Is brain thermodynamics the link between economics and physics?

I've argued that the accumulated wealth of civilization is fundamentally linked to its total rate of energy consumption through a constant. The total historically accumulated value of humanity's inflation-adjusted production -- not just the annual accumulation called the GDP -- rises every year by a percentage that matches the increase in humanity's energetic needs.

But how could this be? The value of stuff is determined by our brains. How do our brains somehow "know" collectively how fast we consume energy? How do we comprehend how a psychological construct like money can be tied to a thermodynamic construct like energetic power? Doesn't economic value go only so far as human judgement?

As a clue, even with no one home and all the utilities turned off, a house still maintains some worth for as long as it can be perceived as being potentially useful by other active members of the global economy. Real estate agents talk about "Comps" for determining the value of a home. Comps are based on the recent sale value of other homes in the neighborhood. Comps were determined by people with brains (though arguably less so in a real-estate bubble) who in turn are connected through social and work connections to other people with brains, and with several degrees of separation, everyone on this planet with a brain. 

Individual brains process a wealth of information from the rest of civilization using extraordinarily dense networks of axons and dendrites. Patterns of oscillatory neuronal activity lead to the emergence of behavior and cognition. Powering this brain activity requires approximately 20 % of the daily caloric input to the body as a whole. Arguably this number is 100% since neither the body nor the brain could survive without the other.  

And we are connected not just to each other but, by definition, all other elements of civilization, including our transport and communications networks. We and civilization also couldn't survive without each other.  Dissipative neuronal circulations along brain networks may implicitly scale with dissipative circulations along civilization networks. Our collective perceptions must reflect global economic wealth. 

Individually, our brains may seem very personal, and a small part of the whole. But they are also connected to each other. They are part of a much larger "super-organism" that includes not just our bodies but our stuff. Our brains collectively march to broader economic circulations along global civilization networks that are sustained by a dissipation of oil, coal, and other primary energy supplies. 

Summing wealth over all the world’s nations, 7.1 Watts is required to maintain every one thousand inflation-adjusted 2005 dollars of historically accumulated economic production. This relationship may seem unorthodox by traditional economic standards, but it may also be seen as a type of psychological constant that ties the physics of human perception to the thermodynamic dissipative flows of energy that drive the global economy.

On the exponential growth, decay and collapse of civilization

Last week I had the fortune of seeing Rogers and Hammerstein's Carousel during a short trip with my wife to New York City. It's a 1940s classic set in a fishing town in New England. Some of the themes are a bit dated to be sure, but then I still love Italian opera which can be totally absurd. This particular Broadway production fittingly introduced Renée Fleming in the role of Nettie - a real treat to hear this world-famous soprano sing.

The plot of the musical contrasts a happy couple with one that is more challenged. For the happier, fisherman Enoch woos his bride-to-be Carrie in a song showing off his good-husband-material ambition:

Enoch

When I make enough money outa one of my boat,

I'll put all of my money in another little boat.

I'll make twice as much outa two little boats,

An' the first thing you know, I'll have four little boats;

Then eight little boats, then a plenty little boats,

Then a great big fleet of great big boats.

All catchin' herring, bringin' into shore;

Sailin' out again, an' bringin' more.

An' more, an' more, an' more!

The first year we're married,

We'll have one little kid.

The second year we're goin'

Have another little kid.

You'll soon be donnin' socks

For eight little feet-

Carrie

I am not enough for another fleet!

Utterly hokey, but presumably this was Rogers and Hammerstein's intention. At least it's clear that Enoch picked up somewhere a basic mathematical mastery of powers of two and the ingredients for exponential growth.

Exponential growth is curious, particularly in the economics literature where it is often presented as a God-given truth without questioning where it actually comes from. In fact, whether we look at boats, fish, or kids, or anything else, exponential growth is subject to fundamental thermodynamic constraints. The rate of exponential growth constantly changes over time as a function of past growth and current conditions, and that rate can evolve from being positive (growth) to negative (decay).

There's a couple of important themes: 

System growth
This is the most basic ingredient of exponential growth. As a system grows, it grows into the resources that enabled its growth in the first place, increasing its interface with its supply. A larger interface permits higher flow rates of the resources thereby allowing the system to grow faster. A bigger fleet catches more fish. As long as fish are profitable, this leads to a bigger fleet yet.

Diminishing returns

Even if a system grows into new resources, growth rates have a natural tendency to slow with time. The reason is that systems compete with their growing selves for available resources so that growth of the interface succumbs to diminishing returns. The more Enoch's fleet grows, the more his own boats compete with with the rest of the fleet for the remaining fish that are there; the bigger the fleet, the more competition. The consequence is that the interface of boats with fish does not grow as fast as the fleet itself so consumption stabilizes.

Discovery

As former U.S. Secretary of Defense Donald Rumsfeld famously put it, there are the "unknown, unknowns... There are things we do not know we don't know.". A system grows exponentially by growing its interface with known resources. Normally, diminishing returns takes over, but by way of this growth, there can also be discovery of previously unknown resources. Early Portuguese fisherman could not easily have anticipated the extraordinary riches of cod to be found in the New World that would propel fish catches skyward.

Depletion

Resources can be depleted if they are not replenished as fast as the ever increasing rate of consumption. In turn, growth of the interface between the system and its supply grows more slowly than it would otherwise.  Enoch catches fish to grow his fleet. But New England fish stocks decline - there are limits to growth.

Decay
Poor Enoch will eventually grow old and his boats and nets constantly need repair. What can't be fixed also slows growth. Exponential growth is still possible if decay is slow enough. But an unpredicted hurricane could wipe out Enoch's entire fleet of boats beyond his knowledge or control, in which case gradual decay can easily tip towards collapse.

Putting it together
Putting all these things together we end up with a mathematical curve for growth known as the logistic function characterized by increasing rates of explosive growth followed by decreasing rates of exponential growth. Growth then stagnates and tips into either slow or rapid decline.

An example of the timeline is shown above, illustrated for the special case where resources are in fixed supply and simply drained like a battery. Resources are consumed by the system; the system thrives on resources but is always consumed by decay. While growth is initially exponential, diminishing returns takes over. Then, during a period of overshoot, the system keeps growing for a time, even as resources and consumption decline, but eventually decay takes over and tips the system into decay and collapse. Critically, there is no equilibrium of steady-state to be had, not at any point.

But, the situation is rarely as simple as a depleted battery. This is because resources can be discovered.  The figure above shows how this works. All the same phenomena are present as in the drained battery scenario except just as the system enters overshoot and plateaus, a new resource is discovered, and the system enters a second period of exponential growth. Eventually decay still takes over, but it does not forbid the system from potentially entering some new phase of growth in the future, perhaps repeating the original cycle.

It's easy to see some of these dynamics at play in our civilization. At least in the U.S., energy has consumption has seen multiple waves of exponential growth, diminishing returns, competition and discovery. Since the mid-1700s, we have progressed from biomass, to coal, to petroleum, each discovery rescuing the U.S. so that it can continue expansion outward of its interface with primary energy supplies. Currently, natural gas and renewables appear to be entering a new exponential growth phase, with coal sliding into decline. 

Similar things can be seen in world population growth going back even further in time: always successive pulses of exponential growth, followed by stagnation, then discovery, and renewed expansion. We are now growing faster than ever.

So what does this mean for us and our future? The thermodynamics and mathematics of how a system grows can be described and predicted provided we know the size of resources and the magnitude of decay.  The problem is that we don't because there are always the "unknown unknowns". That said, we can say with some confidence that there are two main forces that will shape this century, resource depletion and environmental decline: it seems like one of the two will get us.  

So far resource discovery has more than adequately kept civilization afloat. But this cannot continue forever. When will it stop? This depends on this balance between discovery and decay. Discovery of new energy resources seems to be fairly unpredictable. Still, we've been remarkably good at it considering doomsday forecasts of Peak Oil have been overcome by the introduction of shale oil, natural gas, and renewables. Nonetheless, we currently double our energy demands every 30 years or so. Can new discoveries keep pace?  If they can, won't that lead to environmental disaster as atmospheric CO2 concentrations climb past 1000 ppm and we lay waste to the forests, oceans, and ground? 

Unlike diamonds, exponential growth cannot be forever. It just can't. Eventually, something has to give.

  

What's your Carbon Footprint?

Much has been made of the question of how we can reduce our individual impact on climate change. We all of us want to make a difference. If we individually consume less, surely we're doing our part to save the planet.

But I really think the premise is wrong, because in an interconnected world, none of us can be meaningfully separated from the whole. 

Consider the number of degrees of separation between you and anyone else on the planet. This might seem like a pretty hard thing to assess given how many of us there are and in some pretty far-flung places. I don’t know personally anyone in the Papua New Guinea Highlands (to mention some arbitrarily remote location), but I can be pretty sure that it’s not too much of a stretch to suppose my Australian friend has a friend who has been to the PNG capital Port Moresby where he ran across a guy whose cousin occasionally makes trips to the capital to work for “luxury” items to take back to his remote mountain jungle dwelling where he presents them to his wife. 

That would be just five degrees of separation. Certainly, the relationships are pretty far-flung, but it’s like the line from the TV series Breaking Bad,  “I know a guy who knows a guy.” None of us is truly independent of anyone else.

The same principle can apply to all of history. Suppose that an estimated 100 billion people have walked the earth in the last 50,000 years. With each successive generation, each of us is related to two others to the power of the number of generations. Exponentials lead to big numbers quickly: 100 billion people equates to just 37 successive generations. So, it shouldn’t take too great a number of generations before the number of your number ancestors is similar to the number of people living at that time. As evidence, all humans look and act pretty much the same. One way or another, there was sufficient intermingling for us all to have ancestors in common.

So, as a first approximation, we are linked through our social and economic connections to everyone currently alive, and moreover we can be linked by blood and tradition to everyone who has ever been alive.

It seems then that the question should be not what is your carbon footprint but what instead is our carbon footprint, that for humanity as a whole. We are a collective “super-organism” that has evolved over time by burning carbon based fuels to sustain ourselves and to grow. Individually, we may profoundly feel that we can behave as isolated entities; our personal economic choices, in however limited a way, can reduce the collective rate of CO2 exhalation. 

The evidence is against this argument, however.  If we term our collective wealth as the accumulation of all past economic production, summing over all of humanity over all of history, then the data reveal a remarkable fact: independent of the year that is considered, collective wealth has had a fixed relationship to added atmospheric CO2 concentrations. Expressed quantitatively,  2.42 +/- 0.02 ppmv CO2 is added every year for every one thousand trillion inflation-adjusted 1990 US dollars of current global wealth.   

A useful analogy here is to a growing child, who consumes food and oxygen and exhales carbon dioxide. The rate of CO2 exhalation by the child is determined by the sum of all cellular activity in the child. All the child’s current living cells require energy, and all produce CO2 as a waste product. But there are two key things: first, no set of cells can be magically dissociated from any set of others, e.g. the heart from the brain, as they are all interdependent; and second, the total number of current cells in the child is not determined by what the child does today, but rather by child’s past. Over time, the child grew from infancy to its current size, accumulating cells such that prior growth determined the child's current capacity to exhale CO2.

For humanity, it is the same. We currently “exhale” CO2 as a total civilization, but our current rate of exhalation is a collective enterprise by its intertwined parts as they have emerged from past civilization growth. 

So, if emissions are so tightly linked to the collective whole, and all past growth of civilization’s consumptive needs has already happened, entirely beyond our current control, what individually can we do right now?

To further illustrate the problem, let’s look at CO2 concentrations in the atmosphere. To calculate the actual increase in atmospheric CO2 concentrations, one has to consider that the land and oceans absorb a fraction of what is emitted. Estimating carbon sinks is possible but can get pretty tricky. Nonetheless, we can look at the observed relationship between economic activity and atmospheric chemistry to get a sense of what is going on. 

Looking above at the past 2000 years of atmospheric carbon dioxide concentrations, obtained from Mauna Loa in Hawaii and from ice cores in Antarctica, and measured as a perturbation from a baseline “pre-industrial” concentration of 275 ppmv, there is a surprisingly tight power-law relationship with global GDP. For the entire dataset :

log[CO2(ppmv perturbation)] ~ 0.6 x log[GDP(2005 USD)]

Amazingly, for over 2000 years, the relationship between CO2 and economic activity has been pretty much a mathematical constant.

In fact, if we look just at the past 60 years in the above, the relationship is linear and even tighter: since 1950, for every trillion inflation-adjusted year 2005 USD of global economy, the atmospheric concentration of CO2 has been 1.7 ppmv higher.

And, we could turn this around. With an extremely high degree of accuracy, we could estimate the global GDP simply with a CO2 probe at Mauna Loa. In units of trillion year 2005 USD and ppmv CO2:

   

GDP  = 0.58 x CO2 - 174

An atmospheric chemist could easily obtain the size of the global economy within a 95% uncertainty bound of just 1.5%! No need for economists!

Of course, we have to be careful with correlation and causation. And even if the above relationship has worked extraordinarily well for the past 65 years, the underlying basis for a relationship between GDP and CO2 concentrations is in fact rather more complicated. Nonetheless, these data clearly support an argument that what matters for determining the concentrations of this key greenhouse gas are collective human activities. 

The relationship between CO2 and world economic activity has been extremely tight and invariant over a very long time period during which the configuration of humanity has changed extraordinarily. There have been periodic wars,  famines, and global economic crises. We do not consume the same raw materials with the same efficiencies to the same extent now as we did in the past. The mix of wind, solar, nuclear and fossil fuels has been consumed in widely varying mixtures using an extraordinary range of different technologies. 

Yet this tie remains. What is going on? Speculating, perhaps one way to look at it is to consider individually the impact of buying that fuel efficient Prius, or turning down the thermostat. A car or house that consumes less fuel allows for an instantaneously incremental reduction in the demand for fuel. Sounds great. Except, the resources for producing the fuel are still available. If demand drops incrementally, then oil producers reduce prices to increase demand. Cheaper gas is more desirable, and so the collective response of all consumers is to consume more. Ultimately, the net effect on the collective rate of fossil fuel consumption of buying a fuel efficient Prius is zero (or even an increase). 

“No man is an island entire of itself...” We have no individual carbon footprint. We are only “... a part of the main”. We'd like to think our individual actions matter but it is only collectively will they reduce our impact on climate. And this will be very, very hard. As unpalatable as it may be, the only successful climate action will be to dramatically and collectively deflate the global economy.  Unfortunately, this may be a bit like asking that growing child, once it has reached a healthy adulthood, to voluntarily suffocate or shrink back to infancy. 

Is there an alternative perspective that allows for change but is still consistent with the observations? It would be nice to think that our individual or collective actions can meaningfully decouple the economy from changes in atmospheric composition. But how? 

Determinism and the human machine

I've been called a "dangerous nihilist" for trying to show how humanity can be treated usefully as a simple physical object. And a paper by D. Cullenward et al. strongly critical of my work - albeit by totally getting it wrong - referenced this rather funny cartoon, concluding that "Perhaps in the future a particularly brilliant scientist will discover a robust and verifiable means for deterministically predicting energy system dynamics. Until that time, however, the evidence suggests we should err of the side of humility and uncertainty in making projections about the future."

I get it. Treating the collective behavior of humans as simple by-products of the 2nd Law of Thermodynamics, something reducible to one line equations in physics, would hardly be the stuff of Keats or Shakespeare sonnets. Obviously, interpersonal dynamics can feel strong, and those feelings lead to decisions that seem at times to be utterly unpredictable, certainly a far, far cry from the elegant simplicity of an equation in physics.

But I don't see that poor predictability of the human condition is inconsistent with it being strictly deterministic, something that could be, at least in principle, reduced to a mathematical representation. 

The father of chaos theory, MIT atmospheric scientist Ed Lorenz was perhaps the pioneer of this idea. In his seminal paper Deterministic Nonperiodic Flow  he devised a simplified set of equations that represented some key processes in the atmosphere. The details don't really matter, but for edification here's the set up:

Aside from the rather extraordinary genius of representing the atmosphere in such a compact manner, what was so enormously influential was that Lorenz showed how purely deterministic equations - X, Y, and Z at any given time is uniquely determined by where X, Y, and Z start out - are nonetheless unstable and inherently unpredictable. This did not mean the solutions aren't well bounded. That is to say, on Earth, X, Y, and Z couldn't become just anything. It's just that the precise solution of X, Y, and Z  at any point in time could not be predicted very far ahead because even very small differences in the precision of the initial state translated to large differences in some not-so-distant future state. 

Sensitivity to initial conditions means the final outcome can be deterministic but nonetheless unpredictable. Determinism does not mean knowability. 

Of course, this does not mean we give up in despair. Well-bounded solutions do nonetheless exist. Not everything is possible, although we must accept a certain loss of resolution in our results the further out we look. 

Due to the approximate length of the water cycle, we know we can't peer beyond about 10-days in our weather forecasts, but we will accept a 1-week forecast for our weekend planning - albeit with a larger grain of salt than the 1-day forecast for the kids' soccer game. And we can still make climate forecasts that are averaged over space and time: it's not idiocy to claim that summer will arrive in the Northern Hemisphere around May, 2026 even if there's no prayer of saying it will rain in New York City on July 15.

I don't see it as being entirely dehumanizing to treat humanity in a similar fashion. We are wonderfully unpredictable and predictable at the same time. We don't know what exactly the day or year will bring even if we have a pretty good idea. Like the popular quote in the financial world "History doesn't repeat itself but it often rhymes". 

So it is with the approach I've taken to the evolution of civilization. There is no pretense at being able to explain the details at any given time or place, but there is predictability, predictability that can be tested with hindcasts, provided we step back and look at humanity as a whole. Stepping back, we can see farther into the future. Sensitivity to initial conditions yields to bounded solutions that are constrained by the laws of thermodynamics. For example, whatever a Nobel prize winning economist might imagine, we will not decouple the economy from energy consumption and carbon dioxide emissions. It's as physically possible as a perpetual motion machine.

Maybe we are a bit like Don Draper in the opening credits of the Mad Men series, falling deterministically through a series of life events, lacking any real internal control. Watching the series we already know the story: Don is irretrievably trapped by his mysterious past, and will inevitably succumb to women and booze. However, because we never know exactly how, we nonetheless derive the very human joy of watching his agonies as we binge-watch the next episode. Is this Nihilism? Voyeurism? I don't know. But I feel a bit the same about watching the progression of civilization through its own coming struggles with resource depletion and environmental decline.

EIA energy forecasts also spell economic doom?

The last post looked at the Energy Information Administration (EIA) energy forecasts to conclude that the 1% per year global energy consumption growth rate implied that over the course of the next 40 years we will consume as much energy total as the total we consumed in the past 100 years. Of course, we will consume even more if the growth rate continues at 2% per year, as it has in the past decade. If the past century of environmental destruction is any guide, destruction powered by our energy consumption, then the planet will be rather worse for wear in most of our lifetimes.

But what does it imply economically? Agencies like the World Bank and the International Monetary Fund forecast between 3% and 4% global GDP growth in the coming years. Can this be reconciled with EIA forecasts of just 1% for the fuel that power the economy?


A direct implication of the constant relating energy consumption and historically accumulated wealth that I have described is

GDP growth rate = Energy consumption growth rate  + Growth rate of energy consumption growth rate

So just to show that this isn't totally out to lunch, the respective mean growth rates for the 40 year period between 1970 and 2010 are

3.1%/year = 2.0%/year + 1.4%/year

3.1%/year = 3.4%/year. So not perfect, but pretty close, about 10% error. What we see globally is that GDP has been growing faster than energy consumption, but the difference can be accounted for by the second term on the right hand side above, the growth rate of the growth rate, a term I have been calling innovation since it can be related to improvements in energy efficiency.

So, let's now take a look at what the EIA forecasts imply for the future. If energy consumption has been growing at 2.0%/year, and the EIA projects instead a steady 1%/year, then the equation above for GDP growth would read:

1.0%/year = 1.0%/year + 0% per year

1.0%/year. Isn't that something close to a permanent recession? Keep in mind that 1.0%/year is an average value for the world, and that there will be competition among countries for this global constraint. Developed economies tend to grow more slowly than average, so this doesn't sound particularly rosy for those of us who live there. Really, I'm in no position to say what such an anemic growth rate actually looks like on the global economic stage, but it would seem to be well below what most economists would consider desirable. The last time growth stagnated like this over a long period of time was the 1930s. We know what followed.

And meanwhile, even at 1% per year energy consumption growth, we would still consume enough energy to bring about roughly a doubling of pre-industrial CO2 concentrations in the atmosphere, sufficient to blow well beyond the 2 degrees Celsius cap proposed by the Paris Climate Accords.

It seems we can't win!

The EIA forecasts environmental doom?

The United States Energy Information Administration provides projections for how much energy the world can be expected to use over the next few decades. Predicting the future is hard, but I think one has to give them credit for trying. The low, medium, and high economic growth projections shown above are largely just extrapolations of existing trends. Even if there is a curious inflection point around 2030, assuming persistence in trends is not a bad way of going for something as highly aggregated as the global economy.

The simulations use an everything but the kitchen sink philosophy for approaching the problem, representing to the greatest extent possible the myriad forces that drive energy consumption, such as political agreements and national and sectoral competition for a range of energy sources. Just the US macroeconomic module alone has well over one thousand equations.

But, as always, there's more than one way to skin a cat. For my part, I have developed a model for global energy consumption that is almost absurdly simple. It has only a few equations. Nonetheless it manages to produce accurate hindcasts for energy consumption and GDP growth rates with skill scores >90% for a 50 year period between 1960 and 2010 using only conditions in the 1950s to initialize the model.

The key ingredients of the model are only that global energy consumption and wealth can be linked through a constant; that inflation-adjusted global GDP grows global wealth; that the coefficient relating wealth to GDP is a function of past innovation; and, that innovation can be related through thermodynamics to resource availability and rates of decay.

Claiming that civilization can be reduced so simply is admittedly a bit unorthodox. What the model does have going for it is that each of these things is testable and based on physical reasoning.

Of course, the tremendous trade-off with this more holistic view is it offers little to nothing about the details, like how national consumption will change over the coming decades. Understandably, some think it's important to distinguish the U.S. from the rest of the world.

Still, we do still talk about the global economy. And, for an atmospheric scientist trying to link economic growth to climate change, it doesn't matter whether a molecule of carbon dioxide comes from Timbuktu or Trump Tower since CO2 is a long-lived well-mixed gas.

But let's assume that those thousands of equations the EIA uses does get things plausibly right, at least in the big picture. On average, EIA projections see the global demand for energy growing by about 50% over the next 40 years, 0.9% per year on the low end and 1.4% per year on the high end.

Using the aforementioned constant, what is being referred to by others as the Garrett Relation, a trivial prediction of the model I mentioned is that inflation-adjusted global Wealth will also grow by 50% over the same time period.

Some of us might feel a bit disappointed by a real growth rate for our collective assets of just 1% per year, but effectively this is what the EIA projections imply.

I am a bit skeptical they are correct partly because the physically-based economic model also forecasts that there is substantial inertia to existing trends. Between 2000 and 2010, the average growth rate for global energy consumption and real wealth was about 2% per year (although GDP grew faster, closer to 3% per year). A sudden revision downward in growth to 1% would require something fairly dramatic in terms of a reduction to resource availability. If we were to assume for the sake of argument a continuation of the 2% growth rate instead of the EIA's 1%, that would mean that global Wealth would increase by 60% in 40 years.

But whether the increase is 50%, as implied by the EIA, or 60%, as implied by persistence in trends, the future still looks good. Right?

Well, maybe not, at least not for the environment. Even maintaining 1% per year growth will require something that might seem pretty extraordinary: over the course of the next 40 years we will consume as much energy total as the total we consumed in the past 100 years. At 2% growth, that number is more like 140 years.

Energy does stuff. In thermodynamics we call it Work. A lot of stuff has happened in the past century. Consuming the same amount of energy in 40 Years will mean, very roughly, we will do the same amount of Work all over again.

A more advanced concept in thermodynamics is that there is a coupling of energy dissipation with material flows. What this means is that energy is consumed not just to sustain civilization's internal circulations, but to take raw materials from the environment like fish, minerals, and wood. These are used to repair and grow civilization (including by making more of us as people) while leaving behind a big pile of garbage in solid, liquid, and gaseous forms.

We've certainly packed on the pounds over the past century, largely at the sacrifice of the critters and plants on land and the fish in the oceans, while leaving behind an added 100 ppm to the atmospheric concentration of CO2.

What will the world look like when we manage to do the same all over again?

What's in a name?

Professor Richard Nolthenius of Cabrillo College has been referring to the constant relationship between power and wealth in talks and podcasts as the Garrett Relation, or GR for short. Of course, its a bit embarrassing to have one's own name attached to a phenomenon. In my view, the really interesting thing about any phenomenon is what it tells us, and not the much more incidental matter of whomever happened to stumble upon it first. But as Richard has pointed out, it needs a name. Is there better wording? The Power Theory of Value (PTV) Relation seems a bit dry but workable. Or is the Garrett Relation as good as any?