**— The Interplay of Incompressible Dynamics and Emergent Dynamics —**

The Second Law of Thermodynamics (SLOT) tells us that left undisturbed all fluid-like systems ultimately gravitate towards a state of maximum disorder, a state that is referred to as *“Thermal Equilibrium”*. In reality however, the achievement of thermal equilibrium relies heavily on the mathematics of the *Law of Large Numbers (LLN) —* more specifically it relies on the number of independent elements (of the system) being significantly larger than the number of options available to each element.

This means that although fluid-like systems may spontaneously gravitate to thermal equilibrium, they can often *be held away* from this equilibrium by either, the inability to dampen excessive activity, or by the emergence of positive feedback – each of which are different forms of the *Reverse Law of Large Numbers (RLLN)*…

Now what is most interesting about this is that while weak negative feedback in isolation can cause incompressible diversity, and strong positive feedback in isolation can cause the emergence of asymmetric uniformity, and the most interesting stuff occurs from the *interplay* of both.

Positive reinforcement in a system of great diversity, can spontaneously produce surprisingly complex *“Integrated Diversity”*. In other words, the secret to the creativity of natural evolution is that,

*when we have the co-emergence of diversity, complex-integration comes for free!*

**Below is a matrix of system dynamics**

__The Interplay of Entropy__

Now on a slightly different tact, let’s examine how this emergence of complexity relates to the concept of *“entropy”*.

The RLLN has the effect of pulling all fluid-like systems away from thermal equilibrium. In systems of adaptive agents, positive feedback can ultimately reinforce a symmetry break which pulls the system far away from maximum entropy and thermal equilibrium. However we do not always need to be away from maximum entropy to move/be away from thermal equilibrium. Chaos theory has shown us that even at maximum entropy we can still move/be away from thermal equilibrium, because systems with incompressible dynamics produce turbulent-like behaviour and *“information entropy”*.

*“Information Entropy”* is a different type of entropy to *“Thermodynamic Entropy”*. Information entropy is a measure of uncertainty or unpredictability; whereas thermodynamic entropy is a measure of disorder. Complexity is, in fact, a combination of these two types of entropy. *“**Complex Adaptive Systems”* are characterized by the fact that they have *“low thermodynamic entropy, but high information entropy”*; consequently such systems can be considered to be highly organized but unpredictable nonetheless.

**Below is the system matrix reconstituted in entropic terms**

__Modelling____ Complexity__

Those of you who know anything about statistics and probability distributions might have recognized that thermal equilibrium represents the first moment of a probability distribution. Furthermore you might have also noticed that incompressible diversity represents the second moment (i.e. deviations from equilibrium); and positive reinforcement the third moment (i.e. asymmetric skew).

Thus ever-increasing complexity is ultimately heading towards causing the emergence of an infinite number of moments for the probability distribution, which effectively means that *“any form of mathematical or even statistical representation of complexity is virtually meaningless”…*

The inability to mathematically summarize complexity, makes complexity different from every other form of physics. *There are no equations for complexity*. Complexity is basically bottom-up mutually-reinforced diversity which cannot be summarized by any form of mathematics. Complex incompressible dynamics is the universe’s creativity in action, and creativity cannot be modeled for predictions, but it can be simulated for inspiration.

After studying chaos and complexity for so many years, it strikes me that the universe is not (as it so often suggested) purely mathematical, but is more generally algorithmic; and this 21^{st} century will likely be the century that we finally begin to truly reap the benefits of understanding complexity, by exploring and exploiting this *Computational Universe of Algorithmic Creativity…*