Hi Robert !:

Actually, this is a very interesting and (in my view as well) fundamental question, but moreover this is an open question in Physics.

As i commented in your previous question, there are several concepts named as Entropy, in different fields of science, even in fields as information theory or statistical economics (could be).

So, we can calculate the Shannon Entropy, the Statistical Entropy of a system of millions of particles composing a given macrostate, or the von Newmann Entropy of a quantum system or quantum particle, etc ... (just to give some examples)

But the question in theoretical physics regarding Entropy could be more general than the one read in your title. It could be as follows:

" Among all the concepts and definitions stablished for Entropy in different branches of science, coud it be one to be fundamental ? One from all the other Entropies emerge from ? Maybe.

This is today an open question, and theoretical scientists from fields as Cosmology, Quantum Mechanics and Information Theory still try to tackle up.

There are several interesting ideas around the question if there is one Entropy among all the Entropies which may be fundamental.

1. In particular, it exists a strong relation between the Statistical Entropy (or Configurational Entropy): S=kb * log(W) , and the Thermodynamic Entropy, this is why often people relate both with the level of order or disorder of a system. It is not clear how to get from one to the other, since the Statistical Entropy describes macroscopic (or systems composed of many particles) systems but composed of individual components (particles or molecules) which can arrange themselves in different configurations, but there is not sense to define the individual state of each of its component... but at the end this is a macroscopic system, like a thermodynamic macroscopic body, like a table, or a door, which is a classical thermodynamic object which is described by classical thermodynamics. So, while is clear that these two approaches should be strongly related between them, is not clear what is the treshold where we have to stop treating the entropy of the system as statistical and start considering the Entropy like the one of a macroscopic body in thermodynamic equilibrium.

2. Another idea, taken in a different direction from the people who study this question is inside the Information Theory. John von Neumann and Claude Shannon defined a concept, a computable number, when they were working with communications around the times of the Manhatann Project, the Alamo, etc... they just wanted to (more specifically Shannon was the one who discovered this quantity) he wanted a measurement for the lack of information in a comunication, and they called this Entropy... the Shannon Entropy.

Nowadays another direction in finding what is the most fundamental thing in nature... Quantum Mechanics has pointed strongs arguments for Information, QM has generalized the Information so seriously, that many physicist believe that is not the matter particles (e.g. the standard model) the fundamental building blocks of nature, but Information, raw conceptual information. In this view, information is not an energy, or a force, is really the code of the universe, or something like that.

So it is not weird that some people are more interested in Information Theory to help tackle the problem of how the Entropy of the Universe is changing.

3. the von Neumann Entropy is the Entropy in Quantum Mechanics. Is to say, you can assign a finite amount of Entropy to a quantum mechanical particle, or to a quantum system.

From what I have read, other Entropies, like the Statistical Entropy, Shannon Entropy , and even the Thermodynamical Entropy, somehow are though they will be shown up from the formula of the von Newmann Entropy. I guess this makes kind of sense, since Quantum Mechanics is one of the fundamental theories of nature.

4. Finally, there are (in particular since the last decade... we can say since the 2005s or so) with the new ideas developed around the physics of Black Holes. The Shannon Entropy and the Bekenstein-Hawking Entropy have the same meaning fundamentally, they should be fundamentally related, but there still many questions , e.g. At what degree a Black Hole can be treated as a thermodynamic body in equilibrium (speaking from its event horizon to its interior.) I think if theoretical physics in Black Holes continues to be developing, hopefully the Information loss problem of Black Hole will be able to give us clues of what is really Information at the most fundamental level, and if it is true that information is the most fundamental thing ...

And we are skipping to speak of the Arrow of Time, the Second Law applied to the whole Universe, and the beginning of the Universe, and its end... statistical fluctuations and boltzmann brains .

Regards all !

You gave a us a very important question that goes really deep. Honestly, we still do not understand the core principles of entropy and how are its different concepts related.

It is not necessary to repeats what was said by @https://www.researchgate.net/profile/Franklin_Paras_Hernandez but there is a lot to add.

This answer is focused on entropies used in biology and medicine to asses the state and quantify the complexity of systems observed there.

The human bodies are extremely complicated systems created of many components called cells that are regulated by a complicated network of regulatory networks. We are far from grasping all of this complexity.

On the other hand, we are capable to measure at least some biosignals produced by this complexity. What can we do with this? We can apply, and there are already a quite few research done on it, complexity measures in the form of entropy measures.

Why do we use entropy to measure a complex system, which we do not understand? Entropy can asses the complexity of the system without its direct observation and without understanding it. That is critical in biosignals processing.

Statistical physics (Boltzmann) information entropy (Shannon & von Neumann), and subsequent development of computational methods using entropies enable us to design a whole range of new entropy measures that are extremely useful in biosignals processing.

There are existing

* Information ( Shannon) entropy

* Approximate entropy

* Sample entropy

* Multiscale entropy

* Kolmogorov entropy

* Graph entropy

* Algorithmic entropy

* Detrended fluctuation analysis

And many more not show here.

_________

It is my field of research. More information can be found in my RG profile. Review on complex system modelling in medicine from Jan 2019 is a good starting point.

Dear Jiří Kroc :

What you described here has to deal with the branch of Statistical Mechanics called Stochastic Thermodynamics, Am I right ?

What is your view of Stochastic Thermodynamics?

Do you thing ST could be the framework which with we can get further understanding of the complex biological systems you are talking about ? From what I have read about the subject (which is very limited) I think it could be the case.

Stochastic Thermodynamics go deep on the Equilibrium Fluctuations of very small systems, which behave stochastically, coupled to an Active Bath, as opposed to be coupled to a Thermal Bath (which would be the classical thermodynamic equilibrium case): (where you have ideally a Heat source, or a Heat sink in thermal equilibrium).

I imagine a cell moving randomly (Brownian Motion) inside the biological fluid... If so, this cell could be treated as a stochastic particle 100%, coupled to interactions with a medium which will never be in thermal equilibrium. So, everything is random here, or calling it with the more technical term: Stochastic.

I don't know if there will be a definition of Temperature for this cell. But what I'm sure of is that Thermodynamic variables like Entropy, Enthalpy, or Internal Energy would be undoubdtelly Stochastical Variables..., along with other Thermodynamic Variables of the microscopic system.

Best regards !

Dear Franklin Uriel Parás Hernández,

First of all, citing people using @ just do not work in my browser. Hence, sorry to all who should be cited by the name but are not.

To your last answer. I do not know stochastic thermodynamics or possibly under its name. From your description, I see that we speak about different interpretations of entropy. Could you provide us, all readers a good review of physical entropies you are speaking about?

Entropy(ies) used in biosignals processing are derived from information (Shannon) entropy, which was originally developed to estimate the strength of the coding of encoded messages. Shannon entropy which use logarithm having the base of two, is basically measuring the strength of encoding in bits.

Originally, statistic measures developed for description not physical systems were used in biosignal classification. Later, due to problems with them as they require huge data to work properly, novel entropy measures were developed. See the list in my previous answer for examples.

Basically those measures of entropy map the complex systems' values into a limited number of bins (containers) which are used to define probabilities and subsequently are inserted into the entropy equation defined by Shannon & von Neumann

S = \sum_{i=1}^N p_i * log_2(p_i)

Where N is number of bins and p_i probabilities in them. Obviously, system acquiring only one state give entropy equal to zero log(p=0) =0.

This trick enables us omitting of all subtle details of the system. We use just a very crude measure of the complexity. Surprisingly, when this approach is applied in then correct way and an appropriate measure is chosen, we can tell a lot about the actual state of the biological system under observation. That all just from the biosignals.

This gives tremendous possibilities to assess the state of the biological systems (humans) on distance without touching them. This is the path to a highly personalized, predictive medicine.

Dear Jiří Kroc :

You are right ! I understood a different context when you talked about cells, biosignals and complexity. That was my bad. Now I got the context of your comment :)

I started to talk about individual microscopic components inside biological systems, like cells or bacteria inside biological media like blood, water or other biological fluids, and the attemp to asign an Entropy to the trayectories of these particles.

Now, I think I know what you speak about is the connection between information communication and biosignals, in biological networks like the brain or others.

So, yes, as you have commented, this would be more on the context of information theory and complex systems.

On the other hand, I do not have any scientific literature about what I wrote in my first comment. Everything is from general articles, and from academic and general public lectures I have listened on the internet. I'm sorry.

I believe you would have to go separately on each of these specialized fields to find more about the Entropy definition on each individual one, since the context of Black Holes, Statistics, Rigid-Bodies Mechanics, Information Theory, etc..., have their own guidelines and grounds to define what Entropy is.

Regards ! :)

Of course, I have no satisfactory answer to all the questions posed above.

I attache two papers: One is dealing with the connection between phenomenological non-equilibrium thermodynamics and stochastic thermodynamics. The second claimes that entropy is not uniquely defined, or even replacable by better defined quantities. But up to now, the question is open,

if there is a common background of all these different definitions of "entropy".

Dear W. Muschik :

I think Stochastic Thermodynamics is an interesting approach, and it will become used more and more with time, as we try to develop our understanding on microscopic systems immersed on a caotic medium (so called Active Baths, not exactly the case of Brownian Motion). I believe it will help us to understand the behavoir of caotic (or 'random') motion of individual particles in biological media, those encountered in fields like biology and medicine, the complex behavior of individual cells or microorganisms inside our bodies.

Maybe you already know this review from Seifert, but I will leave it here for the interested readers in general of this question.

Regards !