[My Essay]

Entropy and State

Jim Shao

In one of my recent essay, I wrote:

"The degradation of society is a spontaneous process of increasing ¡®entropy¡¯. It can only be restrained or reversed by the continuous work of external forces. Otherwise, beware of that the 'capitalistic' building of democracy and freedom will also collapse, just like a beautifully ordered sand sculpture that is allowed to be impacted by wind and waves. According to the principle of increasing entropy in physics, it will soon degenerate into a desolate and disorderly beach ruin."

A friend of mine kindly reminded me privately that the way you mentioned "entropy" here might be incorrect in physics, because Dr. Xifeng Wu, in his article "Entropy and Order" published last year (http://hx.cnd.org/?p=213252), criticized exactly your such statement.

Wu wrote:

"Indeed, the physical concept and quantity of entropy are often misunderstood by the public and even physics researchers. For example, some popular science articles compare a carefully built sandcastle on the beach with its 'ruins' after being hit by waves to illustrate the former. The degree of order of the system is high, so the entropy value is low; while the degree of order of the latter is low, the entropy value is high. Then it is concluded that humans can reduce the entropy of a system and turn disorder (ruins) into order (Sand Castle); and the violent winds and waves of nature turn order into disorder and increase the entropy of a system.

These are very common and typical misunderstandings about entropy and ordering. In fact, neither the person sculpting the sandcastle nor the waves destroying the sandcastle have changed the entropy and orderliness of the pile of sand at all. "

I searched Wu¡¯s paper and after reading it carefully, I found that Wu might have some wrong concepts in physics, so I wanted to take this opportunity to explain it again.

My understanding is: entropy is a state function, that is, it is a function of the status of the system and has nothing to do with the process leading to that state. The state of a thermodynamic system can be described by parameters such as pressure, volume, temperature and internal energy. For example, the entropy S of a closed system can be written as a function of the real energy values E of each part of the system:

S = ¦²Sa (Ea)

The sand sculptures we are talking about are all in the gravitational field of the earth. The potential energy of a sandcastle is obviously greater than the pile of sand grains after its collapse. These are two states with different internal energies, and their entropy values are of course also different. From the former to the latter, the entropy increases greatly.[Note]

[Note] Imagine a grain of sand falling from a height in the castle due to perturbation. Its potential energy is converted into kinetic energy. Due to frictional resistance and inelastic collision, the kinetic energy is dissipated into heat, causing the temperature of the sand grain and its microenvironment to be slightly increases, and eventually, it stops somewhere at the bottom in a more probable state than before. The collapse of the entire castle is similar to processes such as avalanches or glacier migration. Although energy is conserved, the system entropy increases.

A castle with high energy and low entropy can collapse into a pile of sand, but a pile of sand with low energy and high entropy cannot automatically restore itself to a castle.

Wu believes: "In fact, neither the person who sculpted the sandcastle nor the waves that destroyed the sandcastle changed the entropy and order of the pile of sand at all."

However, Ludwig Boltzmann proved that the entropy S of a system is proportional to the logarithm of its state probability W:

S = k ln(W)

If it is true, as Wu said, that after the wind and waves turn the sand sculpture "castle" into a "ruin", the entropy value S of the two remains unchanged, then the wind and waves will have the same opportunity W to turn the "ruin" back into a "castle".

In other words, if a process connects two states with the same entropy value, then it is a "reversible process", that is, in principle, the sand pile can automatically recover into a castle in the opposite direction with equal probability.

However, things like this kind of "reunion" have never happened naturally in the world. The law of entropy growth is the "second law of thermodynamics". Heat flows from high temperature to low temperature, and particles diffuse from high density to low density. These are irreversible processes unless there is "external force doing work" or "time reversal".

The ideal gas molecule model in thermodynamics assumes that the system is composed of thermally moving particles, and there is no interaction between the particles except elastic collisions. Important statistical laws including the Maxwell distribution can be derived from this model. However, due to the neglect of molecular volume and intermolecular interactions, the inferences of this model are inconsistent with the behavior of real gas under high pressure or low temperature conditions.

Wu's mistake was that he used the molecular model of an ideal gas to discuss the entropy of the sand system, and regarded the sand particles as particles without any interaction, thinking that there was only a geometric relationship between them (configuration). He forgot about gravity, which determines the state of the system.

Even if a sand sculpture is constructed in a closed space with no external gravitational field, the sand particles with mass will still have their own gravitational interaction with each other. Therefore, their energy status is different due to the different configurations, so the entropy value will also be different.

Even for elementary particles whose gravity can be ignored, they still have electromagnetic, strong, and weak interactions with each other. Furthermore, there are distinctions between fermions and bosons, so that even a simple exchanging positions may affect the status of their wave functions.

Therefore, Wu uses non-interacting ideal particles to discuss entropy and only considers their geometric relationships, which is really meaningless.

When talking about "orderliness", Wu also said:

"Let's simplify this problem first. Now there are only three grains of sand. What kind of arrangement is more orderly? A straight line? An equilateral triangle? Or any triangle? No matter how you arrange these three grains of sand (or one million grains of sand), each time a specific configuration is formed. Which configuration has a higher degree of order (not symmetry)? This is subjectively determined by the observer, lacking objectivity, and if using different 'orderliness' standards, the orderliness of a configuration will be different."

"For example, configuration 1 consists of eight numbers: 12345678 (analogous to a sandcastle), and configuration 2: 36821457 (analogous to ruins). Configuration 1 looks very orderly, while configuration 2 looks quite disordered. Many popular science books (even some textbooks) use this example to illustrate what entropy increase is: a system changes from configuration 1 (highly ordered sandcastle) to configuration 2 (highly disordered ruins). And 12345678 are thought to be an orderly arrangement just because we subjectively believe so. If Change to eight colors, such as configuration 1: yellow, blue, black, orange, red, green, indigo, purple, and configuration 2: black, green, purple, blue, yellow, orange, red, indigo, do you still think that the order of configuration 1 is higher than that of configuration 2? "

"In the same way, a sandcastle looks highly ordered, but that's just our human perception. Ruins that have been hit by wind and waves have lost the 'orderliness', but that just was a certain aesthetic interest by many people. However, some modern and postmodern artists believe that ruins created by natural forces are more aesthetically pleasing than man-made sandcastles, because there is a deeper and subtler order hidden in the seemingly disorderly ruins. Statistical physicists believe that both are a specific configuration of a sand pile, and there is no difference in order between the two, just like the previously mentioned configuration 1 (yellow, blue, black, orange, red, green, indigo, violet) and configuration 2 (black Green, purple, blue, yellow, orange, red) like that."

My answers to these arguments are as follows:

Starting from above-mentioned Boltzmann's relationship between "entropy and state probability", in thermodynamics and statistical physics, the correct question is not to ask: "Which state is more ordered?" but to ask: "Which state is more probable?"

To determine the state probability of a system, in addition to reference to geometric configuration and spatial distribution, it mainly depends on relevant physical parameters such as energy, temperature, pressure, and volume.

Entropy is a function of ¡°state¡±, it cannot be said directly that it is a function of ¡°order¡±. The reason why order is related to entropy is because of the second law of thermodynamics, that is: the entropy of a closed system can only increase monotonically to a maximum value unless an external force does work to reverse it to a low-entropy state. Therefore, based on this experience of "having to pay a price", people are accustomed to calling the low-entropy state as "tidy and orderly" and the high-entropy state as "chaos and disorder". This is just a common convention.

Therefore, the "orderliness" of the sandcastle is high because its energy state is high and the entropy value is low. It requires external force to do work to form it. The "orderliness" of the sand pile is low because its energy state is low and the entropy value is high. It is the result of natural collapse. Energy and the change of entropy are physical quantities that can be measured and calculated, and have nothing to do with people's preferences.

Similarly, people think that the arrangement of the numbers 12345678 is more orderly than that of 36821457, because the former is arranged according to the rules of an arithmetic increasing sequence, while the latter is just a combination of random numbers.

While, the spectrum "red, orange, yellow, green, blue, purple" is called orderly, because its wavelength changes from long to short. When there are no human beings, the rainbow produced by the refraction of water vapor after rain will still be the same. It will not be "yellow, blue, black, orange, red, green, indigo, violet" or "black, green, purple, blue, yellow, orange, red, indigo". This has nothing to do with people's 'degree of order' based on their "certain aesthetic pleasure".

Although "some modern and postmodern artists believe that ruins made by natural forces have higher aesthetic pleasure than man-made sandcastles, because there is a deeper and subtler order hidden in the seemingly disordered ruins", They themselves can "appreciate" it this way, but physicists cannot agree.

Finally, leaving aside the issue of physical entropy, I would like to explain a few more words about "using the term of increase in social 'entropy' as a metaphor for society degradation":

When Elon Musk made new policies for Twitter he purchased, he probably did not take into account the weighting factor of "social psychological asymmetry" and simply believed in "freedom of speech" and equality for everyone, that is: he says, she says, then you can also say something.

However, even in fair societies with "freedom of speech" like Europe and the United States, evil scoundrels are often shameless and unscrupulous, spread rumors and lies, and intimidate and threaten. They are stupid and hard-working, gang up to bullying others, have loud voices, talk a lot, and make trouble and messy unreasonably, use tactics to win by quantity. While, in the same time, righteous and well-educated knowledgeable people are often too lazy or ashamed to argue with them, and just sit back and watch. As a result from this unbalance, social public opinion often manifests itself in filthy language, the proliferation of lies, and the prevalence of fallacies, so that the "voice of noise" is far higher than the "voice of reason."

This "social psychological asymmetry" is a "statistical weight" factor that is easily overlooked.

Therefore, in order to prevent the degradation of a society, no matter what, citizens still need to work hard to overcome such "social psychological asymmetry" factors, fulfill their moral responsibilities, bravely stand up and speak out, work together to do good, and fight against evil forces together.

The truth is our eternal pursuit, while lies and deception are the common enemies of society.

Written on October 17, 2023.

Relevant Reading£º

Jim Shao£º"The three-body problem" (April 28, 2024)

Jim Shao£º"Langton's Ants" (August, 2017)