# Philip Morrison, “Counting Heads: A Physicist’s Model of Population Growth” - LNS46 Symposium: On the Matter of Particles

[MUSIC PLAYING]

GRODZINS: I think most people are now having have gotten their coffee and dessert and we would like to get back for the afternoon session give people a chance to maybe stretch their legs and get outside a bit. When I asked Phil if he would do us the honor of giving us a talk after this lunch, he said he would think about it, give me his answer in a few days, and I was quite sure he might think of saying no. I came back and he immediately said, I have two options for you.

And you know what was going through my mind, Phil, was that I've always told Lulu, my wife, that if Phil would read from the telephone book, I'd be very happy to come and hear him because I know that I would learn something. So I really didn't care which one of the two subjects he was going to talk about. He gave me the two titles, I immediately chose the first one, and I'm so pleased. He's going to be talking about theory of population growth. This is, as he will tell you, not his own. He's been very at great pains to tell me that this is the work of Sergey Kapitsa, the son of the Kapitsa you all know, or know of.

And so if people will finish getting the coffee and relaxing, then we'll hear about why it is that maybe our relations will not grow exponentially forever and forever. Bill, delay it for a moment please. Well, let me delay this by just making a couple of-- an observation. It's not even a delay. As the chairman, I've been sitting in on all talks from the beginning, getting up at an ungodly hour of 7 o'clock in the morning, getting here by 8:30, and this meeting has done what we had hoped it would do. We started yesterday morning with, as in Martin's felicitous phrase, physics on a human scale.

Physics that you could get your hands around and understand. Physics with a very different vocabulary. Physics of just 40 to 50 years ago, and how that has changed. I think those of you who have been here through that time will agree with me that this revolution, which is undoubtedly still going on, we're just too close I think, or at least I am too close, to see where it's going. But this revolution has taken us from an understanding that we thought we had and a set of questions that we were addressing, to realms which have just no obvious connection until you come to a meeting like this, and then you see how problems, the questions, the answers have evolved. We're going to hear more of this afternoon, but this gave me the opportunity to say something, which I probably will not have time or even want to say later on. Phil, what a pleasure.

MORRISON: Thank you very much. I hope you can hear me. I undertook to make this talk as an old friend of the lab, possibly even a member of the laboratory nuclear studies for some time, and it has been a delightful occasion. I hasten to point out that what I'm doing here is reporting the work of someone else. It was a remarkable coincidence. While Lee Grodzins was on the phone persuading me with his gentle but compelling manner-- which I'm sure you all know, so I could hardly refuse-- Sergei Kapitsa had entered and was getting ready to show me a paper he had just written, fresh-- he was fresh from Moscow. So it was quite a busy morning. But what he showed me was so interesting at the same time so simple and possibly of great importance. So I thought it was very worthwhile to learn it, which I did, and present it to you.

I hope you've all got this thing. I'll refer-- I'm not to give any slides. I'm going to use and discuss in detail the graphs that are shown here and a few of the equations, which I wrote out, describing the same material as Kapitsa's, but of course, very much abbreviated.

I should say a little bit about him. Possibly, of course, everyone recognizes his name, a famous name in physics, especially in the physics of the Russians and the Soviet Union. But Sergei, who was a important contributor to the study of instabilities in plasma physics, and therefore knows his nonlinear physics quite well, and made some-- most important contributions in earlier days, has for about 20 years now become a major television figure on Russian TV across the entire expanse because he has run for all that time a weekly program on science, an hour a week.

And it's very widely known among all generations everywhere throughout the Soviet Union. In fact, at the current time, I forget which day of the week it shows, but when it shows, it's showing this term, this season, from October/November on till the summer, a study of economics. And nearly every week a new economist from somewhere in the world is produced. They interview him and ask him what he thinks about this, that, and the other thing, develop theoretical structure of economics.

They have had Friedman and Galbraith and the usual suspects. And I would not be surprised if some MIT people have not or will not soon appear. So he was in this country collecting his tapes, so to speak, by bearding these people in their offices to talk to them in order to put it on TV. He's had, of course, every scientific activity on TV in all these years. Very interesting record.

On the side he has some time to do things. And the point which I think I can most easily make is that it's appropriate to do this because it comes out of, really, I think I can say, it comes out of 1946, just like the nuclear lab. Through the then publish-- public announcement of the serious accomplishments of operations, operations research during World War II. In Britain, around the physicist, Patrick Blackett in the United States, around the physicist Philip Morris of MIT. So it's somewhat reasonable we do it. This was a scheme. It was really a program. It's still going. Like many things in physics, it's spun off its own discipline, its own profession, its own institutions. But it really began with a physicist's hope that looking at a structuring or a phenomenon I do not understand very well, if it had numbers connected to it, if you could somehow connect it with numbers, you'd be able to make some semi-quantitative stab at doing something. And that regularization would help him get farther into it, and try to help the practitioners, who knew much more about the process, to see what was going on, and make it work.

This began in wartime around the tactical problems of war. How many times do you have to approach a submarine before you can see it on radar, et cetera. That kind of problem left to the people flying in the field and their senior officers home worrying about them. Doesn't get very big change. Experience is hard to share and hard to generalize.

But it was found that if you got a bunch of physicists, the odd geneticist, physiologist, and so on, to think about it and argue about it in a quantitative way, you could make a theory which was certainly only a caricature of the real dynamics of the situation. These are not laboratory situations. You could put them in a lab, modify the parameters, look all around, do the wonderful analysis that the people can do at the nuclear labs. Much more like the bad efforts of the astrophysicist, especially without their powerful machinery of the day.

So you're looking back at an old astronomy, that's that style of theorizing. Maybe this is the Kepler of population growth. It certainly is not the detailed understanding, which comes from human society and its biological nature.

But I think it is important. It's certainly extremely helpful to a physicist reader to understand what on earth is going on, and some feeling for quantitative control of it. That it is compellingly true, no. No one would claim that. Not Kapitsa, and certainly not any of his readers. It is an effort at a phenomenological theory, which has remarkable support from the degree with which only a parameter or two that makes some sense, gives sensible values, and fit a very complicated set of data quite well.

It looks as though one is onto something simple not because we understand the causes, but because in the aggregate, some statistical, some control situation makes it work that way pretty often. We're looking not so much at causes, as at consequences. And I'll close with a little more of that. What I really want to do is expose what the theory says, and what the data say. I think you'll find it quite interesting. It will only take me another 15 minutes.

On the first page, I put a little mathematics. And I want to make it quite plain that please do not think of the population, global population, n of t, as a solution to an exponential function. That is a straight forward error. It has nothing to do with it, never was exponential. It was only exponential in the mind of the Reverend Thomas Malthus, who was a very good theorist of his day, but didn't control much mathematics. He probably knew two or three functions, and that was one of them.

[LAUGHTER]

And it isn't that way at all. I wrote the equation, because anyone can write that one down. But it's really nothing to do with human population. The reason is, of course, it's parametric. It has a tau in it. So if I allow you to make an arbitrary function out of tau, then I really can't complain.

But if you imagine tau as a constant, then of course, you'd be greatly mistaken. Already in the time of good demographic statistics, where errors are not anything like the order of magnitude, more like 10%, 20%, back to 1800 or so, already the local tau, the instantaneous tau, has varied by a factor of 6 or 8. Now, that's no proper exponential.

So you, in time, in rather short time compared to the species, a big variation in the doubling time, or the e-folding time. So that's really the crudest approximation. Of course, at any point you can define a tau, but that isn't the purpose. The purpose is to integrate somehow, get some picture of the whole story. And therefore, while extremely simple, introducing a characteristic scale, like tau, has never worked and doesn't work now, for the fundamental, the zero order approximation.

Now the next, simpler point which might be tried in this is known to be very powerful. Nowadays we are paying attention to nonlinear systems of all kinds. The next simple thing to do is look for a scale-free or an invariant theory, which depends on ratios alone and doesn't have an absolute time or an absolute number introduced as the zeroth order theory. That's too hard, and it doesn't work.

So let's try a scale-free theory, delta n over m, proportional to delta t over t. No dimensions, no parameters, except dimensionless parameter, which you might hope to make hay with, making things go up, things go down. Obviously, that's a very powerful parameter whatever you do. Mind you, this in a very elementary beginning.

Now at the same time, of course, t-- well, why should we introduce 0 of t? Let's make the 0 explicit. And it's convenient, in general, to put in a t1 for some fixed time date. And it's also useful to say we're ahead of that time date for a reason you'll see the next statement. So we're going to count t as time from 0 BC, say. That would be a good [INAUDIBLE].

Now, the original direction was given to this whole story. The demographers, of course, it was a big subject matter out there, very important, very professional, full of people who master [INAUDIBLE] equations and do wonderful things about projecting the population distribution at one time onto the population distribution the other time, knowing all the age-specific mortalities and all of the age-specific fertilities and calculating beautifully what goes on.

Of course, far ahead is pretty hard for them to do, and their tendency has been not to try to make big projections, but to be guided by strong interpolation theorems and keep things going more or less as they are. That's about the best they can do. And that's how the UN works and does all projections, and on the whole, they're not too bad.

But then, of course, it attracted attention to some system operations research people. And I gave the two interesting data, which are key references here. I like to give them both, because they don't differ much, and yet they are 15 years apart, which is some sign of the stability of this idea. It's only a very simple, naive idea, which Kapitsa takes off from, but I want to put it down as a foundation.

The first one was done by Heinz von Foerster in 1960 and published in Science. And he noticed from the then statistics, maybe 50 or 60 years of world population, that it fit very well to the strictly empirical formula that n was proportional to a constant, of course, a population number, divided by t1 minus t to the 0.99 power, where t1 was 2027, and 0.99 is obviously not the result of the theory, but the result of the parametric fit done by a careful statistician, because I would never have done that.

And Sebastian von Hoerner 15 years later, looking at the same data a little further away from statistics operations research and demography-- von Hoerner is really a radio astronomer-- decided, well, 0.99 invites every physicist to call it 1, so he did. And he only had to modify the date by two years, and this is 50 years in advance of the date.

But I want to call your attention to what the function is. n is a constant divided by t1 minus t, and it plainly diverges at t1. And indeed, that was the point of both of these letters, and that was as far as these people went, which I must say deceived most readers, misled most readers like me, but not readers like Sergey Kapitsa that came around to it, and I suppose other more thoughtful people. Because I felt a theory that leads directly to divergences built in is hardly worth paying attention to. It might be fun to talk about that, but it's not going to happen.

And it didn't occur to me, why don't you improve it by cutting off the divergence? Well, that's essentially what Sergey has done. He introduced a way to cut off the divergence, which of course, was not real. We don't know much about human population. I would say we know one thing. It's never going to be infinite. Therefore, I think it's a fair proposition to change that.

So how does he cut it off? He looks around for a parameter that would be directly translatable into measurable terms, of course. So he looks at the thing, and he cuts it off the way that every Physics 802 student would be able to cut it off, by dampening, in the sense of an oscillator, or putting radiative loss or whatever you want to do with whatever you want to do into any of these resonant-like formulas. It's a resonant formula without the resonance.

As soon as you take in [? dot, ?] it becomes resonant, as I say when I differentiate, I get 1 over t squared, and of course, a still worse [? poll. ?] But nobody expects that [? poll. ?] We all know what to do. We stick in a tau squared, and then the tau square has a wonderful result, because not only does that give you a nice formula that doesn't diverge, but it directly translates the formula into a measurable quantity and taus mainly found by finding n dot max.

When the rate of population increase was much more measurable in the population-- in the modern world, it's all recent data, and it's gathered and summed up, and it's probably not too bad-- you would have n dot equals c over tau squared, and that's when the whole thing begins. The parameter of taus is clearly going to be rather small length, small time.

And now I urge you to look over the page and look at figure 3, the top of the two pages, and the spread. And there I'm skipping, of course, a lot of numerical fitting, lot of discussion about interpolation formulae, about integrations, other things to check it and make it all proper. But I'm just showing Sergey's graph of the results. It's a very nice thing to do.

Here you see the times since 1800 across the horizontal axis and the population units of [INAUDIBLE] heads on the right-hand axis. Now, if you look at the right-hand part near the right hand axis to see the thing in its asymptotic freedom direction, which is easier to read, and there you see there are 1, 2, 3, 4, 5, 6 rapidly rising curves. These are simply different parametric fits, which Kapitsa introduced in order to be able to choose among them the one closest to the data, but to show the general course of the curve.

And then on the same part of the graph, he has drawn two dashed lines. These are the extrapolations done by the rather careful step-by-step, country-by-country, age-by-age, epoch-by-epoch projections of the UN experts, the demographic experts, perhaps the best in the world. And there you are.

So you see, if you look at it, his model 3, somewhere between model 2 and model 3 would split the difference of the UN completely. And he happened to take model 3, because he did that in advance of looking up these data. He did that a priori from just making simple steps in the parameter fit to cover the parameter space. And that's very nice.

So you see, there it is. It's no news to anybody that we are on a curve which rose, rose, rose, went to curve A, which is now-- they're are all the same up to the open circle, which was the present time, 1990, say. And then the five different models begin to differentiate themselves at that point. Follow it up, and you'll see what happens.

Now, it's nothing new that there is going to be a plateau. We all expect that. It's not even very much crystal gazing. Of course, tremendous catastrophes, giant nuclear war-- which I think has disappeared-- a giant asteroid-- which any have not yet appeared-- barred, we will not have grand catastrophic changes.

We cannot control to something like 10% or 20% in the future, because we could imagine famines and great changes in governments or whatnot, but not very big changes, because, of course, most of the people who will bear children in 2030 or something like that, many of those people are now alive in 2020. So we have about 30 years head start, and then the rest will not be too great change if you don't imagine we're trying to push it out very far to the very end of that curve.

So that's quite reasonable. And it's interesting that his curve, which, mind you, is a parametric fit-- it has nothing whatever to do with the statistics-- once you accept the suggestion of von Foerster and von Hoerner, that you could indeed fit with the singularity, the local data, hyperbolic curve, why, go ahead and try what that does.

And so you see curves A, B, and C. And I'd like to call attention three curves in order. A is simply the overall past, which is all fitted by each of these three curves. Since when you're far enough ahead of the resonant time, it doesn't make any difference whether you dampen the resonance or not, only when you've got across, that makes a big difference, and the difference is then shown. So that's A.

B is quite interesting. The upper left-hand corner of that drawing shows the percentage deviation from the statistical number that demographers tell us. And model 3, it has no significance, just described model 3.

But it's interesting to see for comparison how the real curve looks like when you look at it in some detail instead of broadly drawing a smooth n over t-- n in proportion to c over t curve, that's very, very simple, the hyperbola. Instead of just drawing that, we draw the detailed curve, and we look at the difference within the detailed curve and the model curve.

And you see it's tolerably good. It's within a few percent. It looks as though World War I and World War II show up as actual modulations of the human population. And what is most interesting is the rise in decline between 1970 and 2000.

This is something which is not due to this study. It's well known in the empirical data, but it's not called out to people. And it's much better to have this analytic approach to see how it fits into other pictures.

I believe that's been the most important single result. Probably one of the most important political points for the sociopolitical future of the world is governed by that little bump, because that bump is the statement that the relative growth, the percentage growth per year in population, has already passed its maximum and is empirically going downward, which to a physicist with some sense of derivatives and an analytical continuation, if not to somebody who wants the hard data what's actually happening, is a very significant number.

If the derivative not only is going down, but if it's already passed the maximum of derivative, it's bad luck if it's not going to do something like that, if there's any reason at all to expect it. And all these projections do it, both the empirical ones based on projection of age groups and this overall grand parametric one which is so simple, but yet so attractive.

So there you see in curve C the maximum of the growth rate at 2010, just according to the parameters of Kapitsa. Clearly that could be moved by years either way. And then you see the percentage growth is even more interesting, where the maximum has already occurred, which is true in the model and true in the reality.

So if the curve does not show large deviations beyond what we expect, we have indeed begun the decline. Not the decline of the population-- that would take a long time, because people live a long time. Not the decline of the birthrate-- that will take a long time, because tiny infant girls grow up to be mothers, and it takes time. So all those things are going to change, but nevertheless, the first sign the physicists look for I think is the 0 in the first derivative percentage-wise, and that's been passed. An extremely important result.

Sorry?

AUDIENCE MEMBER: How big is tau?

MORRISON: I'll tell you all those things in the next step.

Now you look at the next page, the last page. Turn over the whole thing. Look at the last page.

I want here to discuss something I view as quite interesting, which Kapitsa, as an expert in nonlinear phenomenological theories, has done much more than I would have done I think. It's quite neat. Of course, first you make it dimensionless. We want to look at the singularities.

Now, the obvious singularity we already got rid of by dampening it out. And you see right away that the dimensionless function without cutoff is just k over t. And we take t equals 0 would be the future, and t, going to infinity, being the deep past. And so you see, n goes to 0-- Sorry. N goes to infinity as t approaches 0. That's the bad singularity already gotten rid of.

But in the zeroth order theory, there is another singularity, because of the extreme symmetry between the variable t and the variable n. And there's also trouble, as n equals 0 in the deep past. You should examine that as well. It gives strength to the understanding of the curve to see both of those things happening.

So of course, he proceeds to cut off both things in the same way and to fit the parameters. And the parameters fit extremely well with reasonable values. Answer to the question, tau is about 40 years. It's clear like a human lifetime, because you see at the beginning, it tells you what n dot must be at the beginning.

When n is very small, n dot, the characteristic rate of change in dimensional [INAUDIBLE] is something like 1 over tau, the parameter for the time. And that's what you'd expect. And that makes some sense out of the whole proposition I think.

Now, these are two discontinuities. You don't really carry-- there's no simple analytical function of the whole thing. But he fits them together, two singularities, and it's pretty reasonable to make it work that way. And all this is described on the graph you've not yet looked at. It's very complicated-- well, it's a very rich graph, so I wanted to look at the formulae first. Now you can interpret this graph very nicely.

This second graph, the bottom of the spread, Kapitsa's again, is a double log-- log, log graph. Logarithm time on the horizontal axis, logarithm number on the vertical axis, and the three phases of his curve broken by the two singularities, the one in n and the one in t, which are symmetrical, are both present.

We've already discussed the singularity we're now approaching, which was a move by relaxation. And there it is, 10 to the 10th population, about 10 years before 2010 or something like that. The 0 logarithmic 0 was that of 2010. And then it rises very slowly, asymptotically, to maybe 15 billion.

That's only the parameter set chosen here, number 3. Number 2.5 would be 12 billion. The UN minimum is 11 or 9 and 1/2. So we don't know that. We're not claiming that kind of accuracy. But it's a very reasonable curve.

Now then, look at the curve from 10 or 100 years before 2010, say, from the 20th century way back a million years. It fits tolerably well to that single straight line on log log paper, a power law, a scale-free dependence, as we said. And of course, in n dot, the rate of change of population, that's a 1 over t squared. But if n is like 1 over t, that's the same as n dot is proportional to n squared for that domain.

Now as Watson and Crick said, everybody who knows something about biology knows that the number 2 is very important. And it doesn't seem unreasonable that n dot might in some simple circumstance for a limited situation in the very complex spectrum of human society, might represent some important gradation in that time. And it does seem to fit extremely well over a very long period.

Notice the curves are not-- the point's not tight to the curve. That's quite [INAUDIBLE]. We don't know them very well. And there was a change at that time of species even, and certainly from others to Homo sapiens, something might have happened, and I'm not surprised at those dots are missing. But essentially, they fit tolerably well. Was quite good.

That's curve B, the free power law behavior. Curve C is, of course, the asymptotic behavior. And curve A is the behavior from n equals 0, which can only approximate in this logarithmic curve, n equal 1. You notice a symbol put in n equal 1. And 2 or 3 million years ago when the species we thought to begin, this Venus mirror, the signal for the first woman, is put there, since that must be how the whole thing began.

And here for A has no data at all, except that he has the point to which the A curve goes. And that point he got from the French paleontologists working in the Afar Valley for the last few years, and who say that yes, they think that a number like that of the order of 100,000 a million or two years ago would be a very reasonable number to take.

I think it's quite plausible. I don't think we can claim it's known. They said he fits it to 69,000. That's good enough for me. It helps the parameter description, so nothing too wrong with that.

So there you see it. And these three regimes then are the three regimes of human global population-- rapid growth, when we're only a few, to become a real social being and a proper species; then a power log dependent growth with maybe some wiggles, all the way up to 10 or 20 years ago; and then the beginning of a decline in rate, which will become a real decline until it becomes asymptotic 100 years in the future. That's the picture of this curve, and it is quite interesting.

Now, that's all I really want to say, but the theory you have, the piece of paper, you can take it home and look it over. And I think it's quite interesting. I'll say a little bit about the circumstances and my own view of it, just in a hurry, and then quit.

This paper's been prepared to be delivered to the International Institute of Applied Systems Analysis in Vienna in their forthcoming conference in June of '92. I said that operations research already had its experts, its professions, its societies, and its institutes. And this is one of the best institutes, the one in Vienna, IIASA.

And there it is. They have proceedings and publications that high. So you can look them up at your leisure.

I will say my own feeling on reading this, why I wanted to talk about it. First place, of course, I make it very clear, as I said in the beginning, physicists, even expert non-linear behavior physicists, are not very good at giving the cause of all these things. Dynamics is not known.

This is a kinematic theory of consequences. It's remarkable that it gives such a plausible picture. And I think that's what it's good for. It's an interpolating, hypothesis-suggesting picture to be tested and used as best you can, very gingerly. Nobody would imagine it's accurate or precise or compelling, but it's very interesting.

The most important point I already mentioned. I think that the percentage rate of human population growth has passed. As far as we know, it's empirically passed. There's no reason that it will rise again. It might, but we don't think it will.

The n is proportional 1 over t implies when the situation is far from the singularity, either singularity, that n is proportional to n squared-- sorry, n dot is proportional to n squared, and that's exactly what you want for this twofold nature of human biology, like most other.

You can look at another way. I suggest a [INAUDIBLE] transform, which was also discussed. You can imagine this to be a very large sum, white noise, exponential terms if you like. That will also give you something very similar. I don't want to go into it in detail.

And I remind all the physicians here, probably old enough to remember at least, that fission is exactly the same sort of thing. It's a simple power law. The rate of activity subsequent to uranium fission follows a pretty close to first-power power law, 1.2 shifting to 1.3 or something, for a long, long time after the act.

What is it composed of? It's composed of hundreds of species, nearly all governed by beta decay, all governed by exponential decay. But summing up many exponentials, the Laplace transform naturally leads to something not far from the power law. And if you translate the energy space, you'll see behind it's the energy of beta decay, and that space and what various fragments of nuclei will do. But that goes too far into the dynamics, which we're not able to do in this situation.

But I think it's the same kind of thing. It's the summation of many complex things that have a certain deep similarity, but changing scales all the time. And therefore, our scale-free description is a good first approximation You must avoid the infinities, and that's what this particular procedure does. Like the Higgs boson, it's not guaranteed the only way to describe the situation.

Therefore, I say it's analytical, it's semi-quantitative, it's understandable, just certainly not compelling. But it is good enough to base what I think is considerable hope for working out the giant problems that confront our species.

And I'll finally close by saying simply, though it's uncertain, it seems to me that it's one of the most important single quantitative statements the world could make, because while it's a consequence and not a cause of our actions, the feedback loop of human life is very strong. And therefore energy and food and transport and war and communications-- you name it, the problem is there-- it all is driven or is related strongly to-- not driven by, maybe it drives. It both drives and is driven by the number of people we must accommodate in a finite space, in a finite atmosphere.

And therefore, any effort to reduce it to some kind of plausible reason, especially at this level, simple enough for a physicist, is I think a major contribution. That's why I wanted to spend a few minutes on this occasion to tell you something which is really quite serious, but which comes, nevertheless, as a kind of reduced instruction set at the end of a very long and fascinating conference. Thank you.