Energy and complex systems

1. Energy and complex systems Russ Abbott

2. Dynamical Systems: Attractors, Basins of Attraction, and Limit Cycles 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. • 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 Scholarpediaand Wolfram MathWorld

3. “Far from” (i.e., non-)equilibrium systems • 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. • 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.

4. “Far from” (i.e., non-)equilibrium systems • 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. Dissipative structure: a system that converts usable energy to waste heat, i.e., produces entropy. Persistent (stable, steady-state) dissipativesystem: 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.

5. 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?

6. 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. The state transition step is trivial. It repeats indefinitely. Furthermore, the finite automaton has only finitely many state and must revisit a state. http://www.princeton.edu/~mike/articles/amsterdam/amsterdam.htm

7. 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.

8. Every lawful dynamic system is cyclical Does this also work for continuous transition rules? • (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. How does this relate to the Morowitz theorem?

9. 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.

10. 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.

11. 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.

12. OSI Protocol Layers: what work is actually done? Some work is done packing and unpacking the data. Wrap Unwrap 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.

13. Standard strategy 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. Most machinery packages and uses low level energy to accomplish higher level ends.

14. Standard strategy • Most living things do the same. Most machinery packages and uses low level energy to accomplish higher level ends.

15. Autotrophs and heterotrophs Elsewhere: from an organic source

20. 2050 estimate from 2005 by LLNL

23. Text and image from Wikipedia.org. Dicrocoeliumdendriticum • 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. 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 (Cionellalubrica 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.

24. 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.

25. Energy exchanges http://commons.wikimedia.org/wiki/File:Pollination_Bee_Dandelion_Zoom.JPG 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?

26. T A B Braess’s paradox 12 12 0 C D T Total traffic: 10

27. A system is far from equilibrium • 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.  It is not at an energy equilibrium.  It dissipates energy.  There are energy flows through it.

28. OSI Protocol Layers