Energy and complex systems
Russ Abbott
Dynamical Systems:
Attractors, Basins of Attraction, and Limit Cycles
• Dynamical System: a rule—sometimes required to be an equation—for time
evolution within a state space.
• State space: the set of all possible states of a dynamical system. Circular?
• Attractor: a set of points in a state-space-of-a-dynamical-system (a) that
remains in the set under the rule and (b) that some points approach.
• Basin of [an] attractor: the points that (eventually) approach the attractor.
• Limit cycle: An attractor within which trajectories are periodic.
▫ Why periodic? No external influence or internal randomness? If that’s the
assumption, how can it fail to be periodic?
As a computer scientist, I might think of a
Game of Life grid—or even a computer
running its instruction execution cycle.
These are mathematical objects.
Change is simply assumed to occur.
Paraphrased from Scholarpedia and Wolfram MathWorld
“Far from” (i.e., non-)equilibrium systems
• Nothing on the previous slide required the system to be either “far from
equilibrium” or “dissipative.”
▫ No discussion of what was driving/causing/enabling the system to change.
• [Energy] equilibrium: no energy is available to do work
▫ i.e., to move the system from one state-space point to another.
Is the assumption
 But not fixed, e.g., an orbiting satellite? Gravitational energy is available(?)—but not
now that the rules
to the orbiting body given its (constantly changing) velocity(?)
that govern the
system are powered
 A chemical reaction at equilibrium. Not really static.
(only) by energy?
• Far from [energy] equilibrium: (a) Energy is available to do work.
(b) The equations that govern the system are non-linear
—as they would be if the system were close to but not at equilibrium.
▫ Are equations now required? Why? They weren’t before.
In that case, they are
not “rules” they are
physical processes.
“Far from” (i.e., non-)equilibrium systems
• Dissipative structure: a system that converts usable energy
to waste heat, i.e., produces entropy.
• Persistent (stable, steady-state) dissipative system: a
dissipative structure that acquires energy at more or less the
same rate that it dissipates it.
• Autonomous system: a persistent dissipative system that
controls—at least to some extent—the rate at which it
acquires and dissipates energy—eats & acts.
How is energy stored in biological organisms
so that it can be mobilized as quickly as it is?
Now we are really
focused on energy.
Energy, cycles, and the Morowitz theorem
• A flow of energy through a steady state system that is able to store
energy will lead to at least one cycle. (Morowitz )
• All complex systems have a few basic underlying cycles. Everything else
that happens "rides on the back" of those cycles. (Cliff Hooker)
• Cycle normally brings to mind physical, chemical, biological, etc.
processes that return a system to a previous state: the nitrogen cycle, the
rotation/revolution of the earth, crop rotation, the circadian wake/sleep
cycle, the Krebs cycle, O2  CO2  O2, …
• Why do these cycles matter?
Cycles of states and cyclic processes
Game of Life
The state transition
step is identical from
one transition to the
next.
http://ar.to/2006/02/game-of-life-in-javascript
Turing machine: a finite automaton
and an unbounded tape.
http://www.princeton.edu/~mike/articles/amsterdam/amsterdam.htm
The state transition step is trivial. It
repeats indefinitely. Furthermore, the
finite automaton has only finitely
many state and must revisit a state.
Cycles of states and cyclic processes
• Yet in neither of these is the sequence of states that the system traverses
necessarily cyclic.
▫ It’s undecidable whether the system states will cycle.
• If one looks “one level down” and asks about the states through which the
state transition step pass, the question can be meaningless.
▫ In automata, the sequence of states traversed by the state transition step is not
given. It is assumed that the state transition step occurs atomically. There are
no internal states.
• So we must distinguish between the states through which a system
moves and the (typically simple) transition step that drives it though
those states as it repeats over and over.
Every lawful dynamic system is cyclical
• (Recall) Dynamical System: a rule—typically rules—for time evolution
within a state space. If
▫ a Dynamical System makes discrete transitions according to its rule(s);
▫ the state transition rules are finite;
▫ the state transition rules do not include randomness
Does this also work
for continuous
transition rules?
then the rule application/state transition step is the underlying “cycle.”
• This is the heart of all automata.
• It is also the heart of actual computers as well as virtual machines.
▫ The rule is the instruction execution cycle. Does it cycle? Its overall structure
is cyclical, but it executes instruction sequences that need not be.
How does this relate to the Morowitz theorem?
The function of cycles: to convey energy/work
“upwards”—by performing services
• How are state transition steps powered? What makes
them go?
▫ In computer science we ignore that question—as it is ignored
in the mathematical definition of a dynamical system.
▫ The underlying state transitions are just assumed to happen.
We then draw out the consequences of their occurrence in a
given context.
• In nature energy must be provided for state transitions
to occur.
What good are state transition steps?
• In computer science—and automata theory—the state
transition steps make the computer/automaton do its
thing.
▫ Were it not for them, it would be a paper weight.
• The computer/automaton “rides on the back” of the state
transition steps.
Then what?
• Using software we build more sophisticated cycles/state transitions.
▫ The Java abstract machine.
▫ The (abstract) mouse listener cycle.
▫ Various applications—such as ppt.
▫ In the Game of Life, the “patterns”—like the glider—whose actions and
interactions serve as the state transition steps for higher level constructs
like a Turing machine.
 For an amazing example see http://www.ibiblio.org/lifepatterns/. Open
Primer. Set Speed >Skip to 4 and Zoom to 0.
• More generally, any level of abstraction is powered by the state
transition steps of the layer(s) on top of which it is built.
OSI Protocol Layers:
what work is actually done?
Some work is
done packing
and unpacking
the data.
Unwrap
Wrap
Most of the work is done transmitting the bits.
Application A communicates Data (the higher level operation) to Application B by
marshalling low level energy flows in two ways.
•
At each of the two terminal nodes energy is used to wrap/unwrap the data.
•
At the network physical level energy is used to transmit the data.
Standard strategy
Most machinery packages and uses low level
energy to accomplish higher level ends.
Can we identify and analyze multiple levels
of energy flows in a system?
One simple example is basic power being
converted into motion that is used to carry
materials on an assembly line.
Standard strategy
Most machinery packages and
uses low level energy to
accomplish higher level ends.
Most living things do the same.
Autotrophs and
heterotrophs
Elsewhere: from
an organic source
Trefil, James, Harold Morowitz, and Eric Smith (Feb 2009) “The Origin of Life: A case is made for
the descent of electrons” American Scientist: http://www.americanscientist.org/issues/num2/2009/2/the-origin-of-life/5
Channel theory
Imagine a large pond of water sitting on top of a hill. We know that there are any number of other states—any in which
the water is lower than it is at the top—which have lower energy and are therefore states toward which the system will
tend to evolve over time. In terms of our question, the ”problem” faced by the system is how to get water from its initial
state to any state of lower energy—how to get the water down the hill. We need not think of the laws of physics as being
endpoint directed; rather, they simply adjudicate between states of higher or lower energy, with a preference for lower.
Can we apply the same reasoning to the chemistry of life?
For real hills, we understand not only that the water will flow downward but also many things about how it will do so.
Molecules of water will not each flow down a random path. Instead the flowing water will cut a channel in the hillside.
In fact, the flow of water is at once constructing a channel and contributing to the collapse of the energy imbalance that
drives the entire process. In addition, if we look at this process in detail, we see that what really matters is the
configuration of the earth near the top of the hill, for it is there that the channeling process starts. This part of the
analogy turns out to be particularly appropriate when we consider early chemical reactions.
In the analogy, the “problem” is the fact that the water begins in a state of high energy; the creation of the channel
”solves” this problem by allowing the water to move to a lower energy state. Furthermore, the dynamics of the system
are such that once the channel is established, subsequent flow will reinforce and strengthen it. There are many such
systems of channels in nature—the lightning bolt is an example, although in that case the forces at work are electrical,
not gravitational. …
The establishment of a channel can be seen as a phase transition, similar to the freezing of an ice cube.
Pingualuit Crater Lake, Canada
Marc Adamus
Katmai National Park and Preserve, Alaska.
Citric acid or Krebs cycle
Release energy
Store energy
2050 estimate from 2005 by LLNL
Dicrocoelium dendriticum
D. dendriticum spends its adult life inside the liver of its
host. After mating, the eggs are excreted in the feces.
The first intermediate host, the terrestrial snail (Cionella
lubrica in the United States), eats the feces, and becomes
infected by the larval parasites. … The snail tries to
defend itself by walling the parasites off in cysts, which it
then excretes and leaves behind in the grass.
Text and image from Wikipedia.org.
The second intermediate host, an ant (Formica fusca in the United States) swallows a cyst
loaded with hundreds of juvenile lancet flukes. The parasites enter the gut and then drift
through its body. Some move to a cluster of nerve cells where they take control of the ant's
actions.
Every evening the infested ant climbs to the top of a blade of grass until a grazing animal
comes along and eats the grass—and the ant and the fluke.
The fluke grows to adulthood and lives out its life inside the animal—where it reproduces,
and the cycle continues.
An analysis of the energy flows upon
which it depends
• Its energy (nutrition) is supplied by its hosts.
• It relies on multiple animals for transportation.
▫ Snails eat it, build cysts for it and then excrete it.
▫ Ants eat it and then climb blades of grass.
▫ Grazing animals eat it.
• It lives in a world of abundant energy flows.
▫ To analyze the energy flow upon which it depends
requires analyzing the energy flows of its hosts.
Energy exchanges
• When organisms are mutually
dependent, how are the energy
exchanges?
• How does the energy that bees get
from plants compare to the energy
required to transport pollen?
• For the bee the extra effort is
presumably negligible.
• Is the plant getting its money’s
worth?
http://commons.wikimedia.org/wiki/File:Pollination_Bee_Dandelion_Zoom.JPG
Braess’s paradox
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A system is far from equilibrium
 It is not at an energy equilibrium.
 It dissipates energy.
 There are energy flows through it.
• Why do we not see analyses of energy flows
through complex systems?
• Shouldn’t these be fundamental to the
structure of the system?
Nothing happens without energy flows.
Energy and energy flows are fundamental.
OSI Protocol Layers