you want to call it, right, and I want you to see the how the picture looks, how the phase plane plot looks and based on that I want you to comment whether the origin is a stable one or not, okay, so I would like you to look at the Van der Paul oscillator, just look at the phase plane plots, not do any analysis, not try to find delta corresponding to epsilon and so on and so forth, but I want you to comment just by looking at the phase plane portrait, which is the plot between x and y states, yeah, so this x and y states, yeah, and comment on the stability of the origin, whether it is stable, uniformly stable or not, okay, so one of the important things to remember is that if xe neither stable nor uniformly stable, then it is unstable, okay, this is the definition of an unstable equilibrium, if it's neither stable nor uniformly stable, then it is unstable, okay, all right, excellent, excellent, so I think we saw some rather interesting concepts today, so we were continuing the problem of trying to comment on the stability for a very particular classical system, all right, and we had already looked at stability and we were trying to see if we can also get uniform stability, okay, it turned out that it was not possible, right, and in order to remove the initial time dependence, yeah, on the delta, which is what we require to prove uniform stability, it so turned out that this was not possible because we had to choose the smallest delta and the smallest delta in this case was tending to zero, which is not allowed as per our stability definitions, therefore we could see that the system turned out to be stable but not uniformly stable at the equilibrium, right, which was origin in this case, all right, so great, so this is where we will stop today and let's meet again soon, thank you.