Exercises

1. Consider yourself as a system. Convince yourself that you are an open system.
   (a) Make a list of all your inflows and outflows (dissipative processes).
   (b) For each of the items in (a) think of what would happen (in the short and long run) if the process stopped.
   (c) Do you see why the dissipative processes are as important as the inflows ? Try to imagine a clone of yourself that had no dissipation but only inflows. Could it function ?
   (d) Are you in equilibrium or out of equilibrium ? Why ?
   (e) For life on Earth, what are the ultimate sources of energy that sustain it ?
   
2. Look at the world around you and find an example of a man-made system that is open. Answer the folowing questions with reference to your example.
   (a) Identify the sources of inflow and dissipation.
   (b) Discuss what would happen to the system if either or both of the inflow and dissipation were stopped.
   (c) Do the results of (b) seem surprising?
   (d) Does your system have dynamic structure (compare with the vortex and Benard cell example)?
   (e)What differences does your system have compared to living organisms ?
   
3. Look for some natural but non-biological examples of open systems and and ask the same questions as in the previous question.
4. Try to observe Benard cells in a home experiment (Careful !). See Ref.[6] for some pictures.
5. (a) Does the BZ reaction violate the Second Law ? Explain.
   (b)Belousov could not get his results published because everybody believed at that time that chemical reactions proceed monotonically. Belousov felt offended and never did any work in science again though he managed to publish his findings in the institute journal that did not require a review. Belousov's work was repeated and extended some years later by Zhabotinski. The Soviet Unioun finally awarded both of them its highest medal, but it was too late for Belousov who had passed away 10 years before [7]. Is there a moral here ? Compare with Boltzmann's fate.
6. Repeat the comparison in section(5.4) using the BZ reaction as an example rather than the Benard cells.
7. Explore the numerical solution of the Oregonator model of the BZ reaction using the Maple file MF07 from the CD of Ref.[4]. Try different values for the parameters and note any interesting patterns or oscillations that you observe.
8. Download the computer simulation of a model of chemical waves from Ref.[8] and try it for various parameters. Compare with the experimental results.
9. Look for potential Turing structures in Nature and discuss their similarities, differences, and possible functions or advantages from the perspective of evolution and natural selection.
10. With respect to the ideal gas and the Ising model, discuss those systems from an agent-based approach and a phenomenological approach.
11. (a) Do the Lotka-Volterra equations have an equilibrium state ?
    (b)Use your own software or the Maple file MF04 (that comes with the CD of Ref.[4], to study the Lotka-Volterra equations for some parameters and check the periodicity of the solutions. (Or use the applet in Ref.[].
    (c) Is the equilibrium state reached for generic initial conditions ?
    (d) Phrase your answer in (b) above in terms of stability of fixed points.
    (e) Compare the Lotka-Volterra equations with the Oregonator model representing the BZ reaction.
    
12. In the Lotka-Volterra equations, the little fish population would seem to increase without bound in the absence of a predator.
    (a)Is this reasonable?
    (b) How would you modify the equations to make them more realistic? Hint: Recall the discussion of the Verhulst equation.
    
13. Use your own software, or the Maple file MF06, to study the Rapoport model for the arms race.
    (a) Comment on the results found. (b) Compare the results, where possible, with historical facts.
14. (a) Read and discuss the article in Ref.[12].
    (b) Will humans ever learn from history, or will we forever repeat the mistakes of the past, and maybe annihilate everything ? (c) Remember your answer to (b) when we revisit the issue of conflicts from the point of view of self-organised criticality in a latter chapter.