Entropy, peak oil, and Stoic philosophy

NOTE: Images in this archived article have been removed.

Don’t worry; it doesn’t mean that this will be a very long talk! I brought this apple with me just because I wanted to tell you about Newton’s universal law of gravity. As we know, it seems to be true that he got the idea because he saw an apple falling from a tree (although it may not have fallen straight on his head!).

The fact that apples fall from tree – and that everything else that can fall, does – is an effect of the existence of relatively simple laws in the universe. Many things that we see around us are extremely complicated – or “complex”. Think of the solar system, for instance. There are many bodies of different sizes, moving in different trajectories. But there is a certain logic in it and the logic comes from a very simple law – Newton’s law – which can be expressed as follows:

Before Newton, for a long time scientists could only grumble something about “angels pushing” when asked about what caused planets to move. But if you know the law, you can describe not only the movement of the planets of the solar system, but all sort of bodies, including entire galaxies.

It is not rare to find an underlying simple law that generates complex systems. Think of fractals; Mandelbrot’s set, for instance. Fractals are not just mathematical entities, are common in nature, as well. Or think of models such a Feigenbaum’s bifurcations – they are the result of an extremely simple equation. These are examples of a class of systems that are relatively common in physics. Complex systems resulting from very simple laws. It is one of the beauties of physics that these systems exist.

Now, when we discuss of complex system, of course what comes to our mind is the subject we are discussing today – the economy and what makes it move. This is surely a very complex system and one of the problems economists have is that most of their models just don’t seem to be working very well. Sometimes, economists seem to be still thinking of the “invisible hand” and that looks very much like the angels pushing planets of long ago. But astronomers are not thinking or angels any longer whereas economists…. well, let me not engage in economist-bashing.

2. Newton’s apple in economics

So, let’s see if we can inject some physics into modelling the economy. Can we find something equivalent to Newton’s apple in economics? I think it is possible and let me show the observation that can give us the key that we need to understand how our economy works – considering that it is strongly based on non-renewable resources; crude oil in particular. So, here is this “apple” for crude oil, as Marion King Hubbert published it in 1956.

Note that Hubbert had only data until 1956, the rest is extrapolation. What this graph says is that he expected oil production in the US 48 lower states to behave in a certain way. Did it? Yes, as you can see in this image.

The agreement is impressive, considering that the curve spans several decades. But the main point, I think, is that oil production did follow a certain trajectory. There is a regularity, here. There is some kind of underlying law. And it is not angels – angels don’t extract crude oil (for what we know, at least; one wonders what energy source they use in Paradise…). So, let me show the historical production data for Hubbert’s case as we have them today. It is in Italian, but I think is easy to understand.

Now, I am telling you that this curve can be seen as the “falling apple” that gives us the key to understand the inner mechanisms of the exploitation cycle. Of course, before going on I have to convince you that this is a very general behaviour. All apples fall from trees in the same way and not just apples – also oranges and watermelons; just as do cats and dogs, planes and TV sets and whatever you can think of. Actually, not exactly everything – take a feather and you’ll see that it doesn’t follow Newton’s law. But, of course, you don’t jump to the conclusion that Newton’s law is wrong; of course. It means that – in order to find the inner laws governing a system – you need to make sure that the system is not perturbed by effects that will cloud the effects you are studying. In the case of gravity, you must ensure that the effect of air doesn’t affect too much the fall of an object. In the case of the Hubbert curve, you must make sure that government actions don’t affect too much production. In other words, Hubbert’s law works best in conditions of free market; when people can decide whether to extract oil or not depending on whether they think they can make money or not from the task.

3. The Hubbert law

This said, let me show you a few examples of Hubbert-like curves.

This one is the production of anthracite coal in Pennsylvania, one of the best examples we have of a Hubbert curve. I think it is from this graph that Hubbert got the inspiration to propose a similar curve for petroleum, although I am not aware that he ever mentioned this curve.

This is another example of a Hubbert curve, this time for a mineral not used for energy production: boric acid. These are data that I found just a few weeks ago. The curve is not a “perfect” Hubbert curve but, clearly, the trend is there.

This is another mineral commodity, phospates (From Dery and Anderson). I am showing this one because phospates are a fundamental fertilizer used in agriculture. We could live without oil, but we cannot live without phospates. Here; the curve is not complete, but the tendency is rather clear. 

And here you can see that the curve is the same also for commodities that are not thought of as “minerals”, normally. The Saudis had been extracting “fossil water” from underground aquifers and, for a while, they kept a flourishing agriculture with this water. Then, it was over. Luckily for them, they can import food with the money they make selling oil. But their oil will not be eternal, either.

So, I am asking you to follow me with this idea; that the bell curve is a “natural” behavior of production for non renewable or slowly renewable resources. With “natural” I mean that it is the way the system is expected to behave when there are no strong interferences from political or other kind of perturbations. Then, I said that we should look at the inner mechanisms that make the economy behave in this way. I believe that we don’t need to invent a brand new law, as Newton did for gravity. We already have the laws we need – even though so far we failed to apply them to this case. These are the laws of thermodynamics. Here are the three laws in a simplified form:

And now let me ask you a question: what makes water fall? You’ll say it is gravity; and that is correct. But there is a deeper factor here – this movement is eventually generated by the laws of thermodynamics. Nothing escapes thermodynamic laws. It is a question that I ask to my students, sometimes: how do you explain that water flows down in thermodynamic terms. It is difficult for them to find the answer right away, and yet they have studied thermodynamics. So, let me tell you; water flows down because of the second law – the entropy one.

You may remember from your studies that entropy is related to disorder. In some senses, it is true, but it is a definition that creates a lot of confusion. Think of entropy as heat dissipation. Then, everything that happens in the world is the result of some heat being dissipated – entropy tends to grow. When water falls from a high reservoir, some heat is created. The water at the bottom is slightly hotter than the water at the top – energy must be conserved, so it appears in the form of heat. Slowly, this heat is dissipated to the surroundings and that is what drives the system: entropy increase. The law of entropy is the law of change. Things move because entropy can increase – otherwise everything would stay frozen as it is. An equivalent way of saying that is that things happen because potentials tend to equalize. In the case of a waterfall, we have a gravitational potential difference (or “gradient”). With crude oil we have a chemical potential difference. There are other kinds of potentials, but let’s not go into that now.

Now, maybe it is not correct to say that something happens “because entropy must increase.” Probably, it is more correct to say that the universe behaves in a certain way and that it is convenient for us to describe this behavior with concepts such as “gravity”, “entropy” or “potentials.” These concepts are more useful than those involving angels pushing – or similar ones; such as the invisible hand…. sorry; no economist-bashing, I said. But, in practice, for these concepts to be useful I can’t just tell you, “the economy moves because entropy must increase”. It is true, but we need to go much more in detail. In order to do that, we need some kind of formalism where we can change the parameters of the system and see whether we can reproduce historical data, for instance Hubbert’s curve. That is what I’ll be doing; showing you how Hubbert’s idea can be derived from an interpretation that – ultimately – has to do with thermodynamics. But first let me introduce to you the method known as “system dynamics” which can be used to describe this kind of systems.

Let me show how system dynamics (SD) works by showing a description of a waterfall. Here, actually, it is about a bathtub, but the physics is the same.

The model is made out of boxes, arrows and valves. Boxes are termed “stocks” and arrows are termed “flows”. If there are two boxes connected to each other, a stock may flow into another depending on the potential difference. In general, the concepts of potential or gradients are not so often used in SD. This is a shortcoming, I think. Anyway, I said that I wanted to make a model that describes the economy and produces the behavior that we have termed “Hubbert’s”. In order to do that, a single waterfall is not enough. We need something just a bit more complex – like this three tiered fountain.

The driving forces in the water movement are the same as before – gravitational potentials. Now, this fountain is not the perfect model for what I am trying to do. Use it just as an illustration of the concept that it is a potential-driven system. For modelling an economy, we need a further step: the concept known as “feedback”. That means we have to assume that the flow from one stock to the other does not depend just on the size of the upper stock, but also on that of the lower stock. The model is now more like a biological model. Think of the lower stock “preying” on the upper stock and growing in proportion. Without feedback, we have no growth and the model does not define a real economic system. So, let’s take this further step and describe the model using the convention of system dynamics.

5. A simple model of the economic system

Here, we have a very simple model that has three stocks: resources, the economy, and waste.

Note the arrows that connect stocks to valves. These arrows indicate feedback. But note also that the system is driven by thermodynamic potentials. Essentially, the economy is an engine that transforms resources into waste. Its “fuel” is, mainly, the chemical potential of fossil fuels.

Now, the model is made using a software called “Vensim” which does not just draw arrows and boxes. It “solves” the model, that is it calculates the flows as a function of the initial amounts of stocks and of the parameters of the system (the “ks” here) – those that are basically describing the potentials. Again, let me state that these SD software packages are not thought in terms of thermodynamic potentials. One day, we may have packages specifically defined for that purpose. For the time being, let’s jut keep in mind this point. Now, let’s go on and see how the system works. With Vensim, you can change the parameters in real time and see how stocks and flows change. Here are some results:

The software allows you to solve the model iteratively; you see what happens as you change the values of the constants using sliders. And, here, you already start seeing “bell shaped” curves. We can plot the results in a better way; here is how the three main stocks (resources, the economy, and waste) vary with time.

This is a very, very general behavior – it works for a variety of systems. It describes chemical reactions, epidemics, and even the explosion of a nuclear bomb. I also found that it can be applied to the collapse of empires. In a way, it is something like applying Newton’s law to different systems – you can describe galaxies, planetary systems and spaceship trajectories, all with the same, simple law. Note that here, unlike the case of gravity, we don’t have a physical “force” that pulls together the elements of the system; nothing that you could measure with a dynamometer. But there is a powerful entity that moves the system anyway: entropy.

Now, back to the case of an economic system, you see that the “engine” which is the economy, revs up until a certain time, then it slows down and it falters. Eventually, entropy wins. When all the resources have been transformed into waste, then entropy has been maximized. In the case of the world’s economy, the transformation is mainly from fossil hydrocarbons (CxHy) to CO2 and, of course, the chemical potential of hydrocarbons is higher than that of CO2. The economy is an enormous, three stage chemical reaction.

We could modify the system taking into account many more effects – recycling waste for instance, but let me not go into that. Let’s see, instead, is how the model describes Hubbert’s curve which is the flow rate from the resources stock to the economy stock.

Qualitatively, you see that we do generate “bell shaped” curves. Here, the blue one (“production”) is the one that should be compared to the historical production data for crude oil or other commodities. That is possible, but not sufficient to say that the model is good. What I think is a fundamental test for this model is whether it can fit at least TWO sets of data; if possible more. This is a hard test, as I found out working on that.

In practice, we often have good data for production, but for “the economy” it is much more difficult. Nevertheless, we’ll see that we can find good “proxy” data for that. So, the model can be put  to this hard test and it succeeds. We can test the model on small economic systems that we may assumed to be self-contained. Let me show you an example, whale oil in 19th century. We had already seen the production data earlier on. The question, then, is what could we take as data for “the economy,” in this case related to that subsystem of the whole economy that was engaged in whaling at the time. Unfortunately, we don’t have these data, but we can find a good “proxy” for the size of the whole industry in the size of the whaling fleet. And we see that it works:

There are other examples. Together with my coworker, Alessandro Lavacchi, we published a paper on this subject that shows how even this very simple model can be used to describe the exploitation of non-renewable resources. Here is just another example: crude oil production in the US 48 lower states – the quintessential “Hubbert curve”.

Note that here we have used as “production” not the actual oil production, but the size of oil discoveries. That is because the main effort in oil production is discovery. Once you have found where the oil is, the development process is smooth – almost “automatic” – but it takes several years to go from the first successful find to actually producing something. And, as proxy for the effort of the oil industry we have the number of wildcats, that is of exploratory wells. Note how the industry made a big effort to find oil starting from the 1950s, but basically it couldn’t find much. It is typical, as I said.

Now, to show you what the model can do, let’s use it to extrapolate economic trends to the future. we could take as “production” the total world primary energy production and as “the economy” we use the world’s GDP as a proxy. And here is the result. This is a calculation done together with Leigh Yaxley a few years ago.

As you see, the model predicts that the production of primary energy will peak in a few years from now and then will go down irreversibly. The size of the economy (measured in terms of GDP), curiously, will keep growing for a while; then it will peak and decline as well. Of course, you may be perplexed about these results if you see them as predictions. So, I think I’ll spend a few moments discussing what exactly we aim to do with these models. One fundamental point is that we cannot make predictions of what will happen in decades from now. Maybe it makes sense to say that the world’s primary energy production will peak in four years from now; that is because we have other models that tell us that. But about the world’s GDP peaking in 2044, well, of course you have to take that as a guess. That doesn’t mean that the model is useless. If you ask the right questions to the model, the model will give you useful answers. Otherwise, there holds the rule of “garbage in – garbage out.” For instance, if you are asking, “How can the economy keep growing throughout 21^st century” the model cannot tell you that.

So, from the model you can gain important insights in terms of trends. For instance, if you see the world’s energy production going down and the GDP going up; then you might be very happy because you’ll say that the economy is becoming “more efficient.” But the model tells you that you are not being more efficient, you are simply using previously accumulated resources to keep the economy running. And, of course, you can do that only for a while.

But I do understand that this model is really very simplified. For instance, it does not include renewable resources and it is true that our economy is not completely based on non-renewable resources; even though most of it is. So, the question you may ask now is whether we can do something more detailed. How about adding to the model agriculture, recycling, renewable energy, etc.?

Sure. It can be done and – in fact – it has already been done long ago. The first time it was in 1971 in a work titled “World Dynamics” by Jay Forrester who, by the way, is the inventor of system dynamics. But let’s examine here the more detailed study that was published one year later, in 1972. It was inspired by Forrester’s work and I am sure you have heard of it. It is the “report to the Club of Rome” titled “The Limits to Growth” of 1972.

6. The Limits to Growth

Now, you may have heard that “The Limits to Growth” (let’s call it “LTG”) is an outdated work; that it was all a mistake, that they made wrong predictions and the like. Those are just urban legends. People tend to disbelieve what they don’t like and that is why LTG was so widely rejected and even demonized. I wrote an entire book on the story of “LTG,” it will be published next month, but let me not go into too many details. Let me just say that The Limits to Growth was a very advanced study for its times; it was not a mistake and its predictions were not wrong. In any case, these models are there to show you trends; not to give you exact dates for what will happen.

So, let’s go into some details. Let me show you the structure of the first LTG model, called “World3”. This is a scheme taken from the Italian 1972 edition:

This is a simplified model; it doesn’t reproduce all the features of the original. But it has the advantage of being “mind sized” – it is something that we can grasp and the use of images helps a lot; it is much better than boxes with some label on them. So, as you see, the model can be reduced to a small number of stocks. Here we see them: we have five main stocks; in alphabetic order we have agriculture, industrial capital, non renewable resources, population, and pollution,

Note that, again, this representation of the model does not show the thermodynamics behind. With the stocks arranged as they are in the figure, the potentials that move the system are not evident. Yet, they must be there. Nothing can move without a potential difference that pushes it. So, one thing that we’ll have to do someday is to make these potentials visible in the representations of these models. But, as I said, I am telling you about a work in progress – there is plenty of work in this field that someone will have to do in the future.

Now, let’s examine the model a little more closely. You recognize that there are three stocks which are just the same as those of the simpler model that I showed to you before. Here the stocks are given different names: mineral resources (the stock that was called “resources”), industrial capital (“the economy”) and pollution (“waste”). Then, there are two more stocks; one is agriculture – intended as renewable resources and then there is population. These two new stocks are needed for more detail in the model and, of course, there are many more connections: now the model can describe such things as recycling and the effects of pollutions on the industrial capital. Note also that “renewable” resources may not be absolutely so. Soil is not renewable if it is overexploited – it is called erosion.

At this point, we may go to the results. I am showing to you the data from the first edition of LTG, back in 1972, the main results haven’t changed much in simulations performed 30 years later with updated historical data. So, this is the output of the model for the best data available at the time; that was called the “standard run” (the graph is, again, from the Italian edition; the text is from the 2004 edition)

The labels in the plot are a little too small to be readable, but let me describe these results to you. First, the scale spans two centuries; starting in 1900 and arriving to 2100. We are about at the middle of the graph. Now, look at the “resources” curve (red). It has exactly the same shape as the one that we obtained with the simpler model, before. And the curves for industrial and agricultural production (green and brown), yes, they look very much like Hubbert curves, even though here they are not symmetric. This is due in part to the effect of pollution which adds to the effect of depletion. But it is not a very big change.

And then, of course, you see the pollution curve (dark green) – here a basic supposition is that pollution is not permanent – it is gradually re-absorbed by the ecosystem. So, the pollution curve goes up and then down, following with a time lag the behavior of industrial and agricultural production. Finally, there is population. It keeps growing even though agricultural production goes down; this is because people can still reproduce as long as there is at least some food. Actually, there is no direct proportionality in term of food availability and reproduction rate but, in any case, in the long run the lack of food takes its toll. Population starts going down too. What the graph shows is the total collapse of civilization – our civilization. It is thermodynamics doing its job; it is the way everything in the universe works.