BAK’S SANDPILE
How Stability Creates Instability, and the Tipping Point from Calm to Chaos
September 27, 2021
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GRAINS OF SAND
The most fascinating part of an hourglass is the lower chamber. Grains of sand fall from above to create a pile that grows over time. As sand accumulates, the sides of the mound become steeper and at any point, a single grain of falling sand can cause an avalanche. Incremental inputs that spur disproportionate results are referred to as non-linear and non-equilibrium systems.
Physicists have studied sandpiles to better understand this phenomenon and found that the act of cascading is chaotic, and the timing is unpredictable. Avalanche magnitude varies wildly, sometimes involving a few grains of sand while other times the whole mountain comes tumbling down. Findings from the Bak-Tang-Wiesenfeld experiment (known as the sand pile experiment) suggest these outcomes have less to do with the grains of sand that initiates the sequence of events and more to do with the underlying complex nature of the pile.
SANDPILES AND CRITICAL STATES
Building physical sandpiles is simple enough to accomplish but the behaviour of the pile itself is profoundly complex. Scientists developed a fundamental principle called self-organized criticality to help explain the findings of their experiment which suggests that complex systems tend to self-organize towards a critical state. A system in a critical state means that its structure is hypersensitive to additional input. At this point it is considered unstable. Systems near their critical point contain biases / imbalances that grow in a self-reinforcing process.
Once in this critical state, any incremental change to the system, no matter how small, can ignite a chain reaction that causes significant change. Tension builds up within the system making it increasingly vulnerable over time. This vulnerability combined with additional input inevitably causes the system to collapse but the timing and severity of the chain reaction is impossible to predict.
Sand falling from the upper chamber to the lower chamber creates a pile at the bottom which possesses the characteristics of a complex system. The pile self-organizes as time passes and will inch closer towards its critical state as more sand accumulates. After the critical state has been reached, each incremental falling grain of sand may have no consequence but has the potential to cause a devastating avalanche. It is worth noting that there is a high probability of a small avalanche occurring and a low probability of a large one.
Stability leads to instability. Under normal conditions, small avalanches occur frequently while large ones are few and far between. But the longer a complex system goes without a failure, the higher the likelihood that a low probability, high magnitude event will take place (black swan). As a rule of thumb, the more stable the times, the more self-organized criticality.
INTERCONNECTEDNESS AMPLIFIES SELF-ORGANIZED CRITICALITY
Further, the degree of interconnected within the system also matters. Without significant interconnectedness throughout the system, a grain of sand can fall onto a troubled spot with little consequence. A small avalanche can occur in relative isolation from the rest of the pile, resulting in limited repercussions. However, if multiple pockets of troubled spots exist within the pile and are all interconnected with one another, then a falling grain of sand has the potential to set off a cascading chain reaction, putting the entire structure at risk.
Both self-organized criticality and interconnectedness grow within the system as time without failure passes. The more interconnectedness within the system, the more self-organized criticality. The more self-organized criticality, the more catastrophic the failure. The inevitable impact of self-organized criticality is extended periods of relative calm briefly interrupted by severe bouts of catastrophe.
The illustration involving an hourglass may seem contrived, but this conceptual framework can be applied to other naturally occurring real world phenomena. Crucial ‘tipping points’ can be found in various natural disasters, geopolitical conflicts, social issues, crowd psychology, etc.
SO ANYWAYS
In other unrelated news, there has been an incredible amount of stability in financial markets lately. High complacency, excessive leverage, frothy valuations (the “Buffet Indicator” has surpassed its 2000 peak), rampant speculation (hello six-figure cartoon rock JPEGs), and investors’ collective faith in the Fed Put is downright unshakeable. Stable times, indeed.