Saturday 2 June 2012

30. Bifurcations in Phase Space and the Nature of Evolution

In our day-to-day life we tend to do 'linear' thinking. For example, if the number of guests expected for a party doubles, we plan for twice as much food. But the majority of natural phenomena are governed by nonlinear dynamics.

Consider water at, say, 70oC. Heating it a little raises the temperature by a small amount, and it is still a liquid. Small cause, small consequence. Such linear behaviour continues till the temperature is close to 100oC, the so-called phase-transition point for water at atmospheric pressure. Now a small increase of temperature has a drastic effect: Water changes to steam. This happens because now the system was in a state of unstable equilibrium (like being on top of a hill), and even a small perturbation was enough to make it 'role down the hill', so to speak, with no tendency to come back to the same equilibrium state again. The system has to seek out a new, different, state of equilibrium.

In physics the term 'phase' has a fairly well defined meaning. For example, liquid water and steam are two different (homogeneous) phases of H2O. But when it comes to complex systems, the term 'phase transition' gets used loosely and liberally. For example, V. S. Ramachandran speaks of a 'mental phase transition' in the evolutionary history of our brain:
'Then sometime about a hundred and fifty thousand years ago there was an explosive development of certain key brain structures and functions whose fortuitous combinations resulted in the mental abilities that make us special in the sense that I am arguing for. We went through a mental phase transition. All the same old parts were there, but they started working together in new ways that were far more than the sum of their parts.'
In such contexts, a better phrase than 'phase transition' is 'bifurcation in phase space'. I introduced the phase-space or state-space idea in Part 24. As some control parameter changes in a sustained manner, a situation can arise wherein the system has moved so far away from equilibrium that its potential phase-space trajectory suddenly undergoes a bifurcation: There are two alternative trajectories which can result in a lowering of free energy, and the one actually taken is often a matter of pure chance. Even the tiniest of random fluctuations can make the system choose one trajectory over the other. This is how the unpredictability factor arises in the dynamical evolution of any complex system. [Biological evolution is a subset of dynamical evolution.] And this can happen even for an otherwise deterministic situation.

Try balancing a pencil on its tip. When you release it, the direction in which it falls is purely a matter of chance. This is also an example of unstable equilibrium leading to a new state of stable equilibrium; and there is no going back.
The onset of lasing action is another example of such a bifurcation in phase space, or of a 'generalized phase transition'. After a laser system has been assembled, the lasing action can start only when a certain fine-tuning has been done. The fine tuning amounts to varying a certain control parameter, and at a critical value of the control parameter the lasing action is effected. The onset of the lasing action is rather like the emergence of order (or a breaking of symmetry) at a phase transition (like crystalline ice emerging from liquid water). The ordered-state characteristic of the laser is that in it a coherent emission of radiation occurs.

Lasing action is an emergent phenomenon arising in a complex system. Order emerges out of disorder (or less order) when the bifurcation in phase space occurs.

Symmetry breaking (cf. Part 9) is a commonly occurring adjunct of bifurcations in phase space. Ever since the Big Bang, there has been a huge succession of symmetry-breaking bifurcations, each resulting in the emergence of new states of order, the most spectacular example being the emergence of life out of nonlife. I shall describe some of the crucial milestones in this chain of bifurcations in future posts.

Bifurcations can occur repeatedly if a complex system continues to be driven further and further away from the original state of equilibrium because of a continual input of energy and/or matter. And usually there is a random choice of branch at each bifurcation point. This is what makes it impossible to predict the future behaviour of a complex system. Consider the figure below.
Suppose the present state of the system is found to be d2. From this we can conclude that it must have taken the evolution path b1c1d2. But if the system is at b1 at some instant of time, we have no way of telling that it would evolve to d2. It may as well evolve by the path b1c2 to c2, or the path b1c1d1 to d1.

There is an important lesson in this. Life as we see it evolved along a particular state-space trajectory sequence, from among a huge number of other possible trajectories. Chance played a role at many of the bifurcation points. Therefore, it is impossible to create in the laboratory those conditions from which the evolution would be exactly along the phase-space path that happened to have been chosen by the blind forces of Nature.

That we humans are what we are is a matter of chance. We may as well have evolved to be something very different. Of course, natural selection played its role, and natural selection is not a random process. But the overall chain of processes had chance elements interspersed here and there. It is like wind vs. stagnant air. In both of them the molecules of air move in random directions, but in a wind there is an average velocity which is along a nonrandom direction. Biological natural selection can superimpose an overall direction to the randomness inherent in most of the bifurcation events.


  1. I have a doubt. Is there anything called random or whether the time scale of observation makes it random? Further if we are able to go to higher time resolutions will we be able to get deterministic model?

  2. Suppose you are given a sequence of numbers. A question can be be asked whether the sequence is random or not. Clearly the answer does not depend on the time scale of observation. The second part of your question may be unnecessary in view of this answer. Or perhaps I have not understood the question.

  3. Sir, In modelling physical systems, whether the inability to produce accurate mathematical models requires us to go to random models? In that case, stochasticity may vanish if we are able to produce better mathematical models in future. OR if randomness is always inevitable ? why ?

    thank u

  4. I shall discuss these things when I come to the work of Stephen Wolfram. I think randomness is not always inevitable.

  5. I don't think there is any randomness in the universe. We introduced randomness(or probability) into mathematical models just to model our lack of knowledge to predict the outcome of a process/dynamic model with 100% success. Time resolution has nothing to do with randomness. For example take meteorology. The weather system is completely chaotic. The mathematical models are purely deterministic. The forecast is completely determined by the initial conditions given. You might have heard about 'Butterfly effect'. Randomness plays no role in chaotic dynamics.

  6. A chaotic system is characterized by unpredictable evolution in space and time, in spite of the fact that the differential equations or difference equations describing it are deterministic.

    Nature is governed by the laws of quantum mechanics. Therefore, given the state of a system at some time, the laws of nature determine the PROBABILITIES of various futures and pasts rather than determining the future and the past with certainty. There is randomness in that sense.