Zurek's improvement of the definition of entropy

"A potentially important application of algorithmic complexity to physics was proposed by Wojtek Żurek of the Los Alamos National Laboratory in New Mexico. In order to rid Boltzmann's definition of entropy of its troublesome element of subjectivity, Żurek suggested an almost imperceptible modification of it. Recall that entropy is a measure of missing information about a system. It therefore depends on what an observer happens to know: a smarter being has more information, is missing less, and thus assigns a lower entropy to a system than a more limited creature. To render entropy more objective, Żurek recommended adding a measure of recorded information to that of missing information. The sum of the two remains constant --if you remove data from one column, it reappears in the other. The observant creature thus becomes redundant; only the entries in its notebook or computer memory matter.

But how to access the amount of recorded information? Żurek chose algorithmic complexity as the most natural measure. Accordingly, his new, improved entropy consists of two portions: the conventional entropy as measured by the formula on Boltzmann's tomb, plus a piece that is normally inconceivably tiny, and accounts for the algorithmic complexity of the listing of recorded knowledge about the system. A mathematical description of the size and shape of a vessel containing a gas might be a typical item in the list, while missing information includes the coordinates of a vast number of atoms. Notice that in the hypothetical case that every position and every velocity of every atom is known, the Boltzmann entropy of the system is zero, but the added term --the length of the description of what's known, in binary code-- will be huge, bringing the total entropy back to its previous value. After a hundred years the reek of subjectivity has finally been lifted from the Second Law of Thermodynamics.

In spite of its cogency, Żurek's improved entropy has not gained much support.

Hans Christian von Baeyer, Information, the new language of science, Chapter 12

Simple. It reminds me of the kinetic and potential energy in adiabatic mechanical system, in which the sum of the two stays constant at all times. I also remember vaguely that the history of the total energy followed the same trajectory than that of entropy; only one portion of the energy was first defined and only when the second portion was defined, the energy became a constant of the system and starting to be accepted with its components, as valid quantities.

Notes from The Ages of Gaia by James Lovelock

First, let's start with a word of wisdom:

"The young usually find the constraints of convention too heavy to escape, except as part of a cult. The middle-aged have no time to spare from the conservative business of living. Only the old can happily make fools of themselves."

James Lovelock, The Ages of Gaia, Introductory.

The second point concerns entropy and the fact that has become more and more obvious to me recently, that for any achievement, from the construction of a building to that of an idea, waste is unavoidable and this waste represents the entropy that needs to be rejected for the achievement to be meaningful, to be out of the ordinary and randomness:

"You, as you read these words, are creating entropy by consuming oxygen and the fats and sugars stored in your body. As you breathe, you excrete waste products high in entropy into the air, such as carbon dioxide, and your warm body emits to your surroundings infrared radiation high in entropy. If your excretion of entropy is as large or larger than your internal generation of entropy, you will continue to live and remain a miraculous, improbable, but still legal avoidance of the second law of the Universe. «Excretion of entropy» is just a fancy way of expressing the dirty words excrements and pollution. [...] We animals pollute the air with carbon dioxide, and the vegetation pollutes it with oxygen. The pollution of one is the meat of another. Gaia [Planet Earth] is more subtle and, at least until humans appeared, polluted the region of the Solar System with no more than the gentle warmth of infrared radiation."

James Lovelock, The Ages of Gaia, What is Gaia?

I will write, tomorrow, "Entropy" on the side of my garbage cans.

We keep going with an issue concerning the stability of a system and its level of complexity. Is the more complex a system, the more stable? This is, from what I have heard still a controversial issue. In his book, James Lovelock agrees with the theoretical ecologist, Robert May: the more complex a system, the more fragile and unstable and inversely. This goes maybe against the naive assumption that if a system has a greater diversity, it has a greater chance to handle external perturbation. But May's mathematics prove the contrary: "increasing complexity makes for dynamical fragility rather than robustness". Thus, "the complex natural ecosystems currently under siege in the tropics and subtropics are less able to withstand our battering than are the relatively simple temperate and boreal systems." (R. May, in Theoretical Ecology, cited in The Ages of Gaia, Exploring Daisyworld)

In the chapter Middle Ages, James Lovelock mentions an extraordinary theory that Earth biosphere would be responsible in part to...the plate tectonics:

"The geologist Don Anderson has speculated that the deposition of limestone on the ocean floor [via, for instance, the dying and sinking of Coccolithophores and the burying of their calcium shells] is a key factor in the motion of the Earth's crust. He proposed that sometime far back in the Earth's history, sufficient limestone was deposited to alter the chemical composition of the crustal rocks of the ocean floor near the continental margins. As a result an event, called the basalt-eclogite phase transition by geologists, took place. This transition so altered the physical properties of the crustal rocks that it became possible for the great machinery of plate movement to begin turning."

James Lovelock, The Ages of Gaia, The Middle Ages.

This theory would explain why plate tectonics are not a universal properties of the planets. I have no idea, however, if this idea has been dropped or is still alive. More reading will be needed. But fascinating idea nonetheless.

I will finish on an improved definition of Gaia's theory, that also gives some explanation on the origin of the interaction between the biosphere and its environment. James Lovelock, himself, has corrected a previous definition of Gaia's theory and has re-defined it in his book as follows:

"Living organisms and their material environment are tightly coupled. The coupled system is a superorganism, and as it evolves there emerges a new property, the ability to self-regulate climate and chemistry."

James Lovelock, The Ages of Gaia, Gaia since 1988.

In this definition, the co-evolution of the biosphere and its environment is the key to explain why the self-regulation of the system is a likely property. I know that there is a lot of criticisms against Gaia, even with this improved definition. Although I am also a bit skeptical, I am wondering why there is no such criticism relative to the thermal regulation of mammals. In this case as well, it should be hard to believe that cells can organize at such a higher level that the whole system succeeds in regulating its temperature. If such feast is possible for an organism, given the geological time over which evolution is working, why not for the Earth's system as well?

Entropy = missing information

If you want to learn more about what entropy is and represents, I invite you to this scientific post that I wrote and which summarizes Ben-Naim's book Entropy demystified. I hope it will be of some help.

Some comments on entropy

Definition

Since I reviewed the basic teachings concerning entropy, I see entropy everywhere. The most helpful definition of entropy is the one regarding the state of order. The highest the order, the lowest the entropy. Thus, I declare myself an enemy of entropy, in the sense that I, we, always try to create some order, to put things in order: putting the plates away in the kitchen, entangling an electric wire, etc. Why is it actually easier to put things in disorder and to create entropy than the reverse? One useful explanation to comprehend this difference is to see that a state in order is an improbable state, while a state in disorder is a probable one: the electric wire is more likely to be tangled after so many years than to stay untangled, and there are many ways to put a mess in a room, but only one to put things at their places.

Entropy and time

There is some controversy concerning the relationship between entropy and time. The problem is that the fundamental laws of classical physics (from Newton) are reversible in time; that is whatever happens in one direction (toward to the future) can very well happen in the other direction (toward the past). Thus, as we see a drop of milk in a tea cup spreading and diffusing throughout the volume, we should see all the milk particles to come back and form the initial drop of milk. This is indeed possible according to Poincaré's recurrence theorem, although, because the state of the drop is very unlikely compared to all the states where the milk is spread, the probability that this happens is tiny (but in theory, it could happen!).

On the other hand, the second law of thermodynamics says that for a closed system, the entropy has to increase: the spreading of the drop of milk within the cup is a perfect example of entropy increase. Some, such as Prigogine, argues that entropy carries with itself the so-called arrow of time: because entropy increases, we can make the difference between past and future. But the question then remained, is the second law compatible to the reversible laws of classical physics? Roger Penrose, in his book The emperor's new mind, argues that the second law is not only compatible, but also, contrary to Prigogine's view, that the entropy does not carry the arrow of time with it. Whatever the direction, toward the past or the future, the entropy has to increase within a closed system, in particular within a system where there is no constraint on the entropy. In the case of the drop of milk, although toward the future there is no constraint and the entropy increases indeed, toward the past, there is the constraint of the initial conditions saying that the entropy is low at the beginning: thus, if you run the experiment backwards, there is the constraint that at the end, the entropy is lower than at the beginning. Because of this constraint, the second law does not apply as such and in consequence, the entropy does not itself carry the arrow of time. The arrow of time exists because our system started with a state of low entropy. The remaining question is thus, why and how did we start with a state of low entropy?

Source of low entropy

Both L. Botlzmann and R. Penrose describe the struggle for life as a struggle for low entropy, with the ultimate source of low entropy being the sun. L. Botlzmann writes

"The general struggle for existence of animate beings is therefore not a struggle for raw materials [...] nor energy [...], but a struggle for entropy, which becomes available through the transition of energy from the hot sun to the cold earth."

and R. Penrose says

"We do not need to gain energy from our environment because energy is conserved. But we are continually fighting against the second law of thermodynamics. Entropy is not conserved; it is increasing all the time. To keep ourselves alive, we need to keep lowering the entropy that is within ourselves."

Thus, how do we get this low entropy? The ultimate source of entropy is the sun, and plants are the organisms which are using directly this entropy source, transforming it into molecular structures, themselves ready to be eaten. We, humans, via the food web, are eating plants or animals who themselves eat plants, to get low entropy for our body.

In this aspect, I am then wondering if we can class the food web in terms of entropy content. The plants would get a source of low entropy S_i, some of it would be used such that the entropy content gained in the eaten plant would be actually larger, S_i < S_plant . This process would repeat in that the higher in the food web, the higher the entropy content. In that respect, I would conclude that 1) humans would be organisms with some of the highest entropy (the most disorder) and 2) we should all be vegetarians in order to efficiently get low entropy in our diet. Do you agree with these conclusions?

Why the sun?

R. Penrose also explains why the sun is a source of low entropy and I was very surprised to learn that the reason is nearly a geometrical one. The sun is a hot spot, a small disk of light compared to the entire sky. Because of this geometrical configuration, the energy we receive from the sun has a much lower entropy that the energy sent back to space by earth because this energy is sent into all directions. Thus, if I understand correctly, if the earth was surrounded by many suns so much so that they would cover the entire sky, there would not be any source of low entropy and life would be unable to exist? Of course, one still needs to explain why the sun, itself a compact star and thus a source of low entropy, exists but the explanation goes on with cosmological arguments that I understand much less.

References
Ludwig Boltzmann, The second law of thermodynamics, in Theoretical physics and philosophical problems
Jean Bricmont, Science of chaos or chaos in science?
Roger Penrose, The emperor's new mind