Miracles and Maxwell's demon

The interpretation of miracles and how they are performed is the basis of endless commentary and attempts at scientific explanation.

Chile lightning and volcano ash (photo credit: REUTERS)
Chile lightning and volcano ash
(photo credit: REUTERS)
In Judaism, the topic of miracles performed by the Almighty remains one of the most pivotal and important ones. The interpretation of miracles and how they are performed is the basis of endless commentary and attempts at scientific explanation. It should be noted that there is no universally accepted position on the nature and mechanism of miracles in Judaism. We will expand on that later. 
 
This article is not meant to provide a scientific explanation of miracles. It is merely an attempt to employ the Kabbalah of information and science to outline the governing principles of our world by the Almighty and to consider different perspectives on the nature and mechanism of miracles in Judaism. 
I Interpretation of Miracles in Judaism
Talmudic sages based their interpretation of miracles on the following excerpt from the book of Koheleth: 'What has been will be again, what has been done will be done again; there is nothing new under the sun' (Koheleth 1:9). This stresses that the Almighty gave the Law to the world when creating it and that the Law already provides for the possibility of miracles. For example, the sea, when being created, was told to part at a predetermined moment. The predetermined nature of miracles during the Creation was also speculated upon by Saadia Gaon in his commentary on Koheleth. In his Guide for the Perplexed, Maimonides also wrote that every miracle was planned for during the Creation.
 
According to Nachmanides and Baal Shem Tov, the founder of Chassidism, the Almighty interferes with the world order when performing miracles, i.e. changes the laws of nature that correspond to the name El-him. The interference of the Almighty corresponds to the name HaVaYaH.
II Definition of the Concept of Miracles
The most common definition of a miracle is: 'a phenomenon that defies the laws of nature'. This author considers this definition incorrect for the following reasons: 
 
1) We do not know all the laws of nature.

2) As the laws of nature were set by the Almighty, the above definition may be rephrased as 'a miracle is a phenomenon where the Almighty defies the laws set by Himself', a statement the author deems absurd.

We could define a miracle differently, as an extremely improbable phenomenon that we have never witnessed before. However, this definition would also be incomplete. The seventh Lubavitcher Rebbe wrote: 'As far as modern science is concerned, no one can claim the miracles described in the Torah are impossible. However, it is also absolutely forbidden to reduce miracles to phenomena that are only unique in their low probability'. This statement appears to contain a contradiction. On the one hand, the Rebbe agrees that, scientifically speaking, miracles are 'extremely improbable' phenomena. At the same time, however, he is adamantly opposed to equating miracles with such phenomena. 
 
Let us try to explain this. It is undeniable that miracles are extremely improbable phenomena. However, while essential, this definition is not exhaustive, given that extremely improbable phenomena can occur spontaneously over large periods of time. If you sit a monkey down at a piano and let it slam at the keys chaotically, there is a non-zero (but still extremely slim) probability of it playing a Beethoven sonata once in many billions of years. Note that the monkey would not be breaking any laws of nature by doing so. Miracles, however, occur at the right time and place and serve a concrete purpose. The author thus suggests the following definition of a miracle: 'an extremely improbable phenomenon that occurs at the right time and place and serves a concrete purpose'.
III The Law of G-d and Maxwell's Demon
The Law of G-d 
According to current scientific knowledge, at the micro-level, all phenomena are defined by a set of physical constants and the four fundamental forces: gravity, electromagnetism, and the strong and weak nuclear interactions. A cluster of micro-phenomena can be called an event. This is where we must ask a crucial question: how many micro-phenomena are needed for something to be considered an event?
 
Obviously, an event is a phenomenon that we can detect using our sensory organs or measuring tools. As such, an event is a subjective and relative concept for us, but not for the Almighty.

In the XIX century, scientists discovered a law that governed events without breaking any fundamental laws of the universe. That was the Second law of thermodynamics. It states that in isolated systems, entropy either remains constant or increases. This law, which essentially governs our reality,is axiomatic. It was first formulated in the 19th century by Rudolf Clausius as follows: 'a cycle wherein heat passes from a colder to a warmer body is impossible'. There are other definitions of this law from Kelvin and Gibbs, but they say essentially the same thing. 

The concept of entropy, like many other concepts in physics (such as mass or energy), does not have a clear definition. It can be viewed as the degree of randomness of a system, the number of possible states, or a measure of hiding information. Consequently, different interpretations of the definition of entropy exist in thermodynamics (by Clausius and Gibbs), statistics (by Boltzmann), and information (by Claude Shannon). As stated by Manfred Eigen, one of the 20th century's most prominent scientists, 'the concept of entropy is too broad to be limited to thermodynamics only'. Defining entropy is too broad a problem to be covered in this article. However, within the scope of the specific issue of miracles, let us limit ourselves to the statistical definition of entropy given by Boltzmann, under which the entropy of a system 'S' is equal to
S = klnW, where 'W' is the number of the microstates of the system and 'k' is the Boltzmann constant.
This prompts the question of what states of the system do we mean?
 
To establish this, we will have to delve deeper into physics.

In classical physics, a particle has six degrees of freedom: three spatial coordinates and three velocities (the velocities are usually substituted for momentum).

Thus, the state of a particle at any given time can be described by its position in six-dimensional space. If a system contains N particles, then the state of such a system can be described by its position in 6N-dimensional space. Macro-objects contain a great number of particles, and, accordingly, a phase space has a great number of dimensions. 
 
Let us consider the state of a macro-object (a plain drinking glass for simplicity) at a given time. All the atoms and molecules that make up the glass have certain momentum and coordinates. Such a time sample is called a microstate. However, after a miniscule amount of time, the coordinates and momentum of the particles change and we have a different microstate. This is a continuous process. As such, the position in multidimensional space that describes the state of our glass will be changing continuously. 
 
Let us imagine two people looking at our glass. One, much like us, cannot see any changes in the glass. The other, possessing the magical ability to see atoms and molecules, sees not the glass but a certain structure in a continuously changing state. The state in which the glass appears to the first person is called the macrostate.

Returning to Boltzmann's interpretation, we can say that the number of states in this formula is equal to the number of microstates that represent the same microstate. This gives rise to a logical question: the same' as seen by whom? The answer: by us. Thus, the concept of entropy is subjective and relative, and it depends on what information we can perceive from the system we interact with.
The prominent theoretical physicist Carlo Rovelli speculates on the relative nature of entropy in his book 'The Order of Time'.
However, the issue is far from settled. The question arises of why the macrostate remains unchanged while the possible number of microstates is huge. For example, a system containing N particles has a number of possible states proportional to NN—an unimaginable number.

Manfred Eigen offers an example. Ludwig Boltzmann posed the question of how much time is needed for 1018 molecules contained by a single cm3 of air under a pressure of 1/30 atm to reproduce (repeat) their microstate? The answer given by Boltzmann himself is 101019 years. The time needed for a microstate to reoccur is called the Poincaré time, named after the famous French mathematician. 
 
And here we come across a contradiction. The laws of nature allow for a huge number of microstates, while the macrostate remains virtually unchanged to us. The proportion of microstates representing the permanent aspect of the macrostate is negligible compared to the total number of possible microstates. This can be explained by the following: the Almighty has designed our world so that for us, a great multitude of possible microstates are not forbidden, but are 'unpopulated' in practice. The function governing the arrangement of microstates is a curve with a sharp peak; it drops to virtually zero at even marginal increments away from that peak. 

Hence, when we see a macrostate defined by microstates from the 'unpopulated area', we consider it a miracle. All the while, no laws given to our reality by the Almighty have been broken.

In fact, when creating our world, the Almighty compressed the number of possible microstates greatly, which is why we can interact with specific macro-objects.

Thus, the author concludes that by performing miraculous phenomena, the Almighty is not breaking any laws of nature set by Himself. However, as noted earlier, miraculous phenomena that are extremely improbable differ from other extremely improbable phenomena by occurring at the right time and place and serving a concrete purpose. In the next section, we will talk about how that happens. 
Maxwell's Demon
In 1867, James Maxwell, one of humanity's greatest scientists and the creator of the theory of the electromagnetism, conducted the following thought experiment: let us imagine a container filled with gas and divided by an isolating partition. There is a molecule-sized opening in the middle of the partition. The gas is in a state of equilibrium, and the temperature is equal on both sides of the partition. Maxwell placed a being he called a 'demon' by the opening in the partition. He also equipped the opening with a trapdoor the demon could use to shut it. The demon's actions can be described as follows: when a molecule moving at a lower-than-average velocity approaches the opening from the left side, the demon shuts the trapdoor, and the molecule is elastically reflected back to the left section of the container. When a molecule moving at a higher-than-average velocity approaches the opening from the left side, the demon lets it through to the right section of the container. When a molecule approaches the opening from the right side, the demon does the opposite—blocking faster molecules and retaining them in the right section while letting slower molecules through to the left section. The activities of this 'malevolent demon' lead to the accumulation of slower molecules in the left section, and faster ones in the right. The result is a temperature discrepancy: the temperature of the right section is higher than that of the left one. As Maxwell proposed ideal conditions for his thought experiment, the demon is not doing any work while carrying out this process. As a result of the demon's activities, the entropy of the system decreases in direct violation of the Second law of thermodynamics. 
 
This thought experiment spawned a mass of speculation on the nature of the Second law of thermodynamics in the scientific community, as well as a number of possible explanations. Here it is important to remember that we have not observed any natural phenomena akin to the activities of Maxwell's demon. Speculations on the above thought experiment are much too broad to be fully addressed here, so we will have to settle for a brief summary of them. Explanations of the experiment essentially amount to the following: while the demon does not need to do any physical work to select the molecules with different velocities, it must possess the ability to continuously perceive information about the speed of approaching molecules, as well as enormous memory capacity in order to store this information. One group of scientists, which included Leo Szilard and John von Neumann, stated that the very process of the demon collecting information would lead to increased entropy, thus compensating for the decreased entropy in the gas container. Another group, represented by Roger Penrose and Rolf Landauer (among others), claimed that entropy would increase at the moment the demon's memory is full and has to be reset. The author shares the position of the first group, supporting the opinion that the collection of information leads to increased entropy. 
 
Thus, in order to implement microstates from the 'unpopulated area' (see above), one must possess the ability to process a huge volume of extra information. This is an issue for us humans, but not for the Almighty. We could hypothesize that the Almighty creates not a demon, but an angel in the form of a programme capable of processing an enormous body of information and implementing microstates from the 'unpopulated area'. One possible example of this is the miracle of Chanukah, where this information-processing 'angel' program would create microstates with a considerably lower oxygen intake level, causing oil which usually burns out within a day to stay lit for eight days. 
 
The above speculations allow us to draw an important conclusion: the ability to process extraordinarily large volumes of information makes it possible to create miraculous events without breaking any fundamental laws of reality.
IV Landauer's Principle
A fundamental provision of the Kabbalah of Information proposed by the author in his book 'From Infinity to Man' and a number of articles such as 'What Everything Is Made Of' (lechaim.ru) is the fact that everything ever created, the entirety of reality, is information in one form or another. That is why there is no division between the spiritual and the material at the most fundamental level. 
 
In this regard, the principle proposed by information-processing researcher Rolf Landauer is particularly interesting. The principle states that when executing logically irreversible operations on a computer, a certain amount of heat is released. When logically reversible operations are executed, no heat is released. To clarify, an operation where it is possible to uniquely determine the input parameters using the output results is considered logically reversible. An example of this is a 'no' operation. This means that with an output of 1, the input must be 0. The opposite is also true. An example of a logically irreversible operation is 'or': if the output contains a 0, there are three possible input values, namely, (00), (01), and (10). Here we should point out that Landauer's principle has not been proved. However, it is considered credible among the scientific community. 
 
The importance of Landauer's principle lies in the fact that it bridges completely 'different' states: the mental/logical state and the specific physical state. The author considers this principle to be supportive of the fundamental provisions of the Kabbalah of Information. 
V Miracles and Chaos Theory
The fact of the existence of miracles performed by the Almighty plays a crucial role in speculation on the providence of G-d and the question of whether or not our world is a deterministic one. This is an extremely broad theme, so we will only touch upon it slightly.
In the late XIX century, Henri Poincaré, one of the greatest scientists of the modern era, entered a competition sponsored by King Oscar II of Sweden. The goal of the competition was to determine whether the solar system is stable or can be destroyed at any point. Poincaré made all the necessary calculations and won the competition. However, his calculations included several assumptions and approximations he deemed too minor to have an impact on the ultimate result. When Poincaré finally decided to check his calculation, he was appalled to find out that these minor approximations in the initial positions of the planets caused critical discrepancies in the descriptions of their orbits. However, he made not only a mistake but also a discovery, as this is how he created chaos theory. The core premise of chaos theory is that 'seemingly irrelevant and negligible discrepancies in a system's initial state can lead to great discrepancies in its state during the process of its evolution'. The theory effectively outlined the limits of human abilities, as chaotic systems are non-computable. 
 
Based on this, we can hypothesize that the Almighty can use chaotic systems to govern our reality while hiding from us the very fact that it is being governed. 
 
Completely imperceptible to us, the creation of a 'miraculous microstate' by the Almighty at the micro-level can completely change the course of future events.
Conclusion
In light of the above, the author offers the following findings for readers to assess:
 
1. No fundamental laws of the universe are broken when miracles are performed.
 
2. Thus, the Talmudic sages, Saadia Gaon and Maimonides were correct in asserting that the possibility of miracles was predetermined at the Creation.
 
3. While no laws of our world are broken when miracles are performed, said miracles require the creation of systems capable of processing great volumes of information. Thus, the position of Nachmanides and Baal Shem Tov regarding the Almighty's intervention in the process of performing miracles is also correct.
 
4. The author considers the opinion currently taking shape in the scientific community that mental states are directly connected to specific physical manifestations to be supportive of the fundamental provision behind the Kabbalah of Information.
To purchase Eduard Shyfrin’s book ‘From Infinity to Man: The Fundamental Ideas of Kabbalah Within the Framework of Information Theory and Quantum Physics’ please click here.   
To purchase Eduard Shyfrin’s book ‘Travels with Sushi in the Land of the Mind’ please click here.
This article is part of a collaboration for the Kabbalat Shabbat Project with Eduard Shyfrin.