It is a difficult task for me to translate my work into English, below is a try, it involves some deep topics.
Tackling a Century Mystery: Entropy (2008)
Introduction
Why are we still unable to explain the difficulties caused by a physical concept even after more than 150 years of hard work?
It is a very important milestone in the history of science to introduce the concept of entropy into physics, it was the first time to introduce one-way direction of change into the theory of science, and express irreversibility as the internal property of change. The introduction of the concept of entropy has had an extremely profound impact on the basic view of how science should understand existence and evolution of nature.
However, the introduction of the concept of entropy has also brought serious puzzles into physics, since the heat Q is not a state variable, it is really confusing to define a state function by the aid of a path differential and an inequality. To be exact, in 1854, R.Clausius has only given a symbol without any explanation to the physical meaning of the function S, classical thermodynamics itself cannot explain clearly what entropy is, it can only talking about how the entropy will change.
This is a very strange result difficult to be explain, because the concept of entropy does not seem like to involve the cognitive limits of science in our time, no one can understand what this state function is from classical literatures or any thermodynamic textbook but can only know how the state function will change, is this a perfect result?
The understanding to the entropy today we have mainly come from Boltzmann's statistical theory, in the entropy theorem, Boltzmann pointed out that entropy is proportional to the logarithm of thermodynamic probability, in H theorem, the second law was described to be the state change of thermodynamic probability, this later developed into a very popular view: entropy is a measure of disorder.
Until now, Boltzmann's statistical theory still faces a series of problems, the postulate of equal a priori probability is not applicable to the case when there are interactions within a system, such as multi-phase coexistence in thermodynamic equilibrium, liquid-liquid equilibria in a partially miscible binary system, or the segregation of alloying elements in a solid solution, and the temperature of nuclear spin system, the ordered state of Gibbs free energy, or the gravitational potential energy are some other examples. A very fact is that the postulate of equal a priori probability is not applicable to describe the ordered state of the particle distribution or the energy level distribution, this postulate is only applicable to the case when thermal motion is stronger than interactions. It is quite clear that thermodynamics does not need to consider this postulate, thus, how can one prove that Boltzmann's entropy is exactly equivalent to thermodynamic entropy?
What is puzzling is that we already know that we cannot discuss the second law when ignored the dissipation factors such as friction and viscosity, we also know that these dissipation factors are obviously related to the interactions but not the result of the postulate of equal a priori probability. Whereas, in statistical theory, near independent subsystems or statistical ensemble are the main models to discuss the entropy and the second law, in such ideological system, how can we discuss the dissipation factors such as friction and viscosity?
H theorem have been stumped by the ‘’inversion paradox’’ proposed by J.Loshimidt in 1876, it is also unable to explain the ‘’circular paradox’’ proposed by E.Zermelo based on Poincare recurrence theorem. Boltzmann himself later realized that the H-quantity model as a solution to explain the irreversibility derived from dynamics still has problems that are difficult to explain, it is only a phenomenological model, just as K.R.Poper said: ‘’Boltzmann failed to clarify the state of H theorem, nor did he explain clearly the increase in entropy’’.[1]
Another problem is: since there has been no monotonous function similar to thermodynamic entropy in dynamics, the general mathematical properties of the fundamental laws of dynamics (inversion symmetry) indicate that the dynamic "changes" are reversible. The explanation of the state of thermodynamic probability further revealed the contradictory that the description about the direction of changes and time in thermodynamics and dynamics are different. Since the explanation of thermodynamic probability cannot be applied to describe the direction of microscopic state change, this means that there will be no irreversibility in dynamics in the sense of the second law. This leads to the conflict between the two theoretical systems on the view of nature, in physics, we have been unable to establish the relations on the concepts of the direction of changes and time between dynamics and thermodynamics, the former describes to us a world without evolution, and the latter expresses the evolution as the unfolding of the second law.
Physicists usually regard Boltzmann's statistical theory as a scheme to establish the connection from the dynamics of the basic process to the thermodynamics of the statistical system, but this scheme has not been able to give complete results, and its scope of application has always been ambiguous. The description given by statistical probability shows the view that there is no microscopic explanation of entropy and irreversibility at the basic dynamics level. In this sense, Boltzmann approach leads to a complete separation between basic dynamic process and statistical collective behavior in the description on the direction of change and time, this doesn't seem like a real image of the world around us, for example, we can feel the existence of one-way direction of change from the beep of a radioactive decay counter, but this is not the unidirectionality expressed by Boltzmann's statistical theory, we also can hardly understand how to deduce the irreversible and non-deterministic theory from the reversible and deterministic dynamics.
From the 1850s to the present, in the following 150 years, the contradictory issues concerning the physical meaning of entropy and the different descriptions in the direction of change and time between dynamics and thermodynamics have become one of the core problems in the views of existence and evolution. Since the unidirectionality revealed by the second law of thermodynamics involves the direction of change and the meaning of time, this issue is related to our basic views on existence and evolution of nature. Many scientists believe that the second law is an approximate result, however, though a series of appeasement explanations have been given, the initial problems have never changed in more than 150 years, we still have to repeat the problem again and again that has aroused strong feelings over the past 150 years:
What is entropy?
What are the irreversible changes taking place?
[1] Not the original text.
https://www.researchgate.net/publication/306909219_What_is_Heat_Energy
https://www.researchgate.net/publication/331457885_Heat_Energy_Entropy_and_The_Second_Law_A_New_Perspective_in_Thermodynamics
Entropy (which can only grow) is a very bulky mathematical expression to formulate the second law of Thermodynamics. It is a nightmare for most students. It has some advantages (especially in the pre-computer time) because it was used in graphic analysis methods.
The more descriptive way of expressing the second law is offered by the notation exergy (which can only drop into "anergy" while energy, the sum of both, remains conserved). To your last question: Any irreversibility causes an exergy conversion into anergy. There is much litterature on Exergy.
Best regards
Instead of a direct reply a link to a recent paper of mine on the subject:
download here:
Article Entropy and the Second Law of Thermodynamics-The Nonequilibr...
Stay well, everybody!
Henning
thank you, this is a very interesting document for which I need more time to digest. So far I fully agree with your five observations and the derived second law expressed for entropy. Nevertheless I think that todays engineering students need to be familiar with the notations exergy and anergy and the corresponding formulation of the second law. It is simply more instructive especially if communicating with less educated public and politicians. As a resulting conclusion electric power or potential energy are 100% exergy and the society consumes finally exergy and not energy. Energy statistics identifying energy savings lie to a large extent in this respect.
With best regards
Hans
Hi Hans
Thanks for your comments.
Personally, I am not a big fan of exergy, e.g., in my book I mention it only in passing. However, I discuss work losses to irreversible processes, as well as reversible work all the time, just from the 2nd law. Just a bit less formal. Exergy analysis is too concerned with defining the state of the environment, I'd rather look at processes. Different philosophies, I guess, but leads to the same conclusions. Have a look at my book for examples (if you can get hold of it).
Cheers, Henning
Hi Henning
thanks for your fast answer. You are right, exergy is concerned with environment condition. But in the economically dominating power cycles, gas turbines and steam turbines (with which I am concerned) there are standards defining the reference environment. This is needed also for other reasons. Indeed the resulting conclusions must be the same - if correct.
I appreciate to have a look a your Springer book. May take a few days. Besides exergy I am also reviewing, how the thermodynamic community teaches on polytropic efficiency definitions. Your Book may give me a nail.
Cheers
Hans
Within the kinetic theory of gases, in which only changes in the arrangement of particles are described, Boltzmann had to fail in his claim to universally justify the arrow of time by means of his concept of increasing entropy. Nevertheless, Entropy is the expression for a fundamental law of nature - irreversibility, the difference between past and future - the only expression by the way that physics has found for it. In my opinion, entropy tells us that matter is in permanent transformation in one direction, including elementary particles that emerge (not from nothing), age and vanish (not into nothing).
Hi Grit
exactly that is the approach that I took in my paper and book, where the 2nd law is derived to describe the trend to equilibration, i.e., irreversibility of processes. No need for engines as in the classical Carnot-Clausius approach, no need for statistical mechanics. The statements on engines then are an outcome.
It's a phenomenological approach: We observe processes, and need equations (laws) to make predictions. The 1st law is not enough, we need another equation (law) to describe direction of (equilibration) processes. This must be constructed based on the observation of few phenomena, and then is used to evaluate all processes.
Statistical mechanics yields a deeper understanding of how that approach to equilibrium is linked to microscopic processes, essentially the system exploring the phase space. Boltzmann's kinetic theory does this nicely for ideal gases, and gives an excellent playground for better understanding, e.g. of "local thermodynamic equilibrium", fluctuations, etc.
Best, Henning
Very interesting discussion. But how was the current non-equilibrium state (low entropy) formed. Was it the big bang? Is our understanding of entropy limited to an (tiny?) intermediate phase between big bang and heat death?
cheers, Hans
That's a good question. Clearly it cannot be answered from thermodynamics since the 2nd law knows only the way forward.
:-)
Dear Henning,
Below is an experiment.
Put a big water droplet on a glass plate, then sprinkle some small water droplets near by the big water droplet, then covers the glass plate with a glass shade (sealing airtight), waiting for a few days (maintain an appropriate temperature). What will happen? the small water droplets will migrate into the big droplet and disappear, at same time, the big water droplet becomes bigger, in equilibrium state, there is only one bigger water droplet.
From physical chemistry, we know that the vapor pressure of a small droplet is inversely proportional to its curvature radius, the smaller curvature radius, the bigger vapor pressure, that is why the small water droplets will migrate into the big droplet then disappear, it is also the mechanism of raindrop or dewdrop formation.
This is a spontaneous process different from diffusion, and can be easily explained by thermodynamics, dS>0 or dG
The statistical foundation of entropy may be a mystery.
But thermodynamic entropy is a concept as clearly defined as internal energy. While heat is not a state function, dQrev/T is a total differential, and so no inequalities are needed in the definition of thermodynamic entropy. It is the state variable S, the differential of which is given by dS= dQrev/T. That defines the entropy function up to an additive constant.
But this is true for energies as well. They are also only defined up to an additive constant. Thermodynamic energy is also given only via its differential. The mechanical part of energy is defined by the ability to do mechanical work. For the work function, we again only have a differential, because there is no knowledge about the absolute mechanical work that can be done by a system but only about the work that can be done by a system starting from a given state. So that gives us energy differences only, not the absolute value of any potential that might be involved. By the way, work is not a state function either. However, a total differential may be derived from it via an integrating factor just as in the case of heat. For heat, the resulting thermodynamic function is entropy, for mechanical work, it is volume.
We have dW=-p dV, which is not a total differential. But dV = - dW/p is. The volume is a state function, and its relationship to work is similar to that of entropy to heat.
Dear Hans and Henning,
you write about the current low entropy, big bang, heat death and the way only forward. Perhaps we should also recognize the historical conditionality of these ideas of physics. The big bang theory is a hypothesis and remains a speculation as long as essential assumptions like dark energy or the constancy of the Hubble constant are in doubt. From my point of view there is no one-way street towards heat death. The daily experience of irreversibility and the knowledge of the time arrow can be combined with a universe without beginning and end of time. (please see Matter-energy equivalence, A solution of time paradox of physics). As philosopher already wrote: Time has no beginning and no end; all beginning and all end are in it. This means entropy is the expression we have found for showing for all time the temporal limitation of all states and things.
Dear Kassner,
Now we know that Clausius inequality
dS=deS+diS=δQrev/T+ diS >=δQrev/T.
In mathematics, the state function changes must be independent of the path taken without any exception, δQrev/T is the reversible path dependent, path dependent means that it is not a total differential, so it is impossible to prove that dS is an exact differential by the aid of δQrev/T and the integrating factor T.
Consider the following example
dS=dA+dB.
where S is a state function, if fixed B, then dS=dA is an exact differential in the path of fixed B, but here, dA is a partial differential of dS.
i.e. if we don't consider irreversible path, δQrev/T is an exact differential of dS in reversible path, but δQrev/T is not the total differential of the entropy S. That is why we have Clausius inequality
dS >=δQrev/T.
Dear Tang Suye,
Let me first say, very interesting discussion, thanks a lot!
Regarding your earlier comment, how do you say "the equilibrium state is not the most probable distribution"?. Could you please elaborate on this?
Tang Suye
Sorry, but what you write as the Clausius inequality is wrong.
In the Clausius argument, Entropy is defined, or identified, through dS= δQrev/T , which is path independent--any reversible process connecting initial and end state gives the same entropy difference. The heat δQrev is the heat exchange for such a process, not the actual heat exchanged.
The Clausius inequality then reads, in your notation,
dS = δQ/T+ diS
where δQ is the actual heat exchanged for the process, T is the actual (boundary) temperature of the system, and diS >= 0 is the entropy generation rate.
Hence, in short, Clausius inequality is dS >= δQ/T.
Dear Anirudh Rana,
It is a two-phase system, gas-liquid coexistence, the water vapour is the gas phase, the water droplet is the liquid phase. In equilibrium state, the two-phase coexistence, the molecular distribution in the volumes of the two phases are non equi-probable, so it is not a most probable distribution.
According to the postulate of equal a priori probability, in the state of the most probable distribution, the molecular distribution in the two locals should be approximately proportional to the volume of the two locals, it can only be a gas, only be one phase.
Dear Henning,
For dS= δQrev/T, the subscripts ‘’rev’’ emphasizes that is the reversible path dependent, the reversible path dependent is also one form of the paths dependent.
As you said, ''δQ is the actual heat exchanged for the process, T is the actual (boundary) temperature of the system.''
Since the irreversible heat transfer depends on a temperature gradient ΔT, and the entropy production is equal to Δ(1/T) δQ, we can see that δQ/T denotes a heat transfer δQ at the temperature T, which does not involve a temperature gradient, and does not involve the entropy production, so when δQ through the boundary of the system at the temperature T, the heat transfer process that δQ/T involved is always reversible whether the total process is reversible or irreversible. if one want to emphasize ‘’reversible’’, one can write δQrev/T, if one doesn't write the subscripts ‘’rev’’, that is also no problem.
Is there any difference between δQrev/T and δQ/T? They both denote the same thing, a reversible heat transfer δQ at temperature T.
The inequality dS >=δQrev/T or dS >=δQ/T both are correct.
Why dS=δQrev/T cannot be proven to be a total differential in mathematics?
The real issue is that δQ itself is a partial differential about heat (energy), except for δQ, there is another partial differential about heat (energy), the heat production.
In thermodynamics, energy transfer and energy conversion are the two paths of the state changes in the internal energy, except for the heat transfer δQ, the heat production is another path, which comes from energy conversion, such as the heat effect of mechanics (friction produce heat, etc.), the heat of chemical reaction or the heat of phase change, etc.
Only the sum of the heat transfer and the heat production can define a total differential about heat (energy).
Is my modified interpretation of your last sentence correct?
" Only the sum of the heat import and the internal heat production (by a chemical reaction or by a phase change?) can define together a total differential about heat (energy)". What about import/export of work?
cheers
Dear Hans,
δQ and δW are the two different types of the energies transfer, δQ/T is the entropy of δQ, δW makes no contribution to entropy, so δW/T makes no sense, which cannot be merged with δQ/T, I call δW as the free energy transfer.
Converts heat into work or converts work into heat are the processes of the heat production (negative or positive), it is also the processes of the free energy production (positive or negative).
δQ and δW denote the energy transfer, the heat production comes from the energy conversion.
Combined heat transfer and heat conversion, the entropy can be perfectly proven to be a state function in mathematics without the reversible path dependent.
https://www.researchgate.net/publication/306909219_What_is_Heat_Energy
https://www.researchgate.net/publication/331457885_Heat_Energy_Entropy_and_The_Second_Law_A_New_Perspective_in_Thermodynamics
Perhaps the attached paper may bring some elucidation
Discussion is welcome
W.M.
Is entropy increasing in the universe? What does the temperature of the Microwave background mean? The apparent equality of the potential energy and kinetic energy in the universe seems to imply that although entropy increases with each process, there must be an energy input.
To change the subject a little, a two-dimensional cloud of point vortices is a Hamiltonian system with the property that the total variance of the vorticity distribution about any point is constant. It remains constant even if we have chaotic behaviour in the motion of individual vortices. However, if we add some Brownian motion in a computer simulation then we can represent viscosity, and the expected value of of the total variance increases linearly with time. If the simulation has a time-reversal button, then the variance goes on increasing in reversed time with a new sequence of random displacements.
This is suggestive rather than conclusive. The variance is a stand-in for entropy. It is constant in a pure Hamiltonian system, but increases in the presence of randomness. We have a source of randomness in quantum mechanics, and it is therefore suggested that quantum randomness, amplified by chaotic dynamics, is the mechanism of entropy increase. For many purposes quantum mechanics can be thought of as classical Brownian motion.
@Alexandre Chorin proposed the model of vortices in Brownian motion, while @Reinhold Furth drew attention to the fact that quantum mechanics and Brownian motion both have an Uncertainty Principle. Vortices in Brownian motion represent something more fundamental than a Monte Carlo method.
Thank you Tang for this deep consideration. As a mechanical engineer coming from mechanics with a simplistic mind I cannot answer your question "what is Entropy".
In my view Clausius has introduced entropy as a convenient state variable which played an important role in the development of thermodynamic graphic methods. With current computer tools these entropy based methods can be bypassed. Thermodynamic reversibility is a more fundamental expression and its quantification is exergy and anergy of a system in relation to its environment. This can replace entropy in probably all thermodynamic considerations and it is much more descriptive. The one-way aspect of system changes expressed with entropy can as well be expressed with exergy, which can only remain or drop.
The remaining open issue for me is how exergy came into the universe. This is probably just another form of your question.
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