The Reverse Law & Emergent Dynamics

There are two fundamental forces at work in nature and all evolutionary systems; one is the entropic force of spontaneous decay and disorder (otherwise known as “The Second Law of Thermodynamics”); the other is a universal, and somewhat mysterious, capacity for self-organization and spontaneous emergence…

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Emergence is the spontaneous collective self-organization that can occur in systems of freely interacting parts.  Self-organization is a fascinating subject not least because it seems to be in direct defiance to the all-important Second Law of Thermodynamics (SLOT).

The SLOT is the law of physics that deals with the spontaneous distribution of energy.  In everyday terms the SLOT is simply the fact that cold milk and hot coffee, if left unstirred, will spontaneously mix themselves.  The SLOT states that left undisturbed all systems gravitate towards “Thermal Equilibrium”.

Despite the fact that the SLOT describes what seems to be rather innocuous common everyday behavior, it nevertheless is considered to be one of the most important laws of physics; and the reason for this exalted status is that the SLOT is also the “Law of Maximum Entropy”.

A Quick History of Entropy

In 1850 the German Physicist Rudolf Clausius formulated the Laws of Thermodynamics and in doing so he equated the SLOT with the idea that all spontaneous change is a process of maximizing a thermodynamic quantity which he termed “entropy”, and this maximization of entropy is ultimately achieved at thermal-equilibrium.

During the 1870s Austrian Physicist Ludwig Boltzmann associated the overall state of a system with the internal activity of its atoms.  He put forward a theory that the spontaneous movement to maximum entropy and thermal-equilibrium is in fact nothing more than a system – of freely interacting parts – spontaneously moving to its “most mixed together condition”; which mathematically is its “Most Probable Distribution” (MPD).

Boltzmann theory was suggestive of two things.  Firstly that the SLOT is not actually a real universal law – in the sense that a real law is something that always happens – but is in fact a probabilistic law; in the sense that the movement towards the MPD happens with such an absurdly high degree of probability that it might as well be a real law.  And secondly that maximum entropy – given that it is the most mixed together condition – equates with the state of “maximum disorder”

Although Boltzmann’s probabilistic explanation of the SLOT initially faced hostility – not least because many prominent physicists of the time didn’t believe in atoms – it eventually became the accepted reasoning for why all systems are spontaneously pulled towards thermal-equilibrium; and with it came the general understanding that all spontaneous change ultimately leads to maximum entropy; maximum disorder.

What seems to have been consistently overlooked however (or at least never, mentioned in textbooks) is the fact that the spontaneous movement to maximum entropy relies completely on the absurdly large numbers of particles, in a thermal system, to ensures that the system can fine-tune to equilibrium.  In other words; maximum entropy relies heavily on the “Law of Large Numbers” (LLN)! 


Most people are familiar with the concept that if you toss a coin four times, you won’t necessarily get a 50/50 split of heads and tails: indeed, you could get 4 tails, suggesting (wrongly) that the coin will always land on tails. But if you toss a coin a million times, you will get something close to a 50/50 split between heads and tails.  It is the LLN that ensures that the one million coins tosses will produce an average of 50% heads and 50% tails.

Thermal equilibrium is a consequence of this law of probability; and so anything that weakens the LLN  will also weaken the system’s ability to fine-tune to equilibrium.

In a thermal system billions of tiny particles interact with each other through collisions, but other than that they behave completely independently of each other.  Maximum entropy is achieved in a thermal system when the particles within the system have “maximum independence”.

Complex Adaptive System

Systems where the parts – be they particles, elements, components, entities, agents, organizations, etc – behave independently of each other are actually quite rare.  Many systems are populated by adaptive agents, and such systems have a tendency to reinforce collective behavior.  These systems are best described as a “Complex Adaptive Systems” (CAS).  CAS’s and collective behavior have serious implications for the SLOT!

Random Emergence of Positive Feedback

The LLN ensures the spontaneous movement to thermal-equilibrium in large thermal systems.  However for very small regions within these large thermal systems, there are not enough particles to ensure a “local equilibrium”.  At the very lowest level within these large systems, random fluctuations are occurring all the time which means that local imbalances are constantly and randomly flittering in and out of existence.

Occasionally these random temporary fluctuations can randomly be very persistent.  In a thermal system this is naught but a mere statistical curiosity, but in a CAS it can easily happen that some parts within the system will begin to adapt to these persistent fluctuations; and often such adaptation only serves to amplify the imbalance even further, and in so doing, further extend the fluctuation’s duration.

Thus random local fluctuations can lead to the localized emergence of positive feedback and the associated collective correlated behavior!

Positive Feedback & The Build-up of Viscosity

CAS’s can be thought of as “Fluid-Like Systems” (with varying degrees of viscosity).  In a CAS, collective behavior increases the fluid-like viscosity within this fluid-like system.  In a CAS, collective behavior reduces individual independence; and any reduction in independence weakens the LLN!  This weakening of the LLN causes the system’s behavior to become more “coarse” and jumpy which, of course, weakens the system’s ability to fine-tune to a stable equilibrium.

Consequently excessive collective behavior among the parts within any fluid-like system, will act to hold (or pull) the system as a whole away from equilibrium; by effectively engineering a “Reverse of the Law of Large Numbers” (RLLN)! 

RLLN & Symmetry-Breaking Decisions

It is in the LLN and the RLLN that we see the two potential extremes of agent and system behavior.  Just as a million coins tosses pretty much guarantees an average of 50% heads and 50% tails, and a single coin toss absolutely guarantees either 1 head or 1 tail (but not 1/2 a head + 1/2 a tail); so too the LLN pretty much guarantees thermodynamic equilibrium when the agents are totally independent of each other, and the RLLN absolutely guarantees far-from-equilibrium collective-behavior when the agents are behaving entirely as one.

We can think of this difference in system behavior as being like the difference between two voting systems.  The LLN voting system produces a balanced proportional representation result; while the RLLN, on the other hand, produces a collective choice, “a winner takes all” result.  Thus while the LLN finds the collective average, the RLLN on the other hand ultimately pulls the system as a whole collectively towards making a choice between all of the available options.

All fluid-like systems, will experience a constant tug-of-war between these two extremes.  This constant push and pull can mean that as agent behavior moves from extreme individualism to extreme collectivism, system behavior will transition from self-stabilizing to self-reinforcing.  And ultimately should the RLLN manage to secure the upper hand, it can mean that, an initial random emergence of the collective behavior could in due course actually reinforce a small local imbalance into a collective system-wide “symmetry-break” – and the reason that this is interesting is that a system symmetry-breaking is a system making a decision!

So effectively it is the “RLLN” that is the mysterious source of spontaneous emergence.  “Emergence” is actually the mathematical result of the RLLN driving systems away from the bland uniformity of maximum entropy at thermal equilibrium…

“Emergence occurs in fluid-like systems when positive reinforcement and the RLLN drive the system to make Symmetry-Breaking Decisions!”