Consider an active battery with the physics of $\frac{d P_t}{d t}=K_t-q P_t$ where:

• P is the energy per battery
• $K_t$ is the energy power to charge(+) or consume(-) the battery
• q is the energy loss rate when the battery is active

P is then solved as

$P_t=P_0 e^{-q t}+e^{-q t}\int_0^t {e^{q \tau}K_{\tau} d \tau}$

When the energy power is constant, it is then

$P_t=\frac{K}{q}+\left(P_0 - \frac{K}{q} \right) e^{-q t}$

Note that

• P will tend to $\frac{K}{q}$ as charging time goes to infinity
• The ratio of P to the full capacity from initial empty battery is independent from the charging power: $\frac{P_t}{K/q}=1-e^{-q t}$. The time for reaching 99% is Ln(100)/q.

Classify the energy powers as

• $K_{S u n}$ from the Sun
• $K_{S u n N o n H u m a n}$ the energy from the Sun to non-human activity
• $K_{S u n H u m a n}$ the energy from the Sun to human activity
• $K_{H u m a n N o n S u s t a i n}$ the energy human harvest from non-sustainable source
• $K_{H u m a n}$ energy for human
• $K_{N o n M o n e y}$ energy for human's activity which is not purchased by the money. For example, homestead or feed oneself by a portion of all self-farming harvests.
• $K_{E c o n o m y}$ energy, in additional to the residual for batteries, for human's economy activity which is purchased by or sold for the money.
• $K_{F i n a l C o n s u m i n g}$ energy for final consuming in the economy, for example buying the electricity for a hot bath.
• $K_{G o o d s}$ energy for production of goods and services.
• $K_{M o n e y}$ energy to maintain the battery money sourced from $K_{H u m a n}$. For safety reason to accommodate uncertainty of energy consuming, some battery is always necessary. In this hypothetical society, batteries are even used as the money.
• V is the number of the battery. If the number of batteries can be continuous, the battery volume can be normalized as 1.
• $K_m$ energy power allocated to each battery.

So, it follows:

• $K_{S u n}=K_{S u n N o n H u m a n}+K_{S u n H u m a n}$
• $K_{H u m a n}=K_{S u n H u m a n}+K_{H u m a n N o n S u s t a i n}=K_{N o n M o n e y}+K_{E c o n o m y}$
• $K_{E c o n o m y}=K_{F i n a l C o n s u m i n g}+K_{G o o d s}$
• $K_m=\frac{K_{M o n e y}}{V}$
• $P=\frac{K_m}{q}=\frac{K_{M o n e y}}{q V}$
• Money is then an unhackable energy quota system via proof-of-work because its cost is the same as the battery counterpart and acordingly it is not worthy of faking it. Money's volume is the volume that the energy owners put into an abstract public space to facilitate energy consumption fluctuation of all individuals.
• Typically, a human activity sources its energy from $E_{E c o n o m y}$ and $E_{N o n M o n e y}$. For example, in the production of sea salt with the help of sun light, the boss pays the worker for the collecting job and spends some electricity for salt storage from moisture. So does $E_{M o n e y}$ whose energy sources from $E_{E c o n o m y}$ and $E_{N o n M o n e y}$. All activity involves $E_{E c o n o m y}$ recognizes a cost as the transfer of the ownership of the energy. People who work at the boundary of the economy, such as energy producers and hunters and farmers, are mostly moving $E_{N o n M o n e y}$ into $E_{E c o n o m y}$ and the energy becomes a production/consumption factor of downstream players.
• Proportionally, the higher the energy for a society aka economic growth, the more energy usage of all sectors where the battery number increases to hold the same energy amount per battery or the energy amount per battery increases as well aka money appreciation. The news of higher electricity fee in typical fiat economy is the news of people having less batteries in this energy economy, two interpretations of the same basic fact: poorer.

Let $t_A$ be the consuming time length and $t_B$ be the charging time length. The shorter time length, the more efficient the battery. Some facts about the battery:

• The consuming energy power to drain the battery is $C=\frac{q}{e^{q t_A}-1}P$
• The charging energy power is $K=\frac{q}{1-e^{-q t_B}}P$
• The energy loss in consuming is $\left(1 -\frac{q t_A}{e^{q t_A}-1} \right)P$
• The energy loss in charging is $\left(\frac{q t_B}{1-e^{-q t_B}}-1\right)P$
• The total energy loss is $\left(\frac{q t_B}{1-e^{-q t_B}}-\frac{q t_A}{e^{q t_A}-1}\right)P$

When $t_A=t_B$

• any demanded energy power W for any length of time can be served by a pool of battery, 2W/C batteries among which W/C batteries are charging and W/C batteries are consuming: $\begin{bmatrix}t & t \\s t a n d b y & s t a n d b y\\c h a r g i n g & c o n s u m i n g\\c o n s u m i n g & c h a r g i n g\end{bmatrix}$
• The total energy loss is $P q t=K_m t$
• When qt is small, K and C is around $\frac{K_m}{q t}$ and K is slightly higher and C is slightly lower. Because formula here are established on the ground of a stationary and average sense and day/summer time energy power is higher than that of night/winter time, it is convenient to define the year time as the unit of time. Also, by the above definition of energy power for economy, it means $K_{F i n a l C o n s u m i n g}+K_{G o o d s}=\frac{K_{M o n e y}}{q t}$ or equivalently $K_{E c o n o m y}=\frac{K_{M o n e y}}{q t}$, and $K_{M o n e y}$ is to offset the energy loss of cycle battery who has energy of $\frac{K_{M o n e y}}{q}$ and money velocity $\frac{1}{t}$ or equivalently the whole economy would drain all the inventory of energy battery in time t without charging.
• The energy of a charging battery and a consuming battery is always intact. $\frac{K}{q}\left(1-e^{-1 \tau} \right)+P e^{-q \tau}-\frac{C}{q}\left(1-e^{-q \tau} \right)=P$. Therefore, one full battery is the same as one full battery plus some empty batteries and is the same as one charging battery plus one consuming battery.

When people have some unexpected energy consumption which is not saved off other economic activity, they get the energy from the standby battery provided by those battery owners who are willing to help and charged back later by the borrower who is obliged to save off his economic activity. Denote the following:

• A be the energy that can be consumed from the battery
• B be the energy that is to charge the battery

Then

• $A=\frac{q t_A}{e^{q t_A}-1}P$
• $B=\frac{q t_B}{1-e^{-q t_B}}P$
• $\frac{B}{A}=\frac{t_B}{1-e^{-q t_B}} \times \frac{e^{q t_A}-1}{t_A}$

Assuming the repay never default, it is the compound interest and time preference concept, for examples:

example 1

$t_A=t_B=t$

It follows:

$\frac{B}{A}=e^{q t}$

example 2

$t_A=t$ and $t_B=n t$ and $b=\frac{B}{n}$

It follows:

$A=\frac{b \left( 1-e^{-n q t} \right)}{e^{q t}-1}=\left(e^{-q t}+e^{-2q t}+\cdots+e^{-n q t} \right) b$

example 3

$t_A=n t$ and $t_B=t$ and $a=\frac{A}{n}$

It follows:

$B=\frac{a \left(e^{n q t} -1 \right)}{1-e^{-q t}}=\left(e^{n q t}+e^{(n-1) q t}+\cdots+e^{q t} \right) a$

The activity of lending someone A energy then getting the repayment B energy after t can be done by the physical consuming/charging activity as well as the change of the ownership of the battery. Therefore, the same additional risk-free lending interest rate shall apply to the change of ownership of energy to be arbitrage-free as well. Energy storage is the only medium to move energy across time. Money is the only medium to move capability of purchasing power across time. Currently the major method to storage energy is pumped-storage hydroelectricity which claims A/B can be hopefully 87%. The repayment B can be partially sponsored by others, particularly Mother Nature, who provides part of the energy necessary for the repayment in $K_{M o n e y}$. Let N be the part sponsored by Mother Nature and L be the part responsible by the borrower saved from his steak in $K_{F i n a l C o n s u m i n g}+K_{G o o d s}$, then B = N + L in the above formula. It is possible A/B is 0.87 and the genuine risk-free lending rate is 13% while A/L is 0.995 and the post-sponsored risk-free lending rate is only 0.5% because N is 14% of L. More precisely, $B-A=A\left(e^{q t}-1\right)= t C\left(e^{q t}-1\right)=t K_m$. It all comes down to how much energy people contribute in the $K_m$. If Mother Nature provides all the $K_{M o n e y}$, then the post-sponsored risk-free lending rate can be zero. Many parts of life are cheaper simply because Mother Nature sponsors and the real price will appear when Mother Nature is doomed or people grow up and pay the fee/interest to Mother Nature as well as if she is a real person in the economy.

That purchase of goods by money needs some fee is basically short of energy and to borrow the energy at the time of purchase. This also explains the fee rate is the same as the post-sponsored risk-free lending rate. However, this battery is competing with our body battery, meaning that a borrower might appeal to the body battery of his friends rather than appeal to the money battery if the lending interest rate of the money battery is higher than that of the body battery. Therefore, people may adjust the contribution in the energy for money battery so that the post-sponsored risk-free lending rate is the same as the body's risk-free lending rate aka time preference. After all of the above, the three rates: fee rate, risk-free lending rate and time preference are then the same. The change of the battery ownership is the transaction ledger for energy accounting. The active batteries, consuming and charging, serve as medium to move energy across time. The inactive batteries, off the grid and consuming no energy, serve as saving which the owner can command anytime. Saving is never lending or investment said by Keynes.