Exponential Curve 6
Sustainability, Part 1
For the love of the human
race.
Thursday, December 11, 2014
Our Thesis
We are greatly indebted to Dr. Albert Allen Bartlett (1923-2013),
former emeritus professor of physics at the University of Colorado, Boulder.[1] These are Dr. Bartlett’s ideas, we are merely
reporting them. We have performed a
lengthy analysis of Dr. Bartlett’s “arithmetic” elsewhere.[2]
The rest of Dr. Bartlett’s talk, Parts 6-8 is concerned with
applications, illustrations, and various other anecdotal contributions to this
“arithmetic” which we pass over here. We
summarize the whole series of eight parts by adding Part 9.[3] In Part 10, we went in search of other
contributions that Dr. Bartlett might have for this exponential “arithmetic”
and we found the principle and “arithmetic” of Sustained Availability.[4] We also added the outline for a simple
“arithmetic” of Sustainability under the heading Sustainability Proper.
In his lectures Dr. Bartlett introduces the subject of
sustainability. It turns out that Dr.
Bartlett is one of the world’s leading authorities on sustainability.[5] We
are particularly interested in the reference, The Meaning of Sustainability.[6] If we need to review the crises that prompt
the discussion of sustainability, here are the major sources we used.[7] These were thoroughly reviewed in our
corresponding reports Arithmetic, Population and Energy, Parts 1 through 9.[8]
One cannot investigate either energy policy or energy theory
without a thorough understanding of the exponential curve.[9]
The Meaning of Sustainability
http://www.albartlett.org/articles/art_meaning_of_sustainability_2012mar20.pdf
Sustained Availability
In his paper, The Meaning of
Sustainability, Dr. Bartlett introduces the concept of Sustained
Availability as a means of coping with the energy crisis. He begins with the basic equation:
P = P0 * exp (-kt) = P0
* e-kt
This is mathematically equivalent to the basic
exponential equation, which Dr. Bartlett introduced in Arithmetic,
Population and Energy, Part 1.
y = a * bt = y0 * bt
We see in this new formula that Dr. Bartlett has changed his
notation from y to P to examine estimates of
production, P: this makes no difference to the math at all, it is
nothing more than a label change. The
exponent was positive, and is now negative[10]: positives indicate
growth; negatives indicate decline. The
growth factor b = 1 + r has been replaced by ek:
b = 1 + r is temporarily lost in the process.
“The greatest shortcoming of the human race
is our inability to understand the Exponential [Equations].”[11]
k = –dP/P // dt = –dP/dt * 1/P
As before, the area under the curve represents the total
quantity of a resource. This area can
now be related to k:
k
= 1/A
Any cut will make the resource last longer. If A(t) = 50 years-worth of
something we conclude that a declining rate of 2% will in theory make that
resource last forever.
This
is the minimum rate by which consumption of a resource must be reduced annually
in order to conserve that resource for as long as possible. Greater rates of reduction will produce
better results faster.
This is like the joke about the famished engineer and
mathematician who are only forty feet away from an elaborately set banquet
table, laden with a festal cornucopia of delicious things to eat, all in
quantities that stagger the imagination.
Unfortunately, our pair of heroes is only allowed to advance by half of
the remaining distance every minute. The
mathematician does the math and concludes that he will never get there exactly;
so he leaves in a huff to search for another place to eat. As he is rushing off, the engineer shouts,
“I’ll be close enough.”
Let’s see how that works out. At one minute, he is at 20 feet away; at two
minutes, 10; at three, 5; at four, 2.5; at five, a little bit more than a foot
away. The table itself is more than a
foot wide, and he can easily reach a foot without being ill mannered. For all intents and purposes, he may as well
be seated at the banquet table or standing in the middle of it in only five
minutes.
Even if the table were one hundred feet away, the numbers
would be 100, 50, 25, 12.5, 6.25, 3.125.
In five minutes he’s a little more than 3 feet from his goal. Considering the width of the table, he may be
already touching it.
As a practical rule-of thumb it is usually considered
impractical to continue this calculation more than five times. Engineers consider this to be a practical limit. The mathematician is a little too picky for
his own good. Even in a one thousand
foot room, the distance will be diminished in minutes.
It
is of the nature of exponential functions to devour time.
In the growth model, that property worked against us. Here we are employing that nature to help us:
but, it has practical limits; we cannot keep cutting forever, any more than we
can keep growing forever. How long can
we continue?
From our original equation:
ln(ekt) = kt * ln(e) = ln(2) ≈ 0.693
t = ln(2) / k ≈ 0.693 / k
This is our old friend, the rule of seventy, and if we
multiply both denominator and numerator by 100 we get the formula in
percentages. The rule of seventy
expresses time. Instead of doubling
time, we have halving time, or as the nuclear folks express it, half-life.
In our original example of 50 years, which represented a
necessary decline of 2% (-2% growth): we may now calculate a half-life of 35
years. The mathematician concludes that
we can make that resource last forever.
The engineer says that this is practically good for about five times
that amount or for roughly 175 years.
This
is the minimum rate by which consumption of a resource must be reduced every
year in order to conserve that resource for as long as possible. Greater rates of reduction will produce
better results faster.
On the other hand, 2% is not a draconian cut, so maybe we
can do better in this case. It should be
obvious that a 5% annual cut will make the resource last even longer. The goal of 2% reduction is merely the minimum
amount that will allow a limited resource to approach the behavior of an
infinite or renewable resource. To make
a finite or limited resource into a truly sustainable resource, we would have
to abstain from using it at all. This
defeats the purpose of a resource: the decision not to use a resource at all is
philosophically no different than not having the resource to begin with.
Since we are already in a state of decline; we had better
learn how to manage it.
k
= 1/A, and
t = ln(2) / k ≈ 0.693 / k
These two equations furnish guidelines for accomplishing
such management. This is nothing
new. Long ago, our forefathers knew,
“Waste not; want not.” The following figures
for the United States are no longer current; yet they paint a useful picture of
what could happen.
|
Sustained Availability
|
||||||
|
Resource
|
Reserve (years)
|
Report Year
|
Remaining Reserve (years)
|
Sustainability Reduction Rate (%)
|
Half-life (years)
|
Realistic Expectation (years)
|
|
Current Year
|
2014
|
|||||
|
Coal
|
223.23
|
2008
|
217.23
|
0.5%
|
150.57
|
753
|
|
Oil
|
10.48
|
2012
|
8.48
|
12%
|
5.88
|
29
|
|
Natural Gas
|
14.52
|
2012
|
12.52
|
8%
|
8.68
|
43
|
Sustainability Proper
“Can we transform our
society to a solar-based society which will probably have to be mainly an
agrarian society, while keeping and sharing throughout the world the benefits
of modern medicine and technology?”[12]
We now explore a
new reference.
http://www.resilience.org/stories/2009-11-06/dr-albert-bartletts-laws-sustainability
The Law of Carrying Capacity
0
≤ CC ≡ P * Cpc ≤ 1
Dr. Bartlett hints
at this idea without actually stating it.
Had he lived a little longer, we believed he would have originated this
statement. Carrying Capacity (CC)
is somewhere between zero and 100% (or 1) for the entire planet or any part of
the planet. We only know what CC
is by loading it. The load is the
product of population (P) and Consumption per capita (Cpc). So the loading (P * Cpc)
must always be in balance with CC, while everything we know about
CC depends on how and how much we load CC. Once conditions of 100%
sustainability equilibrium are reached for any fixed location, further growth
in CC cannot take place.
If P increases by a factor of u, Cpc
must decrease by a factor of 1/u.
If Cpc increases by a factor of v,
P must decrease by a factor of 1/v.
Once we have pushed the envelope to its
measurable limits we can replace 1 with a measured quantity (Q)
and attempt to operate somewhere safely within the limits of zero and Q.
0
≤ CC ≡ P * Cpc ≤ Q
Fair share is not a worldwide
constant. People living in the tropics
have different needs than people in the polar regions. Arid climates create different needs than
humid climates. “Therefore, CC
must be maintained in balance both globally and regionally. CC must be tuned, region by
region.[13]
The Law of Caretaking
The Law of Carrying Capacity expresses a worldview devoid of
human contribution. Its fundamental
assumption is that man contributes nothing more or less to the environment than
his bodily waste, and takes nothing more or less from the environment than his
bodily needs, as is the case with any other living animal or plant. Under this constraint, man is incapable of
making either a positive or negative contribution to the equation: man is
merely another unintelligent and irresponsible creature. Since, we commonly believe that this is
untrue, man is both intelligent and responsible; we now look for another,
broader model that expresses the effect of human contribution on The Law of
Carrying Capacity.
The Law of Caretaking expresses the worldview that man
contributes, either positively or negatively to CC: usually through labors as
hunter-gatherers, fishermen, farmers and foresters, or industrial manufacturers. This contribution is the
product of population (P) and Human Contribution per capita (HCpc). Its point is to express
mathematically, ideas expressed in Dr. Bartlett’s Seventh and Sixteenth Laws of
Sustainability. Caretaking, nurturing,
or the older husbanding, all suggest that man has a custodial responsibility to
creation, especially on earth.
A definition of sustainability. Sustainability is the maintenance of a closed
thermodynamic system in a steady state, so that based on any point of
observation all conditions will eventually cycle back to the same exact
conditions that existed at this first point of observation.
0
≤ CC+ HCpc * P1 ≡ P2 * Cpc ≤ 1
0
≤ CC+ HCpc * P1 ≡ P2 * Cpc ≤ Q
We understand that the population of contributors is not
usually the same as the population of consumers: hence P1 ≠ P2. There are an infinite number of ways to break
these concepts out in greater detail.
For example, we might draw a distinction between the contributions and
consumptions of hunter-gatherers, agriculturalists, and industrialists, each
with its own population of involved participants. Optionally we could divide the problem by
particular products or crops. Removing
the “pc” subscript notation, because we understand that all values are per
capita, we might arrive at something like this:
0
≤ CC+ HC1 * P1 + HC2 * P2 + HC3
* P3 + … + HCn * Pn
≡ Pc1 * Cc1 + Pc2 * Cc2 + Pc3 * Cc3 + … + Pcn * Ccn ≤ Q
≡ Pc1 * Cc1 + Pc2 * Cc2 + Pc3 * Cc3 + … + Pcn * Ccn ≤ Q
The only limit to the detail is the ability of the problem
solver to evaluate the problem. Any
number of scales or variations are possible.
Proof
of The Laws of Sustainability
The Laws of Sustainability in any of their infinite number
of forms are a simple and straightforward application of the fundamental laws of
thermodynamics. Since these laws are
commonly taught in high school general science and physics classes as the laws
of conservation of mass, conservation of energy, or conservation of
mass-energy, we do not believe that there is any further need for proof. QED
Relationship
with Dr. Bartlett’s Laws
The Twenty-one Laws. Laws One through Seven and Seventeen are
natural corollaries of either the Law of Carrying Capacity or the Law of
Caretaking. Laws Eight through
Twenty-one provide interesting comment, are sometimes corollary, but add
nothing to our effort to devise a functioning math model, with the following
two exceptions. The Eleventh Law applies
to Sustained Availability and not to Sustainability Proper. The Sixteenth Law (and to some extent the
Seventh Law) adds a new category of human participation which has now been
added to our equation.
The only independent variables named in the First Law are
population and rates of consumption.
These two variables, along with the variable for rates of contribution
are all of the necessary and sufficient conditions in any society. Since the Law of Caretaking incorporates
these variables in a thermodynamically consistent mathematical model, we conclude
that we have derived a mathematically consistent expression of Dr. Bartlett’s First
Law combined with his Seventh and Sixteenth Laws.[14]
The Seventh Law. The Seventh Law like the Sixth Law deals with
the fact that Law of Carrying Capacity only applies to closed thermodynamic systems. Although the earth in its total relationship
to the Sun and to our solar system is not specifically closed, it is effectively
a closed system. Radiation crosses this
boundary in both directions, but this does not appear to be significant in the
present discussion. The Sun must be
included because, it is the final thermodynamic heat source available, and for
purposes of this discussion, must be considered a mathematically infinite
source. It should be clear that people
crossing solar or other boundaries is an unsustainable idea. The same law which must apply to the whole
system, must also be applied to its sub-systems. Importation of people or labor is a violation
of the basic concept, and if it is done the sub-systems must incorporate it in
order to remain thermodynamically closed.
The Eleventh Law. The Eleventh Law also relates to fixed
resources and offers appropriate conservation suggestions. Efficiency improvements only conserve a few
percentage points, and are not sufficient in and of themselves to be a major
conservation contributor. This is not to
say that they are unimportant: their contribution does add up. Sustained Availability is theoretically
infinite, but is practically limited to a few years (Coal: 150 years; Oil: 5-6
years; Natural Gas: 8-9 years). Fixed
energy resources cannot be recycled. Recycling
would apply to things like aluminum, glass, and steel. The best recycling method is simply to reuse
the item with no other recycling process than washing.
The Sixteenth Law. The Sixteenth Law adds the factor of human
contribution. The Law of Carrying
Capacity needs to be modified to accommodate human contributions, either
positive or negative.
Conclusions.
We are indeed being pushed toward the Malthusian
Crisis. We agree that agriculture must
be made a sustainable pursuit, and the defects mentioned by Dr. Bartlett must
be overcome.
We have found all the arithmetic we need to develop a
sustainable community, and culture, the development of this arithmetic on a
larger scale and a broad change in public attitudes will result in a sustainable
society. The growth mentality must be
destroyed and replaced with a conservation mentality. What we have not found is if we have the
means and determination, the grit to actually accomplish this.
k
= 1/A, and
t = ln(2) / k ≈ 0.693 / k,
These two equations express all the arithmetic we need to
conserve our fixed resources for a maximum length of time.
0
≤ CC+ HC1 * P1 + HC2 * P2 + HC3
* P3 + … + HCn * Pn
≡ Pc1 * Cc1 + Pc2 * Cc2 + Pc3 * Cc3 + … + Pcn * Ccn ≤ Q
≡ Pc1 * Cc1 + Pc2 * Cc2 + Pc3 * Cc3 + … + Pcn * Ccn ≤ Q
This is the only arithmetic we need to figure out where we
are and where we need to go to become sustainable. HCn and Pn
can be developed by education, planning, and hard work. Pcn and Ccn
can be managed by education, planning, and immediate judicious belt tightening.[15]
[1] http://en.wikipedia.org/wiki/Albert_Allen_Bartlett
[2] http://swantec-ep.blogspot.com/2014/12/energy-policy-analysis-6-ra.html,
http://swantec-ep.blogspot.com/2014/12/energy-policy-analysis-7-ra.html, http://swantec-ep.blogspot.com/2014/12/energy-policy-analysis-8-ra.html
[3] http://swantec-ep.blogspot.com/2014/12/energy-policy-analysis-9-ra.html
[4] http://swantec-ep.blogspot.com/2014/12/energy-policy-analysis-10-ra.html
[5]
http://www.resilience.org/stories/2009-11-06/dr-albert-bartletts-laws-sustainability,
http://en.wikipedia.org/wiki/The_Limits_to_Growth,
http://www.resilience.org/stories/2009-11-06/dr-albert-bartletts-laws-sustainability
[6] http://www.albartlett.org/articles/art_meaning_of_sustainability_2012mar20.pdf
[7] http://www.albartlett.org/
http://www.albartlett.org/presentations/arithmetic_population_energy.html
http://en.wikipedia.org/wiki/Albert_Allen_Bartlett
http://www.youtube.com/watch?v=umFnrvcS6AQ
[8] http://swantec-ep.blogspot.com/
[10] The
negative, strictly speaking, is a property and part of the k as
is the positive also; it is not a distinct idea. In the original formula this surfaces as r:
for example, -2% results in a b of 98%, which is also a
description of decline. Please note that
k and r are approximately the same thing, but not exactly
the same thing: under 10% the error of approximation is small.
[11]
Dr. Bartlett, oft repeated sayings
[12] http://www.albartlett.org/articles/art_meaning_of_sustainability_2012mar20.pdf
[13] http://swantec.blogspot.com/2014/02/arithmetic-population-and-energy-part-6.html
[14]
We have not quoted Dr. Bartlett’s laws, because we do not know which of them,
if any, are protected by copyright. In
lieu of quotation we suggest that the reader is best served by reading all of
Dr. Bartlett’s Sustainability Laws in their entirety and in their context. Here is the best link to these Laws that we
have found. http://www.resilience.org/stories/2009-11-06/dr-albert-bartletts-laws-sustainability
[15]
For an excellent example please consult the writing and work of Lilienfeld,
Robert, Use Less Stuff. http://www.amazon.com/dp/0449001687/?tag=mh0b-20&hvadid=3483998937&ref=pd_sl_8boe0pihls_e,
and http://www.use-less-stuff.com/
[16] If you have been blessed or helped by any of these meditations,
please repost, share, or use any of them as you wish. No rights are reserved. They are designed and intended for your free
participation. They were freely
received, and are freely given. No other
permission is required for their use.
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