Models of Voltage Dividers, from Victor Niederhoffer

February 21, 2010 |

Passing an asymmetric jagged rock in the middle of a stream today, one was struck with the forces that determine the speed and amount of the water that passes by each fork. The theory of least effort means that the force of the water through each fork will be the same as in a parallel circuit where the voltage in each fork will be the same as opposed to a series circuit where all the water must pass through each point.

I started to think about how to best model this with markets. The flow of money starts by taking many markets together as a stream takes a twig down it until it meets an obstacle. The obstacle might be an announcement or a sponsor touting a position on the media. Then the theory of least effort comes in, and the speed stops as the different markets go their separate ways.

Perhaps a better model would be a circuit with two different voltage sources at opposite ends both starting at different rates and then hitting that asymmetric rock at some point chosen by the higher feeders in the web. It would be interesting to quantify some of these dividers in real life that seem to occur so frequently at the middle of the day. What are readers' thoughts on likely approaches or models?

Ken Drees writes:

In a series circuit the voltage flows through each area of resistance (R) and the resistance is added together to form a total resistance. In a parallel circuit, as described perfectly like a stream hitting a rock and then now flowing down two channels, each channel contains some resistance. Unlike in series R parallel = R1 * R2 / R1 + R2, so the more resistors, the lower the total resistance and more water or current will flow.

So with the stream analogy, in normal markets funds flow in and travel through under this resistance force. In the spring its water level is high and forceful–much market volume — the banks of the stream mark the high point where total funds hopefully are contained — this high volume pressure digs the bottom of the streambed out and over time the banks do or will contain the highest capacity. The 100 year flood will always cause the stream to overflow. Is this idea of maximum flow worth anything market related? Maybe dealing with some form of trading limits or curbs or closures or halts. But, as the funds flow normally and are diverted down one path or the other and into the circuit — think of a leaf floating through–one side generally has less resistance than the other — some leaves go the faster way and some get crowded out and move down the slower path. Total resistance describes the net effect, not the individual effect on one's own water molecule. It seems like a randomness applies to which molecule travels in which channel. Yet in total all can be measured as a force and how this total force is affected by the constraints of the system.

In thinking about each channel, after the diverting point, the stream bed may be dug out deeper in one channel so more initial volume will first go there, then the amount of rocks, debris and resistance in this channel is endured causing a slowing. This series like resistance force pushes back towards the divergence point back up stream and thus moves some of the water into the other channel-which is then endured like a series circuit in itself.

Maybe the crowded and popular trades actually cause their own diminishment in that they can only entice so many molecules into their channel, eventually pushing back on themselves causing other traders who want to get into that channel to be drawn down another less crowded copycat trade. If Apple is getting all the big early action, trades then start to migrate to the apple-like companies assuming the same great things will happen there too. So a water molecule / electrical current with an emotional urge to start with, and then once you are in you get moved along for some piece of time and distance until the circuit is exited.

There are only smooth rocks in a stable riverbed. The sharp-edged rocks quickly get worn down, leaving behind only smooth, Zen and polished ones. One can find market analogies that fit one's temperament if one wants to flow with the current and not against it.

Rocky Humbert, quantitative analyst, speculator and master chef, blogs as OneHonestMan.

Going against the flow also occurs in nature even when the odds don't look very good.

Juveniles of three species of stream-dwelling Hawaiian gobiid fishes are flushed to the ocean after hatching and must climb massive waterfalls (up to 10,000x body length) to return to adult habitats.

Henry Gifford replies:

There is a non-linear relationship between energy and speed of water in this sort of situation. A simplified explanation follows:

For an object such as a bicycle and rider moving through air, similar to the rock "moving" through the water, for a given size and shape (aerodynamic coefficient) the friction increases as the square of the speed, and the energy used increases as the cube (third power) of the speed. This assumes turbulent conditions in the air or water, which is the case in most practical situations such as the rock, a bicycle, normal water pipes, etc. No, no energy is made to disappear– it almost all ends up as heat in the fluid downstream, heating due to friction of turbulent mixing.

A better but simple explanation is available in Bicycling Science, by Witt and Wilson, MIT Press, first published in 1974, still on the shelves at B & N, very good book for understanding a nice mix of practical and theoretical. My father special ordering me a copy when I was 14 much influenced the way I think today.

Analogies between electricity and water flow are common, but with electricity things are much simplified by many resistances being fixed, while in the real world they are often variable. In water flow, practically sized pipes have turbulent flow in them, which means doubling flow increases resistance by four– squared relationship.

This means that solving flows in parallel piping networks requires solving simultaneous equations.

I invented a mathematical method for simplfying pipe flow problems to where they can be solved on the back of an envelope. Explanation downloadable here. Scroll down to "Energy Used by Pumps". Basically, one coefficient is substituted for a whole range of possible flow and friction values, which can be translated to pipe sizes and legnths, which perhaps has some use in markets.

Jim Wynne corrects:

As for "force", if you double the force you double the force. There is no "extra" multiplication factor. Newton's first law of motion says F=ma, or Force equals mass times acceleration. For a given mass, if you double the applied force, you double the acceleration. This is linear. There is no nonlinearity.

If one is referring to a rock lodged in a river bed, or stuck between other rocks, there are lots of factors besides the rock's mass that determine how much force is needed to dislodge the rock. A bigger rock presents a larger surface to the flowing water and will experience a larger force for the same flow rate. If you double the flow rate, to the first approximation your double the force. The nonlinearity comes in when you understand that with too little force, one cannot dislodge the rock. Once the force is large enough, the rock will be dislodged and move down the river until it becomes lodged in another collection of rocks or crevices. This process is very nonlinear. With the force being less than that "dislodging force", the rock stays in place. When the force exceeds the threshold for "dislodging", the rock will move and maybe travel quite a distance before it stops. It might even create an "avalanche" effect, which is a highly nonlinear event.

I don't see what you can learn from this example to apply to markets, which are anything but linear and very much moved by psychological "forces". You can try to use physics as a metaphor for the movement of markets, but I don't think that market behavior is ever linear like Newton's first law of motion.

Financial markets and fluid dynamics share a chaotic nature. For example, the Venturi effect is a concept describing the velocity and pressure changes that occur as a fluid flows across a constriction. Engineers often exploit these pressure differentials to facilitate the movement of fluids, such as drawing fluids into the low pressure side of the constriction, without the need for a supplemental pump.There are numerous similarities between Venturi effect and certain market movements. Conspicuous are the connections between volume and price change magnitude before and after a news driven constriction.

Geologists are always trying to figure out what the forces/environments were by the arrangement of sediments. Depositional environments and the spatial arrangement of sediment sizes are very important for determining the ideal places to drill for oil. It is interesting to note that the form most often seen is dendritic or tree-like in nature.

(Prigogine, 1997; Chaisson, 2001).

Global rules govern evolution toward increasing complexity in open systems:

2) Open systems create dissipative structures to dissipate energy in an effort to minimize gradients.

3) Energy dissipation must be optimized.

4) Energy dissipation transforms energy from one form to another, generally from kinetic energy to heat. In the process of dissipation, entropy is created. Entropy must be transferred from the open system into the surrounding environment in order for the system to grow in complexity and continue to optimized. By optimally transferring entropy to the global environment the system can increase in complexity, the entropy of the global environment increases, and the Second Law is honored.

The dissipative structure must do two things: optimally dissipate energy and transfer the entropy created by dissipation to the surrounding environment. In the world, a single shape optimizes these constraints: the shape of a tree or leaf (Bejan, 2000). Tree structures are all around us: brains, circulatory systems, trees, root systems, clouds, heat sinks, deltas, channel drainage systems, and turbulence (Bejan, 2000) to name a few. All tree structures share common characteristics:

1) they have lowresistance pathways to optimally transport energy to dissipation sites.

2) Dissipation sites are located at the periphery of the structure because that is the optimal location to transfer entropy into the surrounding environment.

3) Low-resistance pathways branch so that the optimal area or volume is utilized for dissipation and the optimally maximum number of dissipation sites at the periphery of the system can be connected to the orifice or energy input site.

Many small dissipation sites are more optimal than a single, large site. We believe that these constraints are the global dynamics that govern the formation and evolution of most clastic sedimentary systems from bedforms to complex bodies such as submarine fans and deltas. It is for this reason that clastic sedimentary bodies have similar shapes: they organize into the shape of a tree or leaf at all scales, and in all environments of deposition, to optimally dissipate energy and transfer entropy.

Gary Rogan writes:

In a series circuit you add up the resistors. In a parallel circuit you add up the inverse of the resistors. What’s also interesting, that in a series circuit the total power dissipated is proportional to the total resistance for a given current, which is what all the resistors have in common. In a parallel circuit the power is proportional to the inverse of the total resistance for a given voltage (which is what the resistors have in common here). The trick in the market analogy is to identify the “force” equivalent to either voltage or current, or for that matter the multiple of the two, the total power, and figuring out how it distributes itself between the obstacles.

`SELECT * FROM wp_comments WHERE comment_post_ID = '4439' AND comment_approved = '1' ORDER BY comment_date`