Appendix: Thermodynamics

The explanations in this appendix are intended for readers without a background in physics. Some technical terms are replaced with simpler expressions to emphasize intuitive understanding.

For a long time, I have intuitively felt that thermodynamics mirrors social phenomena. However, I could not find the reason why I felt this way. Thermodynamics is difficult right from the start. The more I studied it, the more difficult textbooks I encountered.
Heat moves and releases energy to the outside. Alternatively, it takes in energy from the outside, creates heat, and then releases it. I intuitively felt that these inflows and outflows perfectly reflect social phenomena.

In this column, I wanted to explain thermodynamics first, and then share my intuition. However, I realized that I couldn't even explain the basics to others. Explaining thermodynamics exactly as textbooks do does not work for ordinary people. Conversely, I thought that if I couldn't explain it simply, it meant I didn't truly understand it myself. I was just memorizing it, not understanding it.
Therefore, I decided to rethink the essence of thermodynamics so that I could explain it intuitively. As a result, I created my own unique explanation, moving away from standard textbooks. Thanks to this, everything became clear to me. This is also the reason why the explanation of thermodynamics became an appendix at the very end of this column.

First, let’s look at the words used in thermodynamics.

1) About Thermal Energy
We use the symbol Q to represent it. Energy has no fixed shape, but it has a specific amount. If something has an amount, adding more of it increases the total. The linear equation Q = CT, which expresses heat as the product of heat capacity (C) and temperature (T), is the most important concept. This expression applies to all phenomena.
If you increase the capacity (C) of the container that stores heat, the energy increases. However, even if you add more objects of the same temperature (T), the overall temperature (T) does not change.

2) About Work Energy
We use the symbol W to represent work energy, which can be efficiently taken out for use, taken in, or stored. Energy that can be used without leaking away—such as energy stored in a spring, a battery, or kinetic energy in a spinning flywheel—corresponds to W.
Thermodynamics is a science with a long history. It developed in an era when modern electric motors did not exist. Therefore, standard textbooks explain how we can store and release energy by using mechanisms that move in one direction, like a piston.

If there is no waste (meaning we ignore factors like friction), the movement and storage of work energy (W) are reversible. This ideal of "no waste"—or reversible heat transfer—is assumed here. Reversible work energy (W) can be stored and released, and it can be perfectly converted into thermal energy without any loss.
Because it can be converted into anything, W corresponds to "money" if we connect it to society.

3) About Temperature
We use the symbol T to represent temperature. Thermal energy is modeled as a specific amount of "things" at a certain temperature T. This specific amount of "things" is expressed by the term "heat capacity.
"When the number of "things" at temperature T increases, the thermal energy increases. However, temperature T does not change even if you add more "things." T represents the quality of temperature. The "things" that hold this temperature T would be gas molecules (particles) in an ideal gas.
If the "thing" is a solid, it can also be a "state" such as lattice vibration. If we think of "things" as a collection of "states," the term "heat capacity" becomes a scale to measure the amount of "things."
　　　　　　　　　　　　　　
4) About Entropy
We use the symbol S to represent entropy. It shows the amount of states that can store heat. For a gas, it corresponds to the number of molecular states vibrating at temperature T. This means we can express thermal energy Q by counting how many states exist at temperature T.
Therefore, thermal energy Q can be expressed as the product of temperature T and the amount S: Q ＝ ST. In this intuitive model of thermal energy, a "state" is a container for heat, and entropy S is the number or the total amount of these containers.

The term "entropy" is also used to express the complexity of information. Since information entropy represents the number of information states, the term can mean "complexity" or "randomness." At the end of this introduction, I will connect this concept to heat capacity. Even if you do not fully understand it now, I recommend that you just move forward with an intuitive sense of it, and rethink it later.


5) About Heat Capacity
We use the symbol C to represent heat capacity. It shows the amount of containers that hold heat. In junior high school, we learned the relationship between temperature and thermal energy as Q ＝ CT. We memorized the term "specific heat" as the thermal energy needed to raise the temperature by one degree. At that time, specific heat was a specific property of a material, and it was treated as a constant value if the amount of the material was fixed.
In thermodynamics, however, heat capacity is not a constant; we treat it as a heat container whose size can change. This is where it differs from what we learned in junior high school.
Because the equations look so similar, your intuition might tell you that it means something close to entropy. In a gasoline engine that extracts work energy, heat capacity corresponds to the capability of the piston and cylinder moved by the explosion, or the capability of the cooling system during heat exhaust. In a piston engine, since the piston actually moves, it is not fixed; it is the amount of something that holds heat that can change as the piston moves. You can just conceptually accept it as "the ability to hold heat," ignoring the detailed mechanism.

Using the terms explained above, let’s approach a crucial law of thermodynamics—the Second Law, which states that "heat does not naturally move from a cold object to a hot object.“

This law feels like common sense. However, the principles derived from it will make social phenomena visible. But explaining the obvious is always a difficult thing to do. The First Law, which is the law of conservation of energy, is already understood by most people. In contrast, the Second Law of Thermodynamics is the one that is obvious yet so hard to explain.
Writing this reminds me of a question on my graduate school entrance exam: "Prove Ohm's Law: V (Voltage) ＝ R (Resistance) × I (Current)." It is difficult to prove what is obvious. I couldn't give the answer.
Naturally, heat does not move from a cold object to a hot object. This is common sense and obvious. However, if we use external energy to manipulate the system, we can move heat from a cold object to a hot object.

Our everyday refrigerators and air conditioners are machines that run on electricity—which is work energy (W)—to move heat from a cold place to a hot place, creating cold spaces. If we compare the logic of this law to the social phenomena around us, we can see a clear similarity. This similarity explains those social phenomena perfectly. I have already discussed this in Chapter 3 of this column.
However, the textbook explanation of the Second Law of Thermodynamics is quite difficult. This is why I skipped the detailed explanation in the main text and placed it here in the appendix.
Putting it in the appendix doesn’t make the law itself any easier. However, since we have already compared these concepts with social phenomena from Chapters 1 through 3, that knowledge should make you want to read on.

First, let’s begin our explanation of the Second Law of Thermodynamics using the relationship between "truth" and "expression" (the principle explained in Chapter 1).The truth stated in the negative form—"heat does not naturally move from a cold object to a hot object"—cannot be changed into a concrete, visible expression. Therefore, we change it into a positive statement. We express the truth in an affirmative sentence without changing its original meaning. In other words, we use the contrapositive relationship and change the words into a positive expression: "If we supply energy from the outside, we can move thermal energy from a cold object to a hot object." This is still an obvious fact.
If we want to express how much energy needs to be supplied from the outside, it means that without this supply, thermal energy cannot move from cold to hot. By introducing this supply of energy and expressing it positively, we can finally think of a concrete heat engine that actually moves energy. Using that heat engine, we perform experiments to move heat from hot to cold, and from cold to hot, releasing work energy to the outside or inputting energy from the outside. Through this, the truth of the Second Law is converted into an expression that includes the supplied energy. Here, the principle of "truth and expression" from Chapter 1 becomes your power to understand once again.

Standard textbooks of thermodynamics explain heat transfer by using the Carnot cycle, which uses an ideal gas as a working medium.
It transfers work energy to the outside through changes in pressure and volume, moves heat from a hot object to a cold object, and extracts work energy for external use. By using an ideal gas, it performs reversible operations with zero loss. This is a special case that achieves the maximum efficiency of heat utilization. Textbooks explain the relationship between heat transfer and temperature under these conditions. From this relationship, we can derive the formula between thermal energy transfer and temperature for an ideal reversible cycle where entropy does not increase.
Although it is not written in standard textbooks, this reversible model also allows for the reverse transfer of heat by using the work energy that was taken out. Now, looking back, I realize that many people—including myself—found it difficult to understand the whole concept because we did not notice this point.

In reality, there are always losses. Moreover, reversible motion that reverses the direction of time does not actually exist. Heat never moves when there is no temperature difference. I used to feel unsatisfied with the textbook model because it is an impossible, ideal heat engine, and I didn't know the reason why. That is why thermodynamics was so difficult.
Even if you don't understand the true meaning, if you read a textbook 100 times, you will memorize it, and it makes you feel like you understand it. However, if you do not truly understand it, you cannot explain it to ordinary people. After all, ordinary people do not read it 100 times.

When I was struggling again and again to understand the true essence, a sudden flash of insight came to me. I imagined a model of a heat engine that includes "lost energy"—something that the ideal, reversible Carnot cycle completely ignores. The fact that there is energy that is lost and can never be recovered means that things in nature cannot naturally go backward.
If there is lost energy, we can connect it to social phenomena. It corresponds to real-world experiences that happen and can never be undone: "wasted efforts," "ineffective actions," "exhaustion," "irreversible mistakes," "crying over spilled milk," "growing taller," or "getting older." I intuitively felt that with this model, I could finally explain it.
The reason I felt this intuition was because, unconsciously, I was already aware of the close relationship between lost energy and irreversibility.
Once I thought of using a heat engine that includes lost energy, the difficulty of explaining it completely disappeared. Although I started this column from my initial intuition, now that I have reached this final appendix, I have finally understood the most important part that I myself had missed. Now, after all these years since graduating from university, I finally feel a deep sense of relief and complete clarity about thermodynamics.

I have prepared a heat engine model that includes "loss" in the input and output of both heat and energy. This model allows us to explain the relationship between thermodynamics and social phenomena. We can explain it without using the difficult textbook word "entropy." At the very end, once you understand the concept, we will apply the word "entropy" and look back at the model again. When you reach that point, the true meaning of entropy should become naturally clear to you.
At the same time, you will understand the basics of thermodynamics, and you will be able to connect thermodynamics with social phenomena. By understanding social phenomena through your own daily feelings first, and then applying that understanding to thermodynamics at the end, you can truly accept and agree with thermodynamics. For ordinary people, it is impossible to understand thermodynamics first and then try to understand social phenomena. Because this column structures the explanation in the reverse order of standard textbooks, I believe a clear path of understanding has been created.

In the explanation of this heat engine model, the basic concept is how heat moves from a heat source. This fundamental transfer of thermal energy Q is expressed by multiplying temperature T and heat capacity C: Q ＝ CT. To help you picture this heat movement in your mind, I will explain it using the expansion of an ideal gas as a model.
Appendix Figure 1 shows a model where a piston and cylinder mechanism is brought into contact with a heat source to expand the gas. As it expands, it outputs work energy to the outside. This is a mechanism where thermal energy is converted into mechanical work energy for external use. This represents the basic operation of heat transfer. Since this basic operation is a thought experiment, it relies on several assumptions. It is an imaginary motion where "lost energy"—which is unavoidable in reality—does not occur. For example, one such assumption is the imaginary heat transfer where heat moves even though there is no temperature difference. It assumes that this imaginary process happens reversibly, completely independent of time. Appendix Figure 1 illustrates these assumptions, but to help you truly accept it, I will provide some supplementary explanations for Appendix Figure 1.

We will calculate the heat transfer and work during gas expansion using this virtual piston model. Since we have the equation of state for an ideal gas, we will accept this as a given. Here, P represents pressure, V represents volume, and T represents temperature (as found in high school textbooks).

PV ＝ RT
Where R is the gas constant. (To keep it simple, the number of moles n, which shows the amount of gas molecules, is omitted.)

In this model experiment, the system is virtually perfectly insulated, resulting in zero loss. Heat is transferred to the gas while maintaining the temperature T, causing the gas to expand. As the piston moves from position 1 to 2, it does work W on the outside. The law of conservation of energy holds true between the transferred thermal energy Q and the work W.
　　　　　　　　　
Q ＝ W (The law of conservation of energy)

We will calculate the magnitude of the work W caused by expansion. The infinitesimal work dW output to the outside is expressed using the infinitesimal change in volume dV as:
dW ＝ PdV

The work energy W output to the outside is obtained by integrating dW from position 1 to position 2.

W ＝ ∫dW ＝ ∫PdV ＝ RT∫dV/V ＝ RTln(V2/V1)

As the piston changes its position and the structure, the volume of the ideal gas changes, allowing it to receive thermal energy. The structure of the piston functions as something that receives thermal energy through volume change—that is, it functions as a variable heat capacity.

Since the transfer was reversible with zero loss, the thermal energy Q and the work W are in the relationship of Q ＝ W. Changing the expression gives the following equation.

Q＝RT ln(V2/V1)

From this relationship, the thermal energy Q can be expressed as follows, using the temperature T and the heat capacity C due to volume change.
Q＝CT
　　　　
C＝Ｒ ln(V2/V1)

This expression forms a linear multiplication of the heat capacity C and the temperature T for the heat Q, serving as the basic equation for reversible heat transfer. Reversible means that if work energy W is applied from the outside to push the piston back, the heat Q can be returned to the heat source at temperature T.

In the section on thermodynamic terminology, I explained entropy S. There, I explained the following equation, which is the textbook definition:
Q＝ST

It is difficult to understand entropy S when it is just explained at the terminology stage. Assuming it is enough to understand it later, I explained the connection here.
I think you can now visually or intuitively understand it a little through the explanation so far. It means that because the volume change of the thing that holds heat is reversible, you can apply work energy W from the outside to return it to its original form or state.
If the change from 1 to 2 is reversible, it means that every part of the process is reversible and equivalent. If W is dissipated and lost, you lose the energy needed to return to the original state. In human society, this is equivalent to losing funds, growing old, becoming exhausted, or dying. The reason thermodynamics overlaps with social phenomena is that there is this lost energy that can never be reversed.

In the linear expression Q ＝ CT where heat transfers reversibly, the time required for the transfer does not appear. In reality, there is a micro temperature difference between the cylinder and the heat source, as well as resistance from the thermal conductivity of the heat transfer path, so the transfer takes time. However, since this is a hypothetically reversible heat transfer, the temperature difference is infinitely small. Because there is infinitely no temperature difference, the time is infinitely long.

The heat transfer in this model expresses a hypothetical reversible heat transfer where heat moves over an infinite amount of time. If it is reversible, you can push the piston with the output work W, return the piston to its original position, and reverse the heat transfer to restore the original state. A reversible operation that assumes complete insulation, no loss in piston movement, and an infinitely long passage of time for heat transfer does not exist in reality. Therefore, the expression Q ＝ CT serves as the unique basic equation that expresses a time-independent, hypothetical thermal energy transfer. Because it is unique, it could be expressed. If there were many ways, it could not be expressed.

Although the textbooks I read describe this single hypothetical state, they do not touch upon its core essence. To physicists, this essence might have been too obvious to mention. If there had been an explanation about this as a preface, thermodynamics would have been much easier to interpret.

There was a Russian physicist named Landau-Lifshitz. Ⅰ read his textbook on mechanics. The preface of this book was very profound, and I read it many times. Whenever I found a part in the main text difficult to understand, I returned to the preface to try to comprehend it. The greatest meaning of the supplementary explanation is to show that because there is only one unique hypothetical reversible operation, it can be expressed in a mathematical equation.
The equation Q ＝ CT, which represents the hypothetical reversible thermal energy transfer, serves as the basic equation for heat transfer in the heat engine.

Based on the basic equation Q ＝ CT, let us consider a heat engine that extracts thermal energy. There is a conventional heat engine in thermodynamics textbooks—the Carnot cycle heat engine. It was impossible to connect the use of thermal energy with social phenomena using this engine. Although it was a textbook that let me intuitively sense the relationships, it could not be overlapped with social phenomena. It took a long time, but I have invented a new heat engine.

As a reality, there are wastes and losses in society, and there is an irreversibility of time, meaning that once something happens, it can never be undone. The Carnot cycle, a reversible heat engine, does not touch upon this reality. Therefore, I realized that the heat engine in textbooks cannot be used to explain social phenomena. To address this, I developed a heat engine model that incorporates the root of irreversibility—that is, the loss—which corresponds to social losses and aging. This model is shown as FURUMURA's heat engine in Appendix Figure-2.

Let us operate this heat engine model in a thought experiment. Heat is transferred from a high-temperature (T1) object to a low-temperature (T2) object (performing the forward action) to extract energy Wout to the outside. This heat engine model makes thermal contact with the high-temperature heat source at temperature T1 and reversibly absorbs heat Q1. It is expressed by the basic expression as follows:

Q1＝CinT1

Cin becomes the variable heat capacity that the heat engine uses to absorb and store heat. The heat engine model outputs work energy Wout, makes contact with the heat source at low temperature T2, and reversibly releases heat Q2. This is expressed as follows, using the variable heat capacity Cout of the heat engine.

Q2＝CoutT2

At this time, conservation of energy holds true. This is expressed as follows:

Wout＝Q1－Q2

Next, the heat engine model is operated in reverse. It performs the action of extracting heat Q2 from the low-temperature heat source, inputting energy Win from the outside, and returning heat Q1 to the original high-temperature heat source (the return action). At this time, assuming that there is a lost energy L that would have been dissipated, this loss is compensated for to return Q1. As a result, extra energy equivalent to the loss L must be inputted compared to Wout. This is expressed as follows:

Ｗin＝Ｗout ＋L

If the operation is reversible, there is no loss, so it leaves no trace of energy loss on the outside. This is expressed as follows:

Win＝Wout　or　L＝0

In this heat engine model, heat is transferred to extract energy to the outside. When operating the heat engine through one cycle of forward and return heat transfers to return it to its original state, we estimate how much external energy supply is required to compensate for the loss. The thermal energy exchanged with the heat sources and the work energy exchanged with the outside are expressed by the "law of conservation of energy." The relationship of the forward thermal energy moving from the high-temperature heat source to the heat engine and then to the low-temperature heat source is expressed by the basic expression of heat transfer as follows:

Q1（ forward ）＝CinT1　　　 Q2（forward）＝CoutT2　

Wout（ forward ）＝Q1ーＱ２（ forward ）＝CinT1ーCoutT2　

Here, Cin and Cout represent the "variable heat capacity," which is a changeable quantity indicating the capacity of the engine to absorb or release heat. Q1 indicates the thermal energy absorbed from the high-temperature heat source at T1, and Q2 indicates the thermal energy released to the low-temperature heat source at T2.
　
The relationship of the return thermal energy (returning the thermal energy to its original state) moving from the low-temperature heat source to the heat engine and then to the high-temperature heat source is expressed using the same heat capacity as follows:

Q2（ return ）＝ CinT2　　　Q1（ return ）＝ CoutT1　

Win（ return ）＝Q1ーQ2（ return ）＝CoutT1ー CinT2 　

Regardless of the specific mechanical structure or operation of the heat engine, the above expresses the law of conservation of energy.

Using one cycle of operation of this heat engine model, we will estimate the loss of the heat engine and the available energy.

(1) We will estimate the lost energy L compensated for from the outside when operating the heat engine through one cycle.

(2) By compensating for the lost energy in addition to the energy Wout released to the outside, energy Win is inputted into the heat engine from the outside to operate the heat engine in reverse and return the thermal energy to the original high-temperature heat source. When expressing this reverse operation by the law of conservation of energy, the following relationship holds true:

Win＝Wout＋L(lost energy）

The word "loss" was chosen because this energy is compensated for. If one cycle of operation were a reversible operation, there would be no loss, and thus the energy to compensate for it would be unnecessary. If an irreversible operation occurs, it represents energy that feels as though it has been lost, which is why I selected the word "loss.“

Using the capacity of the variable heat capacity of the heat engine model, we obtain the following expression for the lost energy L.

L＝WinーWout＝CoutT1ー CinT2 ーCinT1＋CoutT2 ＝(Coutー Cin)(T1＋T2）

If it is a reversible heat engine with no loss, this is expressed as a loss of L ＝ 0. By transforming this expression, several expressions for a heat engine with no loss are derived as follows:

Cout＝Cin　or　Q1/T1＝Q2/T２　 or 　T2/T1＝Q2/Q1

(2) We will estimate the work energy that can be extracted to the outside from the heat source using the heat engine model.

The heat engine absorbs thermal energy Q1 from the heat source and outputs energy Wout to the outside as work. We will estimate the maximum output energy produced by an ideal heat engine operating reversibly with a loss of L ＝ 0. By substituting the expression for reversible operation, "Q2 ＝ Q1 T2 / T1," the following expression is obtained:

Wout＝Ｑ１ーＱ２＝Q1（１－T2/T1）

By using another expression for reversible operation:
Cout ＝ Cin ＝ C
another expression for the work energy Wout that can be output:
Wout ＝ Q1 - Q2 ＝ C (T1 - T2)is derived.

The higher the temperature T1 of the high-temperature heat source, and the lower the temperature T2 of the low-temperature heat source, the larger the output energy Wout becomes.

If there is no loss L, the energy compensated for from the outside is unnecessary to operate the heat engine through one cycle of transferring thermal energy. Therefore, the reversible operation of the heat engine and the loss L ＝ 0 are expressions with the same meaning.
If the loss L ＝ 0, you would intuitively understand that the available thermal energy becomes maximum.

In an ideal reversible operation with no loss, if the state can be reversed without leaving any effect on the outside, there is no distinction between before and after in the time of one cycle of operation. This means that there is no passage of time for a heat engine operating reversibly. Imagining this concept is necessary for understanding.

(3) We will estimate what happens if the work energy is not output to the outside.

Suppose that the heat engine absorbs energy Q1 from the heat source and releases the heat of Q1 to the low-temperature heat source without outputting any work to the outside (that is, Wout ＝ 0).
This can be expressed as
Q2 ＝ Q1 ＝ Q.
Therefore, by transforming that expression, the following expression is derived.

Wout＝Q1ーＱ２＝CinT1ーCoutT2＝０　or　CinT1＝ CoutT2 　or　 Cout＝CinT1/T2

If no work is produced, the heat capacity C of the heat engine is no longer conserved and becomes altered. . It can no longer return to its original state or structure and cannot be sustained.
By transforming the expression, the lost energy L, which corresponds to the amount of this distortion, is expressed as follows:

L＝ WinーWout＝(Coutー Cin)(T1＋T2）

　＝Q(1/T2-1/T1)（T1＋T2）

　＝Q（T1/T2-T2/T1）
　　　　
The distortion-caused lost energy L increases in proportion to the temperature ratio T1/T2 of the heat sources. When the temperature ratio is large, T2/T1 can be neglected, so the expression L ＝ Q(T1/T2) is obtained.
If no work is done and thermal energy is released from a high-temperature to a low-temperature heat source, the distortion loss L increases in proportion to the temperature ratio (high temperature / low temperature).
To reverse the flow of heat, a large amount of energy is required from the outside to compensate for the loss. This corresponds to a social phenomenon where wasteful benefits that create no value are saved and become dead money.
I had intuitively felt that thermodynamics overlaps with social phenomena, but I myself was surprised while writing this appendix that it appeared in the form of such an expression.

The heat capacities Cin and Cout of the heat engine used so far have been explained as the variable heat capacities of the capability that the heat engine model possesses. They have the same positioning as the specific heat of objects that appears in junior high and high school science.

The difference is that while specific heat is unique to an object and is a constant value, it can change in this model. Looking at the heat engine models in Appendix Figure 1 and Appendix Figure 2, it will be easy to imagine the basic expression used: Q ＝ Cin T1.
The term "variable heat capacity" was used to explain the operation of the heat engine model in an easy-to-understand manner. Now that the explanation has proceeded this far, the meaning of the term "entropy S," which was first explained as a thermodynamic term, begins to be understood. Entropy S appearing in thermodynamics textbooks is defined as follows:

Q/T＝S　or　Q＝ST

Comparing this, you will notice that the variable heat capacity Cin of the heat engine model introduced in this appendix corresponds to the change in entropy S.
The term "entropy" indicates the amount of a "thing" at temperature T, and the changed amount is expressed as delta S.

The variable heat capacities Cin and Cout adopted in Appendix Figure 2 were also used as quantities that change. From here, let us try replacing the variable heat capacities adopted by the heat engine model with the symbols for the amount of change in the entropy of the heat engine.

δSin＝Cin

δSout＝Cout

When the relationships among the transfer of thermal energy obtained in one cycle of operation of the heat engine model, the work energy output to the outside, and the input of lost energy from the outside are expressed using the symbols replaced with the entropy changes of the heat engine, they are replaced with the following expression:
By using thermal energy, temperature, and entropy, the expressions for output energy and lost energy have taken on a beautiful form. Not only are they beautiful, but the expressions have also become suitable for explaining social phenomena.

The expression for lost energy L using the entropy changes δSin and δSout of the heat engine is:

Lost energy L ＝ (Cout - Cin)(T1 ＋ T2) ＝ (δSout - δSin)(T1 ＋ T2) 　　　　　　[A-1]

2) The expression for the heat engine when heat transfers reversibly is:

Lost energy L ＝ 0　or 　Q1/T1 ＝ Q2/T2 　or 　δSout - δSin ＝ 0 　　　　[A-2]

3) The expression for the maximum output energy of a heat engine with no loss is:

Wout ＝ Q1 - Q2 ＝ Q1(1 - T2/T1) ＝ Q1(T1 - T2)/T1 [A-3]

4) The expression for the heat engine with no output energy, because heat transfers to the low-temperature
heat source without outputting anything (since Q1 ＝ Q2), is:

Wout ＝ Q1 - Q2 ＝ 0 or δSout T2 ＝ δSin T1 　　　　[A-4]

5) When there is no output energy, since Q1 ＝ Q2 ＝ Q ＝ δSin T1 ＝ δSout T2, the expression for lost energy L is:

Lost energy L ＝ (δSout - δSin)(T1 ＋ T2) ＝ Q(1/T2 - 1/T1)(T1 ＋ T2) ＝ Q(T1/T2 - T2/T1) [A-5]

Appendix Figure 3 shows an example of energy waste. Through a thin boundary, a small amount of heat transfers from high-temperature substance 1 to low-temperature substance 2.

Assuming that a small amount of heat Q transfers (heat release / heat absorption) compared to the heat capacity possessed by substances 1 and 2, this model does not affect the original temperatures T1 and T2. Substance 1 releases heat Q. Substance 2 absorbs the same heat Q. With no change in shape, heat transfers quietly.

The energy of heat Q that has transferred to the low-temperature substance 2 does not return to the high-temperature substance 1 naturally.
Even though the heat has transferred, the total energy of substances 1 and 2 is conserved.(The model set here is a virtual one and does not exist in reality. This is because heat cannot be transferred without leaving any effect on the outside. Even when simply mixing hot water and cold water, convection occurs, and there are also friction between the container and water, as well as air vibrations, which affect the outside.)

By reprinting expression A-5 for lost energy L when no energy is outputted to the outside, it can be expressed as follows:

　　　　　　　　　Lost energy L＝( Q/T2ーQ/T1 )(T1＋T2) ＝Q(T1/T2ーT２/T１ ）　[A-5]

Since the temperature of the high-temperature substance T1 > the temperature of the low-temperature substance T2, the lost energy L increases in proportion to the T1/T2 ratio.
The term Q/T2 - Q/T1 represents the change in entropy. The increase in entropy increases the energy that prevents reversing (the compensation required for reversing). Furthermore, since the magnification of this increase is in proportion to the T1/T2 ratio, the required energy to reverse increases.

You can intuitively understand that the lost energy required to be replenished to return to the original state matches the increase in entropy. That is, the larger the transferred heat Q, and the larger the temperature ratio (T1/T2), the larger the lost energy, and the larger the increase in entropy. This part is shown in Appendix Figure 4.

The larger the loss, the greater the degree of irreversibility. Also, since being irreversible corresponds to the irreversibility of time, the increase in entropy represents the passage of time. The increase in entropy in this formula is expressed numerically. This overlaps with emotional words such as "aging" or "getting tired," which are "phenomena that cannot be reversed, that is, irreversible phenomena."

By expanding this a little further, let us overlap energy waste with social phenomena. We correspond the temperature T, which is a scale representing quality, to a person's vitality level for living, and the heat Q to the energy of money.

Suppose that money is transferred to a person with low vitality (low temperature). The transferred heat Q (money) through debt or lending does not return to a person with high vitality naturally. If left alone, it becomes neglected.
Also, suppose that taxes collected from active people with high vitality are distributed to the elderly who have no intention of spending money, or to overly wealthy people. That money is not consumed. It happens to become deposits or stored away in cash at home.
Even if money is simply transferred from a certain person to another person who does not have it or does not need it, if it is not used for consumption that takes action to affect the outside, work energy that creates new value will not be generated.
Recently, there was a case where taxes were used to simply distribute money to all citizens, including people in need. A part of it was used for consumption. However, most of it was not used and generated nothing. This fact becomes an example where thermodynamics explained by expression A-5 overlaps with social phenomena.

Let us give another example of a social phenomenon that expression A-5 teaches us. There is a Japanese proverb, "Nasake wa hito no tame narazu" (Kindness is not for the sake of others), which is often interpreted incorrectly. The incorrect interpretation is that showing kindness does not benefit the person because it suppresses their own efforts.
Appendix Figure 4 shows this seemingly incorrect interpretation. That is, it shows that this interpretation is not incorrect.
This is because the law shows that even if you transfer heat energy to an object without taking out any work energy to the outside, it only increases the energy required to restore it.

If we change it into the words of social phenomena, it means that the law of thermodynamics, expression A-5, supports that the interpretation "Benefits are not for the sake of people" is correct.
Even if the interpretation of the proverb is incorrect linguistically, as a real-life feeling, it means that it correctly stated the actual reality.

By reprinting expression A-3 for taking out heat energy Q (Q1) from high-temperature substance 1, reversibly transferring a part of it to a low-temperature substance, and releasing work energy to the outside, it can be expressed as follows:

　　　　　Wout＝Ｑ１ーＱ２＝Q（１－T2/T1）＝Q(T1-T2)/T1 　　　　　　［A-3］　

Since the temperature of the high-temperature substance T1 > the temperature of the low-temperature substance T2, the lower the temperature T2 is, the larger the energy Wout that can be taken out reversibly becomes. This is shown in Appendix Figure 5.

Wout can be used for another heat engine to transfer heat from a low temperature to a high temperature. Saying that there is this reused heat energy that can return to a high temperature is, in society, like a sense of gratitude or obligation. If the sense of gratitude remains, that energy can return to the original high-temperature substance again, just like returning a favor.

Of course, this is a hypothetical concept. I felt that this external release of energy in thermodynamics corresponds to the proverb "Nasake wa hito no tame narazu" (Kindness is not for the sake of others) here as well.

The correct meaning of this proverb is that if you show kindness to others, it will eventually return to you. Since it says that if you help someone appropriately, the benefit will eventually return to you like returning a favor, I intuitively felt that a sense of gratitude was like Wout. This proverb was interpreted in two ways: the incorrect interpretation and the correct interpretation. Thermodynamics supports both of these two interpretations.

As described above, using the FURUMURA heat engine model that incorporates energy corresponding to social loss, we have derived expression A-3 for taking out the maximum energy and expression A-5 for just consuming without using any energy when transferring heat energy from a high-temperature heat source to a low-temperature heat source.

The maximum energy could be used when performing a reversible operation with no loss. Looking at expression A-3 at this time, we can understand that "the difference in quality (T1 - T2) generates vital energy," which was explained in Chapter 2. In addition, expression A-5 shows that the lost energy L is larger in proportion to the temperature ratio of the heat sources (high T1 / low T2) when the maximum consumption occurs.

Based on this knowledge, we can intuitively see social phenomena. If we correspond the vitality of society to the temperature of the heat engine and correspond energy to money, this matches everyday experience.
The reason there is a similarity to social phenomena is that whether the transfer of heat is reversible or irreversible, whether energy is output to the outside, and whether the quantity called entropy increases or decreases are all overlapped with social phenomena.

Also, looking at expression A-3, when T1 ＝ T2, even if there is a transfer of heat Q, the heat release and heat absorption cancel each other out, which is apparently the same as no transfer of heat occurring. In this case, even if there is a transfer of heat from substance 1 to substance 2, since there is no label on heat Q, the changes in state cannot be distinguished. Even if there is a reversible transfer of heat energy, the work to the outside Wout ＝ 0. In addition, at this time, looking at expression A-2, the entropy change (δSout - δSin) ＝ 0. That is, no change occurs in the state.

If we correspond this to social phenomena, we can intuitively understand that even if rich people (at the same temperature) lend and borrow money (heat) among themselves, no sense of gratitude will be generated.
We associated high-temperature substances with rich people， money with heat energy, and available output energy with a sense of gratitude to match social phenomena. However, if we choose the way of correspondence, we will be able to explain various social phenomena with these expressions.

The substances do not have to represent individual people. They may instead represent organizations, companies, or even countries. In the same way, high temperature can be used to represent vitality, the desire for health, ambition, the will to survive, or the desire for conquest. Depending on how we define the correspondence, we may even choose opposite qualities such as envy, poverty, or hostility. In that case, however, the carrier of energy must also be chosen carefully so that the analogy remains consistent.

As described above, I have explained the introductory concepts of thermodynamics using the FURUMURA Heat Engine Model, a framework that can be directly related to social phenomena. Through this approach, I have presented an explanation that is not found in conventional thermodynamics textbooks. When I entered university, I took an introductory thermodynamics course as a first-year student in the general education program. At the time, however, I could do little more than copy the lecture notes, and I never truly understood the subject. Looking back, it is not surprising that I struggled. If I were teaching the same material today, I believe I could explain the concepts that confused me then in a way that students could easily understand. Since the course attempted to cover a great deal of material in a short period of time, my lasting impression was simply that everything was difficult. The distinction between truth and expression introduced in Chapter 1 is a principle that I developed while struggling with mathematics during my second year of high school. This principle later proved invaluable in helping me understand thermodynamics. I also came to realize that it applies to the interpretation of social phenomena, because the truth that exists within people is ultimately expressed through observable events. Even before developing this model, I had intuitively felt that the differences in quality discussed in Chapters 2 and 3 were the source of energy in social phenomena. My attempt to explain thermodynamics in a way that anyone could understand eventually led me to discover why I had held that intuition for so many years. This column, which has continued for nearly two years, now comes to an end. Writing it has been an invaluable opportunity to organize and deepen my understanding of thermodynamics—a subject that had puzzled me ever since my first year at university.

[ Author : Y. F. ]