By the way, Quora has the question: Why do most people find the concept of entropy confusing?

An answer there given by Garvin H. Boyle is interesting and worth reading.

Classical thermodynamics is a phenomenological theory but not a Fundamental Theory in the usual sense, the zeroth law cannot explain what is temperature; the first law indicates that energy is a conserved quantity, but can hardly distinguish energy exchange and energy conversion; the second law indicates that the entropy will be increasing monotonously, but there is no an explicit definition which indicates the physical meaning of entropy, therefore, classical thermodynamics itself cannot explain what is entropy.

Classical thermodynamics is a deterministic theory. In it self it cannot explain the stochastic aspect of entropy.

The first thing we have to understand is that there are many Entropies in nature. This is to say, that in different fields of physics (Quantum Mechanics, Relativity, Statistics, Thermodynamics, etc.) there are different concepts which has been named as Entropy. And they do not mean the same thing.

For example, it is not the same, the Thermodynamic Entropy than the Statistical Entropy.

This is to say that e.g. the Thermodynamic Entropy is a total different thing than the Shannon Entropy, or that of von Neumann Entropy, etc...

So I think, the first misconception we have to remove from ourselves is that we tend to assign the same meaning to all the Entropies we run into in science.

For instance, the Thermodynamic Entropy is a hard concept to wrap our head around, because it was discovered using a phenomenological approach (not derived from maths, laws, or any theorem). Moreover, Thermodynamic is the science of thermal equilibrium of macroscopic bodies, and Entropy is (from its phenomenological view) a property of a system which needs a change of State (from one equilibrium state to another) to be computed. i.e. the absolute Entropy cannot be calculated for a Thermodynamic System in equilibrium. And, as some have already pointed out, Thermodynamics as a science doesn't give a fundamental meaning of the Thermodynamical Entropy, so it is hard to get and intuitive understanding of this property.

But, for the Shannon Entropy (the Entropy in Information Theory), for example, the concept is well defined, and easy to understand: it is the lack of information in a given communication.

Likewise the Bekenstein-Hawking Entropy, which is the Entropy for a Black Hole, is fully well defined, it can be fully calculated for a Black Hole with a classical easy formula.

So, when people ask why the concept of Entropy is so hard to get around. I think what we have to say instead is: "Why the concept of Thermodynamic Entropy is so hard to get around?" ...being specific.

Is not that is hard, but is that different people have some slightly different interpretation for this thermodynamic property.

Why ? Because in Thermodynamics, Entropy is not well defined. When something like this happen, the best thing to do is to interpret the given quantity (Thermodynamic Entropy in this case) to the letter from what its formula represents:

" The Thermodynamic Entropy is the Infinitesimal change of Heat Rate (removed from, or added to the system), divided by its corresponding change in Temperture ". That's it, if someone ask for a ultimate definition of the Thermodynamic Entropy, the most correct interpretation. Just read the formula, and understand what each term represents, and the relationship between them.

I know this definition doesn't sound fancy, nor friendly at all, but is the best definition. Is a definition we wouldn't have to worry of being wrong.

However, people have given different interpretations and several more pleasent definitions to the Thermodynamic Entropy for a Macroscopic Thermodynamic system. Personally, the one I like the most, and feel it explains a more accurate representation of this property is the following:

The Thermodynamic Entropy accounts for the Irreversibilities of the System, in a process (or processes) of a change (or a series of changes) from one equilibrium state to another equilibrium state.

Best Regards !

Hernandez gives a good framework about why it is difficult to grasp the entropy concept. I would add something more from the stand point of some basic facts.

Energy can be classified generally in two categories, one conservative and the other dissipative; and in thermodynamics the former is work the other is heat (q). In the thermodynamic definition of entropy, we use dq/dT. Both q and T are features of random motion of molecules; and the definition of temperature (T) is somehow problematic in the form kinetic energy=(Boltzmann constant)xT, even though we don’t have any better definition of it. The faster the molecules the higher the temperature. This definition is perfectly good for steam at 100 degrees, because water molecules in gas state are free from any attraction and randomly bounce off in collisions, but for water at 100 degrees, what mass should we use? Since liquid water molecules attract each other we do not know what mass to use if we apply the formula for liquids as the molecules are in the form of clusters. Same logic is also true for solids. Anyway, we don’t use that formula for systems if the individual molecules are not freely moving, but assume that the same T can be used also for such systems, because, while measuring temperature of a liquid we are actually measuring the momentum of liquid molecule clusters (not individual molecules) hitting the unit surface area of a glass thermometer.

The great Boltzmann showed us that entropy defined as degradation of energy is not a sufficient definition, but rather it is a special definition. He proved that entropy is a measure of the increase of the randomness in the existing configuration of a system. In fact, the ‘phase space’ term involves the multiplication of coordinate space with momentum space. Its logarithm gives two added terms, one is associated with coordinate space which corresponds to the entropy of work term, and the second is associated with the momentum space which corresponds to the entropy of heat term. Hence, the multiplication (which gives the possibility) when converted into logarithmic terms (i.e. the probability) gives us entropy, and Boltzmann in his H-theorem showed that entropy (i.e. more probable random state) always increases in the universe that is in perfect agreement with thermodynamic entropy.

Boltzmann opened a door to science world that whenever there is a possibility for a system to go into more random states there we can apply the entropy concept. But what is the temperature in such systems if it is not a thermodynamic system? What is the temperature in Shannon entropy systems? For instance, we can find the Shannon entropy of the distribution of musical notes in a song without using a temperature term. In such systems or Shannon’s information systems we don’t need to use hard core physics terms such as temperature, momentum, or energy. Starting with Boltzmann and especially with Shannon we are able to define entropy for non-physical things within the framework of the magic word “information" i.e. Shannon information.

Physical laws cannot be applied to non-physical world, as fundamental conservation laws ‘momentum’ and ‘energy’ cannot be applied to non-physical things. However, in case of entropy, yes it can be used in both physical and non-physical worlds. All physical conservation laws have “equal” sign when expressed in the form of mathematical expression. Except entropy!. When equality sign is used we presume that what our system has some axioms that the system is constructed on those building blocks. In mathematics, probability theory does not have well-defined axioms, and the increase of randomness in the universe as the most concrete physical fact can be expressed only by “>, i.e. greater than” sign. The increase of entropy is the most vivid and the strongest principle of our universe, and it cannot be expressed by well-defined conservation laws, because there is something not conserved.

The names of ages in the historical order “stone, brass, iron, steam, machinery, electricity, atomic, space…” ages deal with some kind of material or energy. The very recent name of our age is “information age” which has nothing to do with material or energy, it is the age of both physical and non-physical (but not metaphysical !) entities.

‘’Energy can be classified generally in two categories‘’, however, in classical thermodynamics, this classification is not for the internal energy, δQ/T is the entropy of the heat exchange δQ, δW/T makes no sense, the heat Q and work W are the two types of the energy in energy exchange processes, the two both cannot denote the state changes of the internal energy, in this sense, we need to define a similar classification for the internal energy.

Can the Internal Energy be classified generally in two categories? This is the real issue, since no such a classification, “Now entropy is a very strange concept without hoping to achieve a complete description.” (I.Prigogine). It is impossible to explain the physical meaning of the state function entropy only by δQ/T.

Similar issue also involves some other concepts, such as enthalpy, Helmholtz free energy, grand potential.

https://www.researchgate.net/publication/331457885_Heat_Energy_Entropy_and_The_Second_Law_A_New_Perspective_in_Thermodynamics

https://www.researchgate.net/publication/333385885_What_is_Entropy

There is a book that among other things raises the question posed above: Entropy: God’s Dice Game (2013) by Oded Kafri and Hava Kafri and attempts to address it.

I have added an article to RG that relates to this question.

Preprint Why not just Q 1 = Q 2 ?

I invented BiEntropy and TriEntropy to try to understand more about Entropy in an informational sense. BiEntropy has been successfully applied in a variety of fields. Together, I have just had a spectacular success in their joint application in prime number theory. Both functions, though real, are quantised in their gradation of Entropy between perfect order and perfect disorder. This may have a physical meaning or correspondence.

Article BiEntropy - The Approximate Entropy of a Finite Binary String

Preprint BiEntropy, TriEntropy and Primality

Regards

Grenville Croll

I have added an article that touches on this issue,

Preprint Entropy 1865, 2019 update