EnergyDensityOfBattery

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Recall the money-energy isomorphism. Suppose a battery charging operation where Pi is the energy of 1.0 battery at time i and there are total V batteries waiting to be charged and the amount of batteries till time i is Vi and the new batteries at time i is vi. The operation at each iteration will cost energy of F batteries itself. In other words:

Therefore,

Taking i to the infinity then:

which simply means P=KT/F

The capital capacity is simply the total energy of all batteries, a reservoir of energy for the economy. Once all the batteries are fully charged, the P can be maintained by adjusting the batteries number proportional to the energy inventory of the reservoir thus no one can gain or loss by holding the money. Effectively, an economy with sound money is an energy-based economy, no matter the economy is expanding or shrinking.

For case when economy size intact just like a constant surface area planet with the solar power, now the system enters the equilibrium state, as long as there is constant energy influx for the civilization, the energy contained in a battery decreases when a consumer uses some energy and the battery is refilled to full when it is recharged. By the viewpoint of netting, the energy is like coming directly from the sun and being consumed by the consumer and the average energy of a battery being intact. Therefore, in this ultimate equilibrium situation, the only energy not dedicate to physical mechanics of the civilization is accounted by the F and its correspondent energy power input K and the battery is much like an abstract token whose energy is never consumed and just a number and the only important property of the token is that it takes P energy to produce one unit and K is literally the cost to establish the trust medium for the society, equivalently the same as the concept of proof-of-work for money.