**- Albert A. Bartlett[1]. This is especially true of exponential growth with short doubling periods. Particularly troublesome are functions that double in under 5-10 years because they wrong foot us. We think the near future is going to be like the recent past just a bit more so, when the growth rate means it's actually going to be radically different.**

*"The greatest shortcoming of the human race is our inability to understand the exponential function."*Then there's the revolutions and technologies that ought to be possible and contributing to some exponential growth function but are actually permanently 30 years out. Like Nuclear Fusion power, AI, Moon bases, batteries that are high capacity, low volume and cheap.

And there's 30 year futures with a date. 2030 still feels like the far future because back in 2000 it was. Except that now we're half way there and it's only 15 years away. So when politicians talk about targets for 2030 (especially about climate change), you'd better ask what they're going to do right now to get there, because it's not that far away any more. It's not just some far future that can safely be ignored for a few years. Take, for instance the recent PR about China-USA agreements on reducing CO2 emissions; That date of 2030 is prominent. If China and the USA have any chance at all of hitting even those relatively modest goals, they had better start going after them aggressively today, not in 5 years time.

So combine short doubling period exponential growth, with a belief in technical fixes that are actually permanently 30 years away, with a belief that 2030 is so far future as to not be worth bothering about right now. Does that look like sleep walking over a cliff with your eyes shut?

[1] Here's another good one from the same guy. "We must realize that growth is but an adolescent phase of life which stops when physical maturity is reached. If growth continues in the period of maturity it is called obesity or cancer. Prescribing growth as the cure for the energy crisis has all the logic of prescribing increasing quantities of food as a remedy for obesity." - Albert A. Bartlett

Doubling periods for compound growth in % PA

3.5% = 20 years

7.5% = 10 years

15% = 5 years

32% = 2.5 years

Start with 1. After 10 years you'll have

3.5% = 1.41

7.5% = 2

14.9% = 4

35% = 16

Start with 1. After 10 years you'll have

3.5% = 2

7.5% = 4

14.9% = 16

35% = 256

If the growth is in production rate and the produced objects are still around, or in consumption rate and you're looking at total qty consumed then the number accelerate faster. eg, start at a consumption rate of 1 per year. Grow that consumption rate by 35% per year, after 10 years you will have consumed 63 in total. or at 7.5%, 16.

Some useful exponential growth equations.Quantity given growth rate r after time period t, where 0 <= r <=1, and

t > 1:

y = e^(r * t)

Shorthand for large numbers, gives the 10^e notation:

ln(y) = e * (r * t)

log10(y) = (r * t)/ln(10)

If you know multiplier & duration, you can find growth rate:

100 * ln(growth_multiple) / duration

E.g., assume population growth from 2 people to 6.6 billion in 6,000

years. What is the average population growth over the period?

r = 100 * ln(6.6 * 10^9 / 2 ) / 6000

= 0.365%

If we assume 2 million years:

r = 100 * ln((6.6 * 10^9 / 2 ) / (2 * 10^6))

r = 0.001095%

Doubling time, given r: Rule of 70

doubling time ~= 70 / (r * 100)

E.g., a 2% growth rate = 35 year doubling time.

The more precise formula:

ln(2) / r

Where ln(2) ~= .693147

Doubling of consumption equals all previous consumptionIf a value is growing at a constant rate, and has always grown at that

constant rate, then the total amount consumed in the

lastdoublingperiod equals all consumption in all prior doubling periods. That is,

total consumption for all time is equal to 2 x consumption in the final

interval.

total consumption = 2x final doubling period consumption

If we know the multiple and want to determine how many doubling periods

are represented:

doubling periods = ln(X) / ln(2)

E.g., if there is 11,514x as much sunlight reaching earth as there is

energy used by humans, then the number of doubling periods represented is:

ln(11514) / ln(2) = 13.49

Combining Rule of 70 with growth ratesGiven a multiple and rate of growth, to find the time in which

consumption will exceed the multiple is:

ln(X) / r

So, if energy consumption grows at 2%/year and there is 11,514x the

sunlight reaching Earth than energy consumed, consumption will match

sunlight in 467.6 years.

Exponential consumption of a finite resourceIf a fixed quantity of resource is being consumed at an exponentially

growing rate r, periods of supply T given resource size n as a multiple

of present consumption can be calculated as:

T = ln(rn + 1) / r

So, for oil with 1668.9 billion barrels of reserves (BP Annual Review,

2013), R/P ratio of 52.9, and 2.2% annual growth:

T = ln(0.022 * 52.9 + 1) / 0.022

T = 35.045 years

or 2013 + 35 = 2048

Note that multiplying a resource 10x will only increase the exhaustion

interval by a much smaller amount:

T = ln(0.022 * 529 + 1) / 0.022

T = 115.27

*Peak consumption calculation8

Given a non-renewable resource growing at some r, the doubling period is

ln(2)/r, and the peak will occur at the resource duration T less one

doubling period:

Peak Time = ln( r * n + 1) / r - ln(2)/r

Cumulative consumptionFind the multiple of present annual consumption required to provide for

sustained growth over some time period. The quantity (as a multiple of

present consumption) is:

n = (e^(r * t) - 1) / r

So, a static stock of energy capable of supplying recent growth rates of

2%/year for the next 1,000 years would be:

n = (e^(0.02 * 1000) - 1) / 0.02

... n ~ 24.258

billionSince global energy consumption is roughly 722 quadrillion BTU

supplying that energy from uranium (80,620,000 MJ/kg) would require 229

trillion tonnes. As thorium, 232 trillion tonnes.

Fortunately that's less than a single earth mass: 3.89e-8, or roughly

0.389e-9, one third of a billionth, of an earth mass.

As antimatter, that's 102.6 billion tonnes.

As nuclear fusion, given 1 kg of hydrogen converts 0.007kg of matter to

energy and e=mc^2, 29,371.64 billion tonnes. About 0.02% of all water

on Earth.

Basic equations from: http://www.consumptiongrowth101.com/Basics.html

One I find fascinating is the rate of generation of data in pure bytes per year. I believe we're close to doubling the rate of total global generation of data in under 12 months. It's quite hard to see how long we can maintain that. And long term storage isn't growing as fast so a lot of that data is being thrown away. However it is having the effect of re-inforcing Gibson's narrow present. The past disappears because you only need a couple of doubling periods for current production to swamp all previous production. And the future gets closer because today will be swamped and disappear in only a couple of doubling periods.

1% PA leads to radical effects in timespans similar to a human life. Anything over 10% PA causes radical change in timespans comparable to political cycles. 50% PA and up is constant revolution. The landscape has changed again before you can integrate the previous changes.

Rates of change and growth are pretty clearly hitting unsustainable rates. Your notes on the relative percentage changes and the periods over which they show effects is well taken. A 1% growth rate requires a lifetime, 2% will show over 35 years or so, 3% in 20 years. And faster rates are just constant disruption.

Vaguely (possibly) related is the "natural" rate of interest, which a few commentators have remarked on. Over the long term, growth over the Industrial Revolution has trended around (IIRC) 3% p.a. Smith and Ricardo saw interest rates of ~6% p.a. Which makes me wonder if there's some underlying basis for either.

Another element of change: I've concluded that we're

notseeing technological change progress at rates as great as their heyday -- say, 1875 - 1925. But rather, therulesgoverning areas dominated by technology are changing, fast enough that it's difficult to keep up. This creates aperceptionof more rapid underlying change, but it's a false perception.It's still highly disruptive.

And yeah, those formulae are quite handy. Your post gave me an excuse to trot them out ;-)

I have a friend who worked on a little corner of the data analysis at CERN on the LHC. The raw data generation rates there are staggering. Part of the problem is about doing data consolidation very early before it's stored and then used for Big Data analysis.

There's clearly been some kickers to data generation in the recent past with the move from text to audio to HD video, and both commercial and gov tracking of everything. I'm not sure if that might be leveling out as we run out of things to track.

As always, then there's China. They're internal use of tech is well past the hockey stick knee but with still quite a bit of head room.

http://www.emc.com/leadership/digital-universe/2014iview/executive-summary.htm

Like the physical universe, the digital universe is large – by 2020 containing nearly as many digital bits as there are stars in the universe. It is doubling in size every two years, and by 2020 the digital universe – the data we create and copy annually – will reach 44 zettabytes, or 44 trillion gigabytes.You can sample from it, or subset it, or search it. Sometimes.

That last is the most troubling element to me: it means that you can produce any story you want by selectively sampling from real data. Allowing you to claim that it is "backed by evidence".

Cardinal Richelieu's wet dream.

http://www.theguardian.com/environment/planet-oz/2014/dec/08/goal-to-end-fossil-fuels-by-2050-surfaces-in-lima-un-climate-documents

I wonder what exponential compound de-growth is required to get close to that goal in 35 years.

I'd also estimate the chances of achieving it as approximately zero.