WEBVTT
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Okay so the title of this first talk is slightly different
than what is on your schedule. This came about because
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of conversations with John Finnegan and happy and the desire
to have several team projects that do not require enormous
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amount of background knowledge about the specifics of the
topic. So so with this, we decided that perhaps the first
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lecture could be almost a month review of some of the tools
that will be needed in the next lectures And I have
00:00:55.174 --> 00:01:02.849
tried to prepare a lecturer along those lines and so
the title of this lecture is the theoretical minimum
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the title is stolen from an exam that is given by the famous
physicist and Landau, who is a great Russian
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physicist, used to give his students an exam, basically to
assess whether it is worth for him to spend time working with
00:01:19.293 --> 00:01:27.729
them or not. And he used to call the exam the theoretical
minimum. And that exam was a thirteen part exam, by the way.
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So it was not a light one. And basically, that exam covers
everything that left land down knows so basically to
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work with you had to know as much as about
physics, and apparently something like twenty students passed
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that exam over the entire history of of Russian physics.
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Now, this is not what we want to achieve here. We are actually
closer to a second level of of theoretical minimum that
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was suggested by Leonard , who is a professor at Stanford
University, also theoretical physics But in fact
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what Leonard Suskin decided to do is to offer some tools and
some background about physics for researchers who want to
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move into physics, and that basically cut down on a
list from something like thirteen to something like six items
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that now you need to know about to do physics.
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The theoretical minimum here, as articulated by John is
that, hey, look, if you like nature, and you would like to
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study it from some perspective of mathematical tools, you are
in the right place. So that is our theoretical minute. But
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just to make sure we are going to
cover some few points on that.
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Okay, so let me before I start, let me introduce the speakers.
We are fortunate to have a group of speakers from outside
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and inside the kit, the cluster of speakers that you see
from outside kit include John from the Commonwealth
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Science and Industrial Research Organization and more
recently, from the Australian National University. John has
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probably worked for the last fifteen years or so. Now, almost
no then, fifteen years on complex system science. He is
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been leading actually the complex system science initiative
in Australia and has saw enormous amounts of projects and
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topics that have dealt with complex systems and he was
kind enough to agree to give an extra lecture instead of my
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seat dispersal lecture on on topics that are pertinent to
complex system science, population dynamics collapses and so
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forth. As you will see, social systems shared a lot with
turbulence, and that is already an interesting phenomenon about
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flows so. John also is perhaps more known to many of
us on his work in fluid mechanics and canopy turbulence. In
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fact, micrometerology as a field was born from plant
science. For obvious reasons, the micro climate does affect
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plant productivity. And for those of you who have studied
some basic atmospheric science and have read the paper by
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on a book of the famous paper that has laid out almost
the foundations of the paper actually starts by
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talking about the substrate on the land surface. Carbon
temperature everything and in fact many of the
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physics that that were working at the Russian Institute were
and a book of work later bifurcated to other topics.
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But they all started working on on turbulence. And my
colleague from Duke University summed up well. He
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said," If you work on turbulence, you are really training
well for everything else. So that was kind of his statement.
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And that is why he has all his students take some of the most
serious turbulence courses. John's contribution basically
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was to take a field that had a lot of empirical results in
terms of transport processes near the canopy atmosphere
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interface and transform it into a field that is now basically
contributing to fundamental understanding of physics, of
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transport phenomena next to interfaces and. I Think this
is one of the areas where environmental sciences actually
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is now in a position thanks to people like John and Microp
to actually contribute to physics of turbulence, rather than
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the other way around Who, will be arriving later in
the course is a professor at the Technical University of
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Denmark. Kari actually finished his Ph D quite recently.
I think in two thousand and ten. And since then, he has
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been doing probably one of the most innovative and leading
work, both experimental and theoretical, on water movement
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and plots how water moves and asylum how water
moves in the flam, how all this network operates, how plants
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respond to pressures based on on very innovative ideas that
are coming from from theoretical physics. And just to show
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you that our field now can be labeled truly as physics. That
would be the sign. If you were able to publish a paper and
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reviews of modern physics. So and that is what
Koreans and just did so in two thousand and sixteen.
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He had a very nice review paper on water movement and asylum.
And the flame in reviews of modern physics. So now we are
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officially in the physics of our research and
certainly Carrie is heading speed, heading a lot of that
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effort, by the way. Also, if you look at Cardiov, this guy
can't publish easily in plant cellar environment. New
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biologists, physical review letters physical view is
actually where Albert Einstein got his first introduction to
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the peer review process. If you are wondering, science nature
proceedings of the National Academy with Greaties. I
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think when he finishes a paper, he debates," should it go to
an ecology literature, or should go to probably a physics
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literature. So a very, very short guy. Now, Wilfrid comrade,
on the other hand, who is here had a very interesting
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trajectory to get to the point where he is. He started
actually. Also, he finished his speech in theoretical physics.
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His Ph D work was on deriving basically analytical solutions
to Einstein's field equations but then discovered
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that if you derived an analytical solution for
field equations there is nothing else to be done.
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So he had to change.
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And at that point, he decided that if he is going to change,
he is going to work on a topic where the equations for the
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problem have not yet been derived so he connected with
Anitaos and with Anita, he basically discovered that there
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is a whole field in ecology that is still struggling to even
come up with the basic equations to describe the mechanisms
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in it. And I have to say that Wilfrid has done a fabulous job
at the speed heading. A lot of the initiatives that try to
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understand why plants look like that. Can we learn about the
past climate, from the shape and the function of what we
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think plants have done or are doing now. So basically,
Wilfrid, the search connects quite a bit to climate
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reconstruction but from a physics physiology perspective
and because of his background, it is always fun to
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chat with Wilfrid. He will find something different, even
if you give him a quadratic equation. It will give you an
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insight that you did not think about when, when you walk to
talk to him about that. Kostantin O Man is is a professor at
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Polytechnic of Tournament in Italy. Costa has done his
Ph D, actually in Aberdeen, and experimental hydraulics
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with Vladimir Nikola, one of the best labs actually to to
do experimental hydraulics. And he focus quite a bit on a
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similar problem to John Finnigans but rather than look
at plants, he was looking at gravel beds and in fact
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if you talk to hydrologists, a major question is, what is
happening in the zone between streams and the land
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surface and that actually is where a lot of biochemistry
happens. Temperature gradients happen. And in fact, a lot
00:09:08.502 --> 00:09:16.150
of life is impacted by that zone. So Kstantin has been
working extensively on figuring out the connection between a
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free, moving fluid and what is happening inside the gravel
bed. And he later extended his work to focus on basically how
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life fish and so forth capitalize on the exchange processes
within the zone to enhance the chances of survival and
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so forth. So he is going to be giving
two lectures as well on on this topic.
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Stephan is one of our own from Duke University
What. Can we tell you about Stephanie he finished. He
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finished his Phd. And the day he defended, he had a paper in
science. So, you know, and it was about the composition of
00:09:52.996 --> 00:10:00.893
leaves or something like this. I don't remember but
basically Stefano has studied with who is perhaps
00:10:00.893 --> 00:10:08.751
one of the leading experts on and what Stefano has
brought to the table is innovative tools about stochastic
00:10:08.751 --> 00:10:15.823
processes, dynamical systems theory, physiology
hydrology and he is able to merge them all together to
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attack some of the toughest problems in the field. And he is
been recently moving more and more towards the use of
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as a constraint on flows so basically, how would dimensionless
numbers that are set by basically constrained the
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entire biochemical cycle budgets in landscapes as big
as the planet or as small as the root zone. So this is a
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snapshot of the external speakers. Now we can go to our own
from we will start with Matthews murder. Matthias
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finished his Ph D with Thomas Folkan. And just like Lebanono
has the theoretical minimum. If you want to work with
00:10:55.853 --> 00:11:03.556
Thomas folk, and you will better study the energy balance
closure problem. Ok, there is just no way around it. You
00:11:03.556 --> 00:11:10.319
have to do something there. That is the theoretical minimum
of of Tamas spoken, but Matthias was able to do way more
00:11:10.319 --> 00:11:17.759
and he was able to basically set the stage for a lot
of the data processing machinery that is now being used in
00:11:17.759 --> 00:11:25.457
flux net thanks to his effort. And then he decided that he had
enough of this. So he went to Canada that the post office
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came back. And it is safe to state that Matthias is probably
now one of the leading experts on computational methods on
00:11:32.206 --> 00:11:38.108
flows on complex terrain where the biosphere and the
atmosphere are talking to each other, or or intersecting. And
00:11:38.108 --> 00:11:45.844
unlike some of the work that we have done cheapishly with the
flat surfaces, you know, and so forth. Actually, materials
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is dealing with the real systems. And so and both. He is
bringing innovative measurements as well as simulation tools to
00:11:52.931 --> 00:11:59.079
these problems. And basically, the last five or six years,
the simulations that are coming out of group are just
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pretty impressive. I mean, it was unbelievable
in terms of what is being done now.
00:12:04.309 --> 00:12:10.249
Okay? Moving to nadine Nadine
I'm. Sorry. I forgot the two dots.
00:12:10.990 --> 00:12:21.460
Where is Nadine Nadine actually finished her Ph D with
Nina from Switzerland. Nina is probably one of the
00:12:21.460 --> 00:12:29.410
top notch grassland ecologist. I would say it is safe to say
that. And crop land ecology actually. And she did a lot of
00:12:29.410 --> 00:12:37.544
work on carbon stocks. Carmen flexes then went to one of
the better kept secrets in U s Academia, Oregon State
00:12:37.544 --> 00:12:45.082
University, where she worked with and many others in
that group and continued work on how climatic perturbations
00:12:45.082 --> 00:12:53.782
impact water and carbon flexes in the soil plant system. And
Nadine very much in lined with with Matias, is bringing
00:12:53.782 --> 00:13:00.920
innovative measurement techniques. She is actually heading
the greenhouse that you see when you enter K all the
00:13:00.920 --> 00:13:08.217
experiments that you see there, all these high tech experiments
are actually Nadine rules @unoise@ but. At the same time
00:13:08.217 --> 00:13:14.776
Nadine has several papers where she is using @unoise@ innovative
models to also understand @unoise@ the coupling between
00:13:14.776 --> 00:13:23.629
radiation micro climate, soil processes and plant responses
to them. And she has delineated an area at which is
00:13:23.629 --> 00:13:32.451
looking at physiological stress, basically how elevated
temperature or reduced or moisture content impairs a plant plant
00:13:32.451 --> 00:13:32.969
operation.
00:13:34.080 --> 00:13:41.083
We are leaving the best for last, just alphabetical order.
Hans Peter Smith and Happy has mentioned our meeting at
00:13:41.083 --> 00:13:49.795
the Flux Net, but he didn't mention a couple of other things
that I'm going to fill the gaps. For when we met happy
00:13:49.795 --> 00:13:57.979
there was only a table discussion about how to interpret
and Dave , among others, was suggesting that
00:13:57.979 --> 00:14:06.980
maybe we should be using footprint models, and went to proceed
to describe what the footprint model is to a more general
00:14:06.980 --> 00:14:14.038
audience. And and Department of Energy Managers who were
at that meeting and then this guy from the room that
00:14:14.038 --> 00:14:21.605
nobody has seen before asks a question about the footprint
models. I was a very deep question, and we were looking at
00:14:21.605 --> 00:14:24.799
this guy saying," Well, this. Well, who is this?"
00:14:25.500 --> 00:14:34.444
and that Dave Hollywood says," And who are you?" and and
perhaps Dave did not link the two that the model he was
00:14:34.444 --> 00:14:37.050
presenting is actually Hans Peter Smith's mobile.
00:14:37.200 --> 00:14:46.629
And happy was the person asking him about his model. So so
that is how we got to know how, in the first meeting of the
00:14:46.629 --> 00:14:52.991
Department of Energy But of course, footprint models now
are widely news at that time. That was not the case
00:14:52.991 --> 00:15:00.128
and the idea of connecting what was on the land surface was
what you typically measure in the atmosphere was was an open
00:15:00.128 --> 00:15:06.024
problem, but it was definitely needed, because flux towers
were proliferating everywhere, and being able to link them to
00:15:06.024 --> 00:15:13.161
what actually the the sensor sees on the land surface was an
open problem, but happy already had the solution. So it was
00:15:13.161 --> 00:15:19.242
just he was always ahead of everybody else. And by the time
we appreciated the solution. Everybody was talking about
00:15:19.242 --> 00:15:25.090
aggregation and integration and so forth. But by then,
have you already had papers on that. So you get the trend.
00:15:26.259 --> 00:15:33.129
So this is the group of speakers that are going to be offering
lectures I. Cannot say that when I was a graduate
00:15:33.129 --> 00:15:39.110
student, I had the opportunity to interact with this prolific
group of speakers. So certainly take advantage of it. Any
00:15:39.110 --> 00:15:49.617
questions about the speakers okay okay so now we
are back to the theoretical minimum and the lecturer, and
00:15:49.617 --> 00:15:58.150
so thinking about projects for, for, for group effort. We
thought that it might be important to have at least several
00:15:58.150 --> 00:16:05.559
topics that do not require high technical skills about the
science of the topic. But perhaps if we can frame these
00:16:05.559 --> 00:16:12.527
problems in more general settings that require some
mathematical skills, but at the same time, not technical skills
00:16:12.527 --> 00:16:19.881
about the specifics of the problem, then perhaps one might
argue that sustainability, the research, is probably a good
00:16:19.881 --> 00:16:27.483
one to start with. And perhaps a major question confronting
sustainability research today is to what extend our planet
00:16:27.483 --> 00:16:33.517
with a finite environmental resource base can accommodate
faster than growing faster than exponentially growing
00:16:33.517 --> 00:16:40.307
population. This is, of course, a question that is on
everybody's mind @unoise@ a report after report from the
00:16:40.307 --> 00:16:45.678
United Nations are attempting to understand whether the
planet can handle @unoise@ faster than exponential growing
00:16:45.678 --> 00:16:52.727
population so you are now wondering, what are the connections
between what we are going to talk about and this problem,
00:16:52.727 --> 00:16:54.979
they will become apparent. So be patient.
00:16:56.000 --> 00:17:02.654
Now, these concerns are often attributed to Reverend
Thomas Robert Maltus. His famous essay on the principle of
00:17:02.654 --> 00:17:08.929
population basically suggested that if the population is growing
exponentially, but the food supply is growing linearly.
00:17:08.930 --> 00:17:17.578
There is a problem and that is the problem is basically
attributed to him but by no means he was the first
00:17:17.578 --> 00:17:25.376
to appreciate this problem. In fact, the Dutch were ahead
of the Brits on this one early attempt to estimate how much
00:17:25.376 --> 00:17:33.151
the carrying capacity of the planet is were done by a very
famous Dutch microbiologist, by the name of Van .
00:17:33.151 --> 00:17:40.738
I Hope I'm pronouncing his name correctly. Now that day when
hock is attributed to is attributed to perhaps creating the
00:17:40.738 --> 00:17:48.050
field of microbiology and using microscopes to study actually
microbes so he was among the first to do that.
00:17:48.490 --> 00:17:57.192
He is also famous for being friends with Bermuda, the
famous touch painter. So, but on it on his own in terms of
00:17:57.192 --> 00:18:04.922
accomplishments in two thousand and four. The Dutch population
was surveyed about who they think is the most influential
00:18:04.922 --> 00:18:13.673
Dutch person So to speak. And when Lewin Hok ranked
number four ahead of Johan . For those of you who know
00:18:13.673 --> 00:18:21.829
who Johan Crawley is , which is speaking a lot about the
Dutch. Okay, that they actually value the scientist over a
00:18:21.829 --> 00:18:23.309
football player or something.
00:18:24.400 --> 00:18:31.368
Now, van we will describe now what he did came up with
an estimate that the carrying capacity of the planet that
00:18:31.368 --> 00:18:38.336
was in one thousand, six hundred and thirty two origin of
just in the 1600s, came up with an estimate that the planet
00:18:38.336 --> 00:18:44.720
can handle thirteen billion people at that
time. Ok, now you may wonder how he did that.
00:18:45.380 --> 00:18:48.339
The crafty double. This is what he did.
00:18:49.210 --> 00:18:56.477
He said that if we assume that the inhabited part of the
earth is as densely populated as the Netherlands, Holland at
00:18:56.477 --> 00:19:04.052
the time, which had about a million people. And of course,
if we were managing the land on the planet as efficiently as
00:19:04.052 --> 00:19:04.709
the Dutch.
00:19:05.380 --> 00:19:11.869
And since the Netherlands was thirteen thousand
times smaller than the inhabitable land mass.
00:19:12.319 --> 00:19:21.840
He estimated that the planet can handle thirteen billion
people as simple as that. Ok so where does that estimate
00:19:21.840 --> 00:19:28.269
sit relative to the most modern estimates that we
have from the United Nations in terms of projections.
00:19:29.079 --> 00:19:37.086
Well, here is one. This is the most recent. I literally
downloaded this graph maybe two weeks ago and it
00:19:37.086 --> 00:19:45.351
shows on the X axis time and on the Y axis, the population
in billion and it shows roughly the trends from one
00:19:45.351 --> 00:19:49.139
thousand, nine hundred and fifty to
two thousand and roughly sixteen.
00:19:49.490 --> 00:19:56.760
And then it offers a bunch of projections, and these projections
have some prediction interval uncertainties, depending
00:19:56.760 --> 00:20:06.168
on what you assume as an intrinsic growth rate and you
can see that the projections are beginning to saturate or
00:20:06.168 --> 00:20:15.104
flatten at about maybe eleven billion or so. And I've put for
reference the prediction by van hugs in red line at
00:20:15.104 --> 00:20:22.198
the top now to go back to Malta. So we don't leave the guy
alone. Maltus actually predicted that the world
00:20:22.198 --> 00:20:28.714
population would add a billion every twenty five years. If
we analyzed the record that was published by the United
00:20:28.714 --> 00:20:36.791
Nations you find that, in fact, we are adding two billion
people every twenty five years. So certainly double of
00:20:36.791 --> 00:20:43.968
what Malta has predicted interesting so. How is the
carrying capacity determined in generals either through
00:20:43.968 --> 00:20:50.494
projections and so forth the world, the work horse
model actually comes from the work of for their rules
00:20:50.494 --> 00:20:58.028
and in it this is probably known to everyone, but it
is worth putting on the board in it. He argues that the
00:20:58.028 --> 00:21:03.590
population can be exponential, but that exponential, at some
point, faces some limitations that are density dependent.
00:21:03.589 --> 00:21:10.259
And so you could see basically that the density dependent
correction is here. The exponential growth is here.
00:21:10.579 --> 00:21:18.260
And the outcome of this analysis is that you get a population
that saturates at some carrying capacity K. Now you could
00:21:18.260 --> 00:21:25.023
write this equation a little bit differently. And there are
many forms of it in the literature. But one form is shown
00:21:25.023 --> 00:21:30.236
here where you have a proportionality constant multiplied by
the population, multiplied by the carrying capacity minus
00:21:30.236 --> 00:21:38.076
the population. And when you see a differential equation, it
is always interesting to ask the obvious question, what is
00:21:38.076 --> 00:21:48.099
going to happen if I wait for a very, very, very, very long
time for what happens typically if you wait for a very
00:21:48.099 --> 00:21:52.510
long time, you are likely to reach
steady state or equilibrium.
00:21:53.690 --> 00:22:00.747
Okay, so it could say that fine the obvious
point to start studying differential equations is the
00:22:00.747 --> 00:22:01.490
equilibrium points.
00:22:02.549 --> 00:22:08.690
And these are typically viewed as special cases of of
they are special cases of steady states, even though
00:22:08.690 --> 00:22:14.831
in the dynamical systems literature, the usage of steady state
and equilibrium is a little bit loose but, from a
00:22:14.831 --> 00:22:19.917
thermodynamic perspective, there is a difference and we
can discuss that. If you are interested in the difference,
00:22:19.917 --> 00:22:25.076
a system that is an equilibrium must be in steady state. But
the opposite is not to a system that is in southern state
00:22:25.076 --> 00:22:27.110
need not be in equilibrium.
00:22:28.619 --> 00:22:36.815
So I'll leave it up to you to see how we might figure out
which is which, if I give you some information, of course, at
00:22:36.815 --> 00:22:42.485
equilibrium, the rate of change goes to zero. And suddenly,
rather than studying a differential equation. Now you are
00:22:42.485 --> 00:22:49.263
studying the properties of an algebraic equation. Well,
that that is easier and basically what happens at
00:22:49.263 --> 00:22:57.969
equilibrium is that you get the rate going to zero, and you
get two possible solutions that satisfy this. This algebraic
00:22:57.969 --> 00:23:06.333
equation. One of them is the population going to zero here.
If I multiply, if P goes to zero, this quantity all becomes
00:23:06.333 --> 00:23:14.995
zero or if we go to K. Also, it makes this derivative
zero right, okay? fine, so fine. We found equilibrium
00:23:14.995 --> 00:23:22.490
points. Now, what the next thing you do in any theoretical
analysis is you try to assess whether these equilibriums are
00:23:22.490 --> 00:23:29.988
stable or unstable, meaning that if you reach this equilibrium
point and you kick the system a little bit around
00:23:29.988 --> 00:23:36.720
this equilibrium point. What happens do I go back
to equilibrium, or do I jump to another state.
00:23:37.509 --> 00:23:46.849
So to do that, we have to figure out ways to assess the
stability of equilibrium and this is all preparation for
00:23:46.849 --> 00:23:56.189
John Stock I think so as we shall see in a
minute that extinction and the logistic equation turns out
00:23:56.189 --> 00:24:03.186
to be an unstable equilibrium. So the minute you create
population, somehow @unoise@ then, you are going to propagate
00:24:03.186 --> 00:24:10.798
all the way up to the carrying capacity the carrying
capacity turns out to be a stable equilibrium. And the way
00:24:10.798 --> 00:24:17.414
you typically assess that is by what is called linear stability
analysis. And the way you do this is you basically start
00:24:17.414 --> 00:24:24.419
with the dynamical system. In this case the population
and you are equated to zero, and you evaluate the
00:24:24.419 --> 00:24:31.333
equilibrium points then what you do is you introduce
small perturbations from equilibrium. So in this case, delta
00:24:31.333 --> 00:24:35.059
is some small perturbation around
the equilibrium point pea star.
00:24:35.400 --> 00:24:43.140
In the case of the logistic equation, peace style was either
extinction or the carrying capacity. So it is very too.
00:24:43.380 --> 00:24:52.217
Then what you do next in this analysis is you expand this F
of B around the equilibrium point, and you butcher all the
00:24:52.217 --> 00:24:59.045
high order terms in the tailor series expansion. So you had
left with the first order. Taylor City's expansion of F Ofp
00:24:59.045 --> 00:25:06.147
around the equilibrium point F of P star and if you do
that, you will find that the second correction from P Star
00:25:06.147 --> 00:25:11.290
is the Ddp of this function evaluated that the
equilibrium multiplied by the perturbation.
00:25:12.720 --> 00:25:23.892
Okay, so we have just neglected the high order terms. And
that has an important implication. Basically, we are assuming
00:25:23.892 --> 00:25:26.960
that this derivative art equilibrium is finite.
00:25:28.660 --> 00:25:36.547
This could be a problem that will put a monkey range in
linear stability analysis. But let us proceed. Let us assume
00:25:36.547 --> 00:25:38.809
that it is finite.
00:25:40.089 --> 00:25:48.815
So let us now shift a little bit from the logistic equation
and keep the discussion in the most general sense. So let us
00:25:48.815 --> 00:25:55.965
assume that X is our state variable, and Afghax represents
the derivative of X with respect to time, and we have found
00:25:55.965 --> 00:26:03.736
the equilibrium ecstasy. So basically now we perturb ex star
by Delta. And if you differentiate basically
00:26:03.736 --> 00:26:11.488
find that it is the plus the externality and of
course, the extardity is zero equilibrium and that has
00:26:11.488 --> 00:26:19.273
to be the tailor series expansion of F effect. So that would
be a perfect star, plus the derivative of X would respect T
00:26:19.273 --> 00:26:25.859
X evaluated Eggstar multiplied by the perturbation multiplied
by all the high older terms. And that has to be zero
00:26:25.859 --> 00:26:32.454
now because we are at equilibrium F of exterior zero. And
the externality is, by definition, zero. That is how we
00:26:32.454 --> 00:26:39.806
evaluated the equilibrium so we get rid of those. We
get rid of this, and we are left with is now the
00:26:39.806 --> 00:26:46.360
derivative of X evaluated at Exeter multiplied by Delta. So
this is a first order differential equation that describes
00:26:46.360 --> 00:26:54.808
how the perturbation will grow in time and the direction
of that growth is going to be dependent on what this
00:26:54.808 --> 00:27:03.256
derivative will do at equilibrium so if it is zero, you
have a problem ok linear stability analysis is
00:27:03.256 --> 00:27:08.089
going to choke a little bit. There are
ways around it, but it will choke.
00:27:08.710 --> 00:27:14.931
If it is negative, then you have something that looks like
the Delta Dt is minus some number multiplied by delta. And
00:27:14.931 --> 00:27:21.150
the solution of this is a negative exponential in time. So
you know that the perturbations are going to decay in time.
00:27:21.160 --> 00:27:27.803
And so we can call the equilibrium stable, because whatever
we do, if we per turtle with it is going to relax back to
00:27:27.803 --> 00:27:36.645
equilibrium. On the other hand, if if Prime of Exter is
positive, something else happens now. If, we perturb the
00:27:36.645 --> 00:27:43.276
system in fact, these perturbations grow, and very soon they
will stop abiding by linear stability analysis but we
00:27:43.276 --> 00:27:50.575
know one thing that these perturbations are going to move us
away from this equilibrium point. We are not going to relax
00:27:50.575 --> 00:27:59.303
back to this equilibrium point. Again, this whole analysis
is making the assumption that the derivative of Ff X, this
00:27:59.303 --> 00:28:08.870
genetic function that describes the derivative of X with
respect to time is finite at equilibrium so. Let us now
00:28:08.870 --> 00:28:16.437
specialize again to our logistic equation and see if we can
recover the intuitive results that we know about it, that
00:28:16.437 --> 00:28:18.959
basically extinction is unstable, carrying capacity stable.
00:28:19.349 --> 00:28:26.831
If we evaluate the derivative of F of P that is given here.
With respect to P, we find that this derivative is alpha
00:28:26.831 --> 00:28:33.549
multiplied by Kay minus so. If we insert the extinction
equilibrium we get Alpha Kate to be positive. So
00:28:33.549 --> 00:28:40.149
that is clearly unstable, right? because if he goes to
zero, we are left with Alpha Kay. And that is a positive
00:28:40.149 --> 00:28:46.936
derivative. Whereas if we go to population reaching carrying
capacity, we have K minus two K. So we get a negative
00:28:46.936 --> 00:28:53.237
number here and that equilibrium is unstable ok
stable sorry because the derivative is negative. So
00:28:53.237 --> 00:28:57.169
basically, we just showed that this is the case.
00:28:58.220 --> 00:29:06.163
Fine there is one more thing that you can learn
about this analysis, which is a relaxation time scale to
00:29:06.163 --> 00:29:12.048
equilibrium and that is another important concept.
Relaxation, timecade. Basically, you could show very quickly
00:29:12.048 --> 00:29:19.142
that one over the derivative of evaluated that equilibrium
does describe, as you might expect, the time scale at which
00:29:19.142 --> 00:29:27.058
you decay, or you move away from equilibrium. And again, in
the logistic equation. If you plot the value of a frame of P
00:29:27.058 --> 00:29:34.206
Star at the equilibrium the carrying capacity, you find
that you would relax the equilibrium at a rate of one over
00:29:34.206 --> 00:29:42.455
our, but in general, in any dynamical system you
could think of a prime of peace. Tar, where P start is the
00:29:42.455 --> 00:29:50.434
equilibrium point as almost a relaxation Timescape now.
We come to the discussion about, basically, what is going
00:29:50.434 --> 00:29:58.099
to happen to the population in the future. Now, the United
Nations analysis has taken a short time view of about fifty
00:29:58.099 --> 00:30:05.556
years or so to understand or project what will happen to the
population in the future and perhaps use the data from one
00:30:05.556 --> 00:30:12.366
thousand, nine hundred and fifty to two thousand and sixteen
or so to try to estimate the carrying capacity of the
00:30:12.366 --> 00:30:18.189
planet, depending on different growth projections but.
If you take a much longer viewpoint of human population.
00:30:18.200 --> 00:30:27.188
And that is shown on the graph here, where the Y axis is the
world population. The X axis is here. Now we have to speak
00:30:27.188 --> 00:30:36.024
the ability to reconstruct world population, starting from
almost one idea roughly and if you look at the human
00:30:36.024 --> 00:30:44.258
population since that period of time up to today. It looks
that it is growing faster than exponential, which brings the
00:30:44.258 --> 00:30:51.329
sustainability question that we started this lecture with
into focus. This is exactly what we mean. How can we explain
00:30:51.329 --> 00:30:57.403
the faster than exponential growth in human population. If
we are anticipating to see carrying capacity restrictions
00:30:57.403 --> 00:31:05.308
kicking in this point was not whist by the famous physicist
Heinzvong forester And he had a wonderful paper in one
00:31:05.308 --> 00:31:11.960
thousand, nine hundred and sixty, where he actually brought
this point up. He said," Hey, there is something fishy. If
00:31:11.960 --> 00:31:17.200
you have faster than exponential growth and human population
that cannot be compatible with the logistic equation. The
00:31:17.200 --> 00:31:22.439
fastest growth you could see in the logistic equation
is exponential. So if something is growing faster than
00:31:22.439 --> 00:31:29.875
exponential. It means that the whole assumption of your
logistic equation and any projections from it comes into
00:31:29.875 --> 00:31:38.269
question right okay now the paper was quite catchy,
very stylish, also very well written and very humorous.
00:31:38.279 --> 00:31:45.388
If you read it. And I'll explain a little bit the humor as
we go along, he titled," This was science." Yeah, this is
00:31:45.388 --> 00:31:52.797
when the days were, you know, you could you could publish a
paper and science and have a little bit of fun and still do
00:31:52.797 --> 00:31:59.836
serious work. So the paper was titled " Doomsday Friday,
the thirteenth November, A, D, Two, O, Two, six. At this
00:31:59.836 --> 00:32:05.919
date, human population will approach infinity if
it grows as it had grown in the last two million.
00:32:07.619 --> 00:32:14.518
Now, the crafty devil, this guy hines one forester. I mean,
with few data points, he was able to do miracles. It was a
00:32:14.518 --> 00:32:21.334
little bit like Wilfrid Conrad. In that sense, he gave him a
few data points. He could do a lot and incidentally,
00:32:21.334 --> 00:32:27.333
the co authors with with with Hansbong Forest. There happened
to be two undergraduate students who were taking his
00:32:27.333 --> 00:32:37.301
class. So just okay so he actually decided that you
know what, to really try to understand the limitations of the
00:32:37.301 --> 00:32:44.059
logistic equation. There are two ways to look
at it as as in life. You could be an optimist.
00:32:45.490 --> 00:32:51.214
So he was trying to interpret what it means if the population
is growing faster than exponential @unoise@ which, is
00:32:51.214 --> 00:32:57.128
pretty much what the people and stable is it? Ofs do yeah.
They look at the two end members, and they come up with a
00:32:57.128 --> 00:33:02.323
mixing moment. So his argument is that we can identify the two
end members, then we can figure out everything in between
00:33:02.323 --> 00:33:02.560
here.
00:33:03.099 --> 00:33:09.417
So he pitied the pessimist against the optimist. Basically,
in this paper and he said," If you were a pessimist,
00:33:09.417 --> 00:33:14.179
you would say," Hey, sooner or later, the carrying
capacity is going to kick in. Environmental degradation is
00:33:14.179 --> 00:33:20.017
happening we are going to be able to sustain less and
less, and still at steady state, we are going to reach a
00:33:20.017 --> 00:33:25.429
carrying capacity pretty fast. That is going to depreciate,
even though we don't see it now in the human population, but
00:33:25.429 --> 00:33:31.443
it is looming in the background. So that was his specimen
view. He actually made it sound like this view is so boring
00:33:31.443 --> 00:33:34.840
that it is not even worth exploring
beyond what I have just said.
00:33:35.160 --> 00:33:42.746
The more interesting one, according to him, is the optimist.
What is the optimist? the optimist, basically will make the
00:33:42.746 --> 00:33:49.866
claim that the carrying capacity can increase with population
due to technological innovations. If, ah, you think about
00:33:49.866 --> 00:33:59.163
the problems now on the planet, you could distill them pretty
much to water, food, energy and disease. I mean, we all
00:33:59.163 --> 00:34:07.611
know what the big problems are. I think it is safe to say
that they are pretty much decided upon and we all know
00:34:07.611 --> 00:34:13.737
that they are interlinked. They are not entirely independent
problems. The challenges, how do you draw the boundaries
00:34:13.737 --> 00:34:22.099
around the inquiry of these connections. But but I think
it is safe to state that we are well aware of what the big
00:34:22.099 --> 00:34:23.849
problem is facing humanity are.
00:34:24.199 --> 00:34:31.310
And so he said that if you are an optimist, you could say,"
Well, every time we have had carrying capacity
00:34:31.310 --> 00:34:37.730
limitations kicking in. We have seemed to have found a
technological solution to overcome it. So that is the optimistic
00:34:37.730 --> 00:34:38.010
view.
00:34:38.260 --> 00:34:46.471
So if it was food limitations, we found agriculture. We
found fertilizers. If it is cold and cold health. We found
00:34:46.471 --> 00:34:54.642
vaccines, we found antibiotics, energy. We have fossil fuel.
We have fire you name it. So it seems that every time we
00:34:54.642 --> 00:35:01.271
hit a bottleneck, the population as a whole was able to
find the technological innovation and eliminate that bottle
00:35:01.271 --> 00:35:08.861
like. So that is the optimism, which is pretty much what you
would see now, in most political debates, no more or less
00:35:08.861 --> 00:35:11.819
you could distill it to these two simplified views.
00:35:12.360 --> 00:35:18.191
So then let us explore what happens if we pursue the line
of arguments of the optimist where does that take us
00:35:18.191 --> 00:35:20.219
and that is what warm forest suggested.
00:35:20.980 --> 00:35:27.003
He said that," Ok, well, if this is the case, then a macroscopic
description. So we are not really concerned about a
00:35:27.003 --> 00:35:32.960
society in this area, or that area, or whatever. We are
looking at the whole globe, collectively, our collective
00:35:32.960 --> 00:35:40.479
knowledge, how it gets transferred how we do things
that is not really the issue here but collectively we
00:35:40.479 --> 00:35:46.912
have seen that the carrying capacity limitations have been
overcome because of the population growing. So the more
00:35:46.912 --> 00:35:53.929
people we have, according to the optimist, the more probability
we have to find the technological fix to our limiting
00:35:53.929 --> 00:36:01.648
carrying capacity okay so fair enough that is a
very optimistic view so he said," If you are really,
00:36:01.648 --> 00:36:08.314
really optimistic, then the carrying capacity will scale with
the population raised to some power and that
00:36:08.314 --> 00:36:15.506
power would be bigger than one. So that was the most optimistic
you could get. According to him, you cannot get better
00:36:15.506 --> 00:36:18.779
than this. Okay, so where does that take us?
00:36:18.909 --> 00:36:26.756
so he decided to go back and revisit the logistic equation.
So he said," If you go back to the logistic equation with
00:36:26.756 --> 00:36:33.051
this model of the carrying capacity that is now not constant,
but grows with human population, because you are able to
00:36:33.051 --> 00:36:39.160
find all these technological fixes. It grows with the human
population, then the logistic equation reduces to this form
00:36:39.160 --> 00:36:45.917
we have rather than being a constant proportional
to the carrying capacity here. It is actually dependent on
00:36:45.917 --> 00:36:52.823
the population raised to some exponent delta, plus one minus
the term that is actually causing the reduction population
00:36:52.823 --> 00:37:00.785
p square. Now he said," If death is bigger than one. So this
exponent here is bigger than one. And as the population
00:37:00.785 --> 00:37:07.688
increases this term that is here this time that is
here will dominate the solution, because this term is, say,
00:37:07.688 --> 00:37:13.867
cubicle or fourth power or fifth power. This term is
reducing the population as quadratic, for sure, for big
00:37:13.867 --> 00:37:22.174
populations. The bigger the power, the more dominant that
is going to be okay. Fair enough. So let us do some
00:37:22.174 --> 00:37:28.217
simplifications. We are going to get rid of this term and
just leave the big time when we are looking at the
00:37:28.217 --> 00:37:29.509
difference between these two terms.
00:37:29.869 --> 00:37:36.241
So he did that, and basically you could simplify that for
large populations, the logistic equation now looks more like
00:37:36.241 --> 00:37:43.253
that. D the rate of change of a population with respect
to time scales, with the population raised to some delta
00:37:43.253 --> 00:37:51.528
plus one. And even I can solve this differential equation.
So if you do that, you find that this solution turns out to
00:37:51.528 --> 00:37:57.699
be the exponent itself Delta multiplied by this proportionality
constant, our prime time, Steve minus some integration.
00:37:57.710 --> 00:38:03.879
Constant race of the minus one over delta,
which we can write a little bit this way.
00:38:04.909 --> 00:38:09.799
Okay, so now we have something here that
looks like time in the denominator.
00:38:10.329 --> 00:38:18.287
Fine. Let us try to get rid of this integration constant
by imposing some plausible condition, say we know the
00:38:18.287 --> 00:38:22.270
population at some time equal to zero, say one eighty.
00:38:23.579 --> 00:38:31.303
So let us see that Pe of zero is penal. So that could be a
reference point that you pick but, not the Garden of
00:38:31.303 --> 00:38:37.863
Eden because that would be small populations so
then becomes when you impose this initial condition,
00:38:37.863 --> 00:38:44.768
and now you have peanut as the initial population becomes
actually shown here and automatically you see what is
00:38:44.768 --> 00:38:47.529
going to happen now with the optimistic solution.
00:38:48.929 --> 00:38:55.899
There is a chance that if T goes to peanut
race of the minuscule divided by Delta R.
00:38:56.269 --> 00:39:08.391
The denominator of this quantity will go to zero, raised to
some positive exponent, and one over zero is interesting. So
00:39:08.391 --> 00:39:14.910
at finite times, at finite times, we
can have an explosion in population.
00:39:16.130 --> 00:39:26.463
This is known as finite time singularities. The concept
is not new if you talk to engineers, they already knew
00:39:26.463 --> 00:39:34.336
about critical time to failure. But it was an interesting
idea in terms of population dynamics, that the population, in
00:39:34.336 --> 00:39:43.194
fact, as bomb forester, hinted that in the title of the Paper
of Science that, you know, this is, in fact, an explosion
00:39:43.194 --> 00:39:51.056
in finite time of population. Now, what does that really mean?
When we say our population cost in twenty one
00:39:51.056 --> 00:39:58.142
the universe cannot handle that. No, what is happening? well,
this is a little bit of an interesting discussion. You
00:39:58.142 --> 00:40:05.787
actually never reach the critical time. What happens is that
the system crashes and new dynamics have to emerge. Okay,
00:40:05.787 --> 00:40:13.082
so the history of of the population dynamics that was
describing the time evolution of the population after T C
00:40:13.082 --> 00:40:21.067
will no longer hold okay. It is a little bit like the Big
Bang. Yeah, if you think about the big bank, you could say,"
00:40:21.067 --> 00:40:25.999
Hey, the universe started." thirteen point six
billion years ago for what was there before.
00:40:27.750 --> 00:40:32.190
Who knows, there was no time time
was created after the big back.
00:40:33.269 --> 00:40:40.070
There was no space so. The whole question is wrong.
What was before the big bank. You cannot ask that question.
00:40:40.079 --> 00:40:46.255
There was no time similarly after the
critical time the dynamics of that population are no longer
00:40:46.255 --> 00:40:50.039
described by this differential
equation. Does that make sense?
00:40:50.730 --> 00:40:59.889
okay, so it appears, therefore, that yes, when you reach
this finite time singularity, or this critical time, the
00:40:59.889 --> 00:41:08.349
dynamics have to change basically, and new dynamics or new
regime change to bother from our Us. Politicians when they
00:41:08.349 --> 00:41:14.480
want to change regimes and countries the same idea,
but that is external rather than indogenous.
00:41:15.389 --> 00:41:24.427
Okay, so now what did bomb forester calculate based on
what he knew out of curiosity. He used the dataset, fitted
00:41:24.427 --> 00:41:32.020
the data set to his power law model and estimated that
the critical time would have been two or two, six.
00:41:33.610 --> 00:41:40.940
So that is a calculation Now. You could say okay,
we understand that two or two? six. But what about this?
00:41:40.949 --> 00:41:41.460
thirteen, November.
00:41:42.489 --> 00:41:44.229
Where did that come from?
00:41:44.750 --> 00:41:52.600
certainly the population dynamics was not that precise
now I can only offer a conjecture okay.
00:41:52.989 --> 00:41:56.429
Von forester was born on November thirteenth.
One thousand, nine hundred and eleven.
00:41:56.750 --> 00:42:04.136
And if you do the calculations, it turns out that November
thirteen, two or two, six would have been his birthday
00:42:04.136 --> 00:42:13.570
so it gives you a sense of the style of the paper he
snuck it in science. You have to give him credit for that.
00:42:13.730 --> 00:42:21.449
Okay, so with that there are still a few questions that
were pretty entertaining, that was asked by one forester.
00:42:21.460 --> 00:42:30.264
It didn't stop here, didn't stop here. There is more to come.
Okay, but just to summarize the outcome, he says," Hey,
00:42:30.264 --> 00:42:39.992
look, if you are an optimist or a pessimist. If you are a
pessimist, you are Maltusian by profession. Now your job is to
00:42:39.992 --> 00:42:47.895
be negative. If you are an optimist, you are by heart
that was the conclusion of the paper but there
00:42:47.895 --> 00:42:55.673
is actually more so he said," Hey, as much as it is fun to
use mathematical models to predict the future. It is equally
00:42:55.673 --> 00:43:02.355
fun to try to go back in the past and he actually asked
an interesting question. He said," If we use the doomsday
00:43:02.355 --> 00:43:10.053
equation to go back all the way to the Garden of Eden.
How long would it take us okay how long would
00:43:10.053 --> 00:43:18.423
it take us to go back to a population of two time reversal
clever guy so he didn't miss this point. He
00:43:18.423 --> 00:43:26.985
did not miss it. Okay so he used the original
doomsday equation and reversed time and said," Okay, well,
00:43:26.985 --> 00:43:34.578
what would be the time to reach two. Adam and Eve say, you
know, and turns out that you would need more than twenty
00:43:34.578 --> 00:43:35.100
billion years.
00:43:35.699 --> 00:43:43.652
Now, if the age of the universe is thirteen point six billion
years we don't know what is happening with way in
00:43:43.652 --> 00:43:51.892
thirteen point six. And and the creation of the Garden of
Eden. But the question is, why? and he actually had a very
00:43:51.892 --> 00:43:54.050
good answer. Nothing misses him.
00:43:54.539 --> 00:44:01.198
This is all coming from some twenty data points about the
human population of it. Just to show you what twenty data
00:44:01.198 --> 00:44:08.022
points can do in the hands of hindsown forester i'm
sure even Wilfrid will agree. This is impressive so he
00:44:08.022 --> 00:44:14.588
actually says well, you know to actually reach a phase
where the carrying capacity depends on the population. You
00:44:14.588 --> 00:44:18.089
need at least a population base,
because you need information storage.
00:44:18.219 --> 00:44:24.240
You need actually the population to grow to a sufficient
size before you could invest in scientists and research. You
00:44:24.240 --> 00:44:30.562
need to store food. You need to be able to have a certain
infrastructure in place before the carrying capacity actually
00:44:30.562 --> 00:44:33.869
grows as a function of the power law of the population.
00:44:34.280 --> 00:44:39.769
As a project there are two interesting
things that we could ask before we go next.
00:44:40.550 --> 00:44:46.162
First, how stable are those calculations of critical times.
This is an important problem, because basically what you are
00:44:46.162 --> 00:44:53.248
saying is that if you have a system, if you have any any
dynamical system that is growing as a power law with an
00:44:53.248 --> 00:44:55.020
exponent that is bigger than one.
00:44:55.949 --> 00:44:59.620
This system will experience a
crash, a finite time singularity.
00:45:00.840 --> 00:45:07.329
But how stable is the calculation of this final time cigarette.
If I give you ten points, we give you twenty points. Do
00:45:07.329 --> 00:45:14.197
I get the same answer? or is this critical time seems to
be a very moving target so one of the things that we
00:45:14.197 --> 00:45:20.200
would like to explore together is I've put the data set of
von forester, and more recent updates, up to one thousand,
00:45:20.200 --> 00:45:26.749
nine hundred and ninety eight on the web and so we can
actually play around fitting mold that is like the one of
00:45:26.749 --> 00:45:32.752
on Forster and see if we start adding every year from one
thousand, nine hundred and ninety eight, or even starting from
00:45:32.752 --> 00:45:38.119
one thousand, nine hundred and sixty. We start adding
incrementally, year by year, information about the population, and
00:45:38.119 --> 00:45:45.869
we track how well do we reach the critical to how stable is
the critical time is it a moving target every time we
00:45:45.869 --> 00:45:52.273
add a little bit of data points, the critical time also
shifts. It becomes almost like chasing ghosts so we are
00:45:52.273 --> 00:45:58.933
going to do a little bit of that, but we are going to do this
not just using bon foresters equation, but also some other
00:45:58.933 --> 00:45:59.199
approaches.
00:46:00.880 --> 00:46:09.411
Now, the topic of von forester was was picked up by another
physicist, by the name of Uh D S Ornett and the D. Sarnett
00:46:09.411 --> 00:46:16.241
has worked with the European Space Agency for a while and
was looking at material failure. Okay? because, uh, they
00:46:16.241 --> 00:46:20.329
didn't want a repeat of Nasa's Uh
explosion in nineteen eighty six.
00:46:21.409 --> 00:46:29.015
So basically, has looked at material failure
for a while and I noticed that material failure,
00:46:29.015 --> 00:46:37.250
especially fatigue, sheer crack propagations in materials
tend to grow as A as a power law too. But there are some
00:46:37.250 --> 00:46:44.333
interesting oscillations around it. And then when he revisited
von paper, he fitted a power law to the dataset
00:46:44.333 --> 00:46:49.973
that that he had. And that was up to while the paper was
published, two thousand, and so probably up to one thousand,
00:46:49.973 --> 00:46:51.080
nine hundred and ninety nine.
00:46:51.829 --> 00:46:57.159
And he found that this critical time that Bon forester has
calculated the stores when it suddenly becomes two. Thirty
00:46:57.159 --> 00:47:04.186
so that is the first thing that it could be a moving
career, but more important, more important for our analysis
00:47:04.186 --> 00:47:11.536
now is that the residuals around the power law are not random.
Now, if you have ever taken a course in regression
00:47:11.536 --> 00:47:20.408
and you do a regression. The first thing you check is whether
the residuals are random, right? do they pass a certain
00:47:20.408 --> 00:47:22.179
randomness test.
00:47:23.400 --> 00:47:33.830
Seems to have noticed that these residuals are clearly
structured. They seem to actually be even periodic, right?
00:47:33.840 --> 00:47:43.389
if you stare at this graph for a while, you almost see some
periodicities around this power law. Does that make sense?
00:47:44.610 --> 00:47:47.949
why is that? so he was curious.
00:47:48.079 --> 00:47:56.728
He, of course, suggested that, well, there may be a more
general theory than that should be able to predict the power
00:47:56.728 --> 00:48:04.377
law and the oscillations around the power law is this part
of a more general framework that allows us to actually see
00:48:04.377 --> 00:48:08.550
something more about the system
than just the power law," it said.
00:48:09.599 --> 00:48:16.393
And he tried to say," Okay, well, can there be more general
theory? and the answer is qualified. Yes, we will actually
00:48:16.393 --> 00:48:19.860
try to find unit in this summer school.
00:48:21.710 --> 00:48:30.191
So the idea that did is as follows he said," Okay,
let us take the population. And that seems to scale as a
00:48:30.191 --> 00:48:33.619
powered law with the critical time
minister raised to some exponent.
00:48:33.980 --> 00:48:40.439
Then he asked," What happens if this exponent is complex
meaning it has a real part and an imaginary part.
00:48:41.300 --> 00:48:48.796
Now, this did not come just out of the blows. He knew what
he was looking for. And I will tell you why he knew what he
00:48:48.796 --> 00:48:54.351
was looking for. But let us proceed that. Yes, maybe a complex
exponent is a more general description than just assuming
00:48:54.351 --> 00:49:03.829
the exponent is real. Okay? @unoise@ he didn't say why it was
complex. Just to be clear, he did not offer an explanation
00:49:03.829 --> 00:49:10.979
why he would choose this exponent here to be complex. But let
us let us humour him a little bit. And we proceed with the
00:49:10.979 --> 00:49:17.633
derivation that he had and assume it complex. What would that
take us? so the idea is that you take this critical time
00:49:17.633 --> 00:49:23.939
minister race to this power. Z and you break the Z
into a real part, and an imaginary part the real part
00:49:23.939 --> 00:49:31.913
being baked at imaginary by being omega and I squared
here is minus one. Yeah, then what you do is you write this
00:49:31.913 --> 00:49:39.601
quantity here? yes, as exponent of the log of that quantity.
Basically, we have just used the obvious fact that if you
00:49:39.601 --> 00:49:47.734
have a number, you take it slog at them, then you exponentiate
the outcome. You get back the same number can five log
00:49:47.734 --> 00:49:56.731
five eight with a log five, you get back five
then what he did is he used, of course, the well known
00:49:56.731 --> 00:50:05.280
property of the log that log X to the A is a log X. And then
he basically separated this A into its real part, Tennessee
00:50:05.280 --> 00:50:12.435
majority parts. So you have the beta log of T, C Minnesty
and the I Omega log of D, C minister. This is all in the
00:50:12.435 --> 00:50:21.451
exponent. Yeah, then you notice something else that each of
the A plus B can be written as each of the A times each of
00:50:21.451 --> 00:50:31.286
the B and that is what we did in this step so now
the analogue of Dc minister, plus I omegal log of Tc
00:50:31.286 --> 00:50:37.819
Minnesota can written as into the Betal lock. This
multiplied by exponential of log of these humanities. So
00:50:37.819 --> 00:50:45.309
all I have just done here is simply used the property that E
to the apron, V is equally eight times into the big .
00:50:45.320 --> 00:50:53.402
And then what he did is he brought the beta up here so
that is this part and he used the well known formula
00:50:53.402 --> 00:51:01.630
that he to the is plus yeah
and that is what you have here so far. So good. That is a
00:51:01.630 --> 00:51:08.310
quick review of complex numbers in case you forgot them. That
is why I decided to put the steps rather than just
00:51:08.310 --> 00:51:15.228
show you quickly the result ok so. Of course we
are seeing the population in real time. So we don't see the
00:51:15.228 --> 00:51:21.337
imaginary part. We just see only the real parts. So we
take the projection of this solution on the real axis, and
00:51:21.337 --> 00:51:27.459
basically you get rid of the eye sign of megaty
but the cosine omega of log T C minister will survive.
00:51:28.159 --> 00:51:35.050
So in this case, if you have a population that is growing
as a power law with an exponent Z that is complex.
00:51:35.599 --> 00:51:45.337
The solution still has a power low component but it
will be multiplied by the cosine of Omega log of Dc. Modesty
00:51:45.337 --> 00:51:55.047
and that is what is generating the log. Periodic
oscillations around that power law according to are
00:51:55.047 --> 00:51:59.309
you falling asleep? or this is exciting enough.
00:52:01.179 --> 00:52:09.844
Okay, so with this