Energy and complex systemsRuss 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?

Paraphrased from Scholarpedia and Wolfram MathWorld

These are mathematical objects. Change is simply assumed to occur.

As a computer scientist, I might think of a Game of Life grid—or even a computer running its instruction execution cycle.

“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.

But not fixed, e.g., an orbiting satellite? Gravitational energy is available(?)—but

not to the orbiting body given its (constantly changing) velocity(?)

A chemical reaction at equilibrium. Not really static.

• 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.

Is the assumption now that the rules that govern the system are powered (only) by energy?

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.

Now we are really focused on energy.

How is energy stored in biological organisms so that it can be mobilized as quickly as it is?

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?

What requires energy?

•Any far-from-equilibrium system—if it is to stay far from equilibrium.

•Any agent based model that does anything interesting.

Cycles of states and cyclic processes

http://www.princeton.edu/~mike/articles/amsterdam/amsterdam.htm

http://ar.to/2006/02/game-of-life-in-javascript

Turing machine: a finite automaton and an unbounded tape.

Game of Life

The state transition step is identical from one transition to the next.

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

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.

Does this also work for continuous transition rules?

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.

• 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.

What good are 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.

Most of the work is done transmitting the bits.

UnwrapWrap

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 strategyMost 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 theoryImagine 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

Katmai National Park and Preserve, Alaska.

Marc Adamus

Citric acid or Krebs cycle

Release energy Store energy

2050 estimate from 2005 by LLNL

Dicrocoelium dendriticumD. 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

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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