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You do not need statistical physics to define entropy. In fact, entropy as a concept was introduced long before statistical physics was a thing. Being able to d

You do not need statistical physics to define entropy. In fact, entropy as a concept was introduced long before statistical physics was a thing. Being able to do thermodynamics without invoking anything related to statistical physics is very valuable tool in mathematical physics, as it leads to definitions and proofs that rely on way fewer assumptions. People should not claim they understand entropy if they understand only the statistical physics way of introducing it. The deep value of the concept of entropy comes from understanding how the two completely unrelated definitions (in thermodynamics and in statistical physics) are equivalent. Edit to add: In statistical physics you rely on the idea of equivalent microstates. In thermodynamics you do not introduce anything about the microscopic structure of the system under study (an extremely valuable way to generalize your results to cases not covered by typical stat physics assumptions), rather you introduce the zeroth law of thermodynamics as an axiom.

lisper 4y ago
> entropy as a concept was introduced long before statistical physics was a thing That's true, but the historical development doesn't always make for the best pedagogy. Quantum mechanics, to take my favorite example, is vastly simpler than it is commonly made out to be [1]. It just took about 80 years for this to be understood. The development of thermodynamics followed a similar path because thermodynamics predated the widespread acceptance of the atomic theory. Indeed, statistical mechanics played a major role in getting the atomic theory accepted. But it does not follow that the thermodynamic treatment is superior in any way. I challenge you to explain to me what entropy actually is in fewer words than I used without reference to atoms or microstates. I'll bet you can't even explain what "temperature" is under those constraints, and that is much simpler. --- [1] See https://en.wikipedia.org/wiki/Quantum_Computing_Since_Democritus https://en.wikipedia.org/wiki/Quantum_Computing_Since_Democr... chapter 9
- krastanov 4y ago
  You are right about pedagogy. But "understanding" requires much more than giving one simple definition without discussing the consequences of the definition. Your (statistical) definition of entropy is perfectly reasonable and fairly intuitive and it should be taught first (as you alluded twice and as it is relatively common these days). But "understanding" entropy involves understanding multiple facets of it and why it is useful, not just knowing the easier definitions. Hence, it seem unreasonable to claim "understanding" entropy without having familiarity with the independent thermodynamic definition of entropy and *why are they equivalent*. That would be like saying one understands programming language theory while being familiar only with object oriented programming. Using atoms and microstates to explain entropy makes the explanation shorter but it also makes it *more* complex because it now depends on the definition of *atoms and microstates*. The thermodynamics explanation of entropy does not actually require more words, but crucially, it requires fewer concepts (heat, useful work, and heat transfer). Concerning the development of thermodynamics: defining entropy *without* statistical physics requires *fewer* assumptions about the physical system. Statistical physics did not make thermodynamics simpler, nor is it necessary to explain thermodynamics. To your example about quantum mechanics: this is different because statistical physics is not some better more-fundamental explanation of thermodynamics. The two are separate sets of axioms with equivalent explanatory power. Concerning the definition of temperature: are you aware that there is an implicit assumption about temperature scales when defined in the statistical physics way? That is much more explicit in the thermodynamic treatment. And it is why I brought up the zeroth law of thermodynamics: it is where temperature is defined. If you think temperature is easier to define in the statistical physics setting, you are simply hiding a ton of assumptions under the rug, which is not a very good way to build a mathematical theory. I am curious what do you claim is a simple statistical explanation of temperature? If it is something about average energy, that is a *wrong* explanation of temperature, valid in only a few special circumstances. As to actually giving you the thermodynamic definition of temperature: The joke is that it is whatever the thermometer says it is. The rigorous version of that joke is: According to the zeroth law of thermodynamics, if bodies A and B are in thermal equilibrium (no heat transfer) and bodies B and C are in thermal equilibrium, then A and C are also in thermal equilibrium. Use this equivalence property to tag all bodies that belong to the same equivalence class. This tag is your measure of temperature. It is an arbitrary (ordered) scale. If you want to, you can then use the second law of thermodynamics to pick a neater canonical scale. Finally, the definition of entropy: enumerate the (macroscopic) variables describing your system and draw them as a coordinate system; compute the value of ΔQ/T as you travel through some path in that coordinate system; the second law of thermodynamics says that value is zero on a closed path, thus we have a potential function; name that potential function entropy. Mathematically, in many aspects that is both a simpler and a more powerful definition. One could hardly claim to understand entropy if they do not understand the enormity of the previous paragraph and its relationship to the independent statistical definition. TLDR: "Understanding" entropy is about understanding why these vastly different definitions (thermodynamic and statistical) are equivalent, not about being able to recite one or the other. Especially given that statistical physics is not more fundamental than thermodynamics (unlike special relativity which is indeed more fundamental than Newtonian gravity)
  - lisper 4y ago
    > But "understanding" requires much more than giving one simple definition without discussing the consequences of the definition. But I did discuss the consequences: my definition (well, Boltzmann and Shannon's) leads directly to the second law, and in a way that is much more general and intuitive than the thermodynamic approach. What, for example, is the entropy of a cryptographic key? On the thermodynamic approach that question doesn't even make sense to ask. It's a category error because the thermodynamic definition relies on the concept of temperature and cryptographic keys don't have temperatures. > Using atoms and microstates to explain entropy makes the explanation shorter but it also makes it more complex because it now depends on the definition of atoms and microstates. Complexity is in the eye of the beholder. The classical billiard-ball model of atoms is perfectly adequate to explain the second law -- both the thermodynamic and information theoretic version -- and most people have no trouble wrapping their brains around the idea of "lots of tiny billiard balls wiggling around". > I am curious what do you claim is a simple statistical explanation of temperature? Higher/lower temperatures mean that the billiard balls are, on average, moving faster/slower relative to each other, all else being equal. And yes, I know that is not entirely accurate, but neither is the idea that atoms are billiard balls in the first place. This model is perfectly adequate as a first-order approximation. It allows you to describe the qualitative features of a heat engine purely in terms of the Newtonian mechanics of colliding billiard balls, which most people have a pretty good intuition for. No, it's not 100% accurate in the thermodynamic case, but the benefit is that it is obvious how to apply the concept of entropy defined in statistical terms to computation and information. I think that's a worthwhile tradeoff. Knowing how to calculate the efficiency of a heat engine is not very useful to most people in today's world. On the other hand, knowing how to estimate the entropy of a password or cryptographic key is very handy. > The joke is that it is whatever the thermometer says it is. Yes. But that joke conceals a deep truth: actually describing (let alone defining) thermodynamic temperature without reference to atoms, and without resorting to the thermometer joke, is very, very hard. (It is analogous to trying to explain quantum mechanics without reference to entanglement, which is how it was done for decades. The problem with that approach is that it leaves you waving your hands about what a "measurement" is, because measurements are entanglements. Once you understand that simple fact, all of the mysteries of QM simply evaporate.) > if bodies A and B are in thermal equilibrium (no heat transfer) And how can I tell if A and B are in "thermal equilibrium" without resorting to the joke? (BTW, your definition as you've given it is actually wrong. I can have two systems that are different temperatures but with no heat transfer between them simply by separating them with a perfect insulator. So now you have to incorporate that into your definition somehow, which means you have to define "insulator", and I don't see how you're going to do that without going around in circles.) > It is an arbitrary (ordered) scale. No, it's not. If it were arbitrary, the concept of specific heat would be meaningless. Ultimately you need to make a connection between heat and motion. That is, after all, the thermodynamic project. You can convert heat into motion, and you can convert motion into heat, but the latter is much, much easier than the former. Why? The statistical answer is: because heat is motion. It's a particular kind of motion. But if you deny yourself a referent to the things that are moving in the case of heat (atoms) then you deny yourself access to this simple straightforward explanation of the fundamental question of thermodynamics. > "Understanding" entropy is about understanding why these vastly different definitions (thermodynamic and statistical) are equivalent I agree, but I think it's much easier to start with the statistical definition and show how the thermodynamic one follows (because that is straightforward) than to start with the thermodynamic definition and try to extend it to non-thermodynamical systems. I've never seen that done successfully. There's a reason that the atomic theory is a thing.

lisper 4y ago

> entropy as a concept was introduced long before statistical physics was a thing That's true, but the historical development doesn't always make for the best pedagogy. Quantum mechanics, to take my favorite example, is vastly simpler than it is commonly made out to be [1]. It just took about 80 years for this to be understood. The development of thermodynamics followed a similar path because thermodynamics predated the widespread acceptance of the atomic theory. Indeed, statistical mechanics played a major role in getting the atomic theory accepted. But it does not follow that the thermodynamic treatment is superior in any way. I challenge you to explain to me what entropy actually is in fewer words than I used without reference to atoms or microstates. I'll bet you can't even explain what "temperature" is under those constraints, and that is much simpler. --- [1] See https://en.wikipedia.org/wiki/Quantum_Computing_Since_Democritus https://en.wikipedia.org/wiki/Quantum_Computing_Since_Democr... chapter 9

krastanov 4y ago
You are right about pedagogy. But "understanding" requires much more than giving one simple definition without discussing the consequences of the definition. Your (statistical) definition of entropy is perfectly reasonable and fairly intuitive and it should be taught first (as you alluded twice and as it is relatively common these days). But "understanding" entropy involves understanding multiple facets of it and why it is useful, not just knowing the easier definitions. Hence, it seem unreasonable to claim "understanding" entropy without having familiarity with the independent thermodynamic definition of entropy and *why are they equivalent*. That would be like saying one understands programming language theory while being familiar only with object oriented programming. Using atoms and microstates to explain entropy makes the explanation shorter but it also makes it *more* complex because it now depends on the definition of *atoms and microstates*. The thermodynamics explanation of entropy does not actually require more words, but crucially, it requires fewer concepts (heat, useful work, and heat transfer). Concerning the development of thermodynamics: defining entropy *without* statistical physics requires *fewer* assumptions about the physical system. Statistical physics did not make thermodynamics simpler, nor is it necessary to explain thermodynamics. To your example about quantum mechanics: this is different because statistical physics is not some better more-fundamental explanation of thermodynamics. The two are separate sets of axioms with equivalent explanatory power. Concerning the definition of temperature: are you aware that there is an implicit assumption about temperature scales when defined in the statistical physics way? That is much more explicit in the thermodynamic treatment. And it is why I brought up the zeroth law of thermodynamics: it is where temperature is defined. If you think temperature is easier to define in the statistical physics setting, you are simply hiding a ton of assumptions under the rug, which is not a very good way to build a mathematical theory. I am curious what do you claim is a simple statistical explanation of temperature? If it is something about average energy, that is a *wrong* explanation of temperature, valid in only a few special circumstances. As to actually giving you the thermodynamic definition of temperature: The joke is that it is whatever the thermometer says it is. The rigorous version of that joke is: According to the zeroth law of thermodynamics, if bodies A and B are in thermal equilibrium (no heat transfer) and bodies B and C are in thermal equilibrium, then A and C are also in thermal equilibrium. Use this equivalence property to tag all bodies that belong to the same equivalence class. This tag is your measure of temperature. It is an arbitrary (ordered) scale. If you want to, you can then use the second law of thermodynamics to pick a neater canonical scale. Finally, the definition of entropy: enumerate the (macroscopic) variables describing your system and draw them as a coordinate system; compute the value of ΔQ/T as you travel through some path in that coordinate system; the second law of thermodynamics says that value is zero on a closed path, thus we have a potential function; name that potential function entropy. Mathematically, in many aspects that is both a simpler and a more powerful definition. One could hardly claim to understand entropy if they do not understand the enormity of the previous paragraph and its relationship to the independent statistical definition. TLDR: "Understanding" entropy is about understanding why these vastly different definitions (thermodynamic and statistical) are equivalent, not about being able to recite one or the other. Especially given that statistical physics is not more fundamental than thermodynamics (unlike special relativity which is indeed more fundamental than Newtonian gravity)
- lisper 4y ago
  > But "understanding" requires much more than giving one simple definition without discussing the consequences of the definition. But I did discuss the consequences: my definition (well, Boltzmann and Shannon's) leads directly to the second law, and in a way that is much more general and intuitive than the thermodynamic approach. What, for example, is the entropy of a cryptographic key? On the thermodynamic approach that question doesn't even make sense to ask. It's a category error because the thermodynamic definition relies on the concept of temperature and cryptographic keys don't have temperatures. > Using atoms and microstates to explain entropy makes the explanation shorter but it also makes it more complex because it now depends on the definition of atoms and microstates. Complexity is in the eye of the beholder. The classical billiard-ball model of atoms is perfectly adequate to explain the second law -- both the thermodynamic and information theoretic version -- and most people have no trouble wrapping their brains around the idea of "lots of tiny billiard balls wiggling around". > I am curious what do you claim is a simple statistical explanation of temperature? Higher/lower temperatures mean that the billiard balls are, on average, moving faster/slower relative to each other, all else being equal. And yes, I know that is not entirely accurate, but neither is the idea that atoms are billiard balls in the first place. This model is perfectly adequate as a first-order approximation. It allows you to describe the qualitative features of a heat engine purely in terms of the Newtonian mechanics of colliding billiard balls, which most people have a pretty good intuition for. No, it's not 100% accurate in the thermodynamic case, but the benefit is that it is obvious how to apply the concept of entropy defined in statistical terms to computation and information. I think that's a worthwhile tradeoff. Knowing how to calculate the efficiency of a heat engine is not very useful to most people in today's world. On the other hand, knowing how to estimate the entropy of a password or cryptographic key is very handy. > The joke is that it is whatever the thermometer says it is. Yes. But that joke conceals a deep truth: actually describing (let alone defining) thermodynamic temperature without reference to atoms, and without resorting to the thermometer joke, is very, very hard. (It is analogous to trying to explain quantum mechanics without reference to entanglement, which is how it was done for decades. The problem with that approach is that it leaves you waving your hands about what a "measurement" is, because measurements are entanglements. Once you understand that simple fact, all of the mysteries of QM simply evaporate.) > if bodies A and B are in thermal equilibrium (no heat transfer) And how can I tell if A and B are in "thermal equilibrium" without resorting to the joke? (BTW, your definition as you've given it is actually wrong. I can have two systems that are different temperatures but with no heat transfer between them simply by separating them with a perfect insulator. So now you have to incorporate that into your definition somehow, which means you have to define "insulator", and I don't see how you're going to do that without going around in circles.) > It is an arbitrary (ordered) scale. No, it's not. If it were arbitrary, the concept of specific heat would be meaningless. Ultimately you need to make a connection between heat and motion. That is, after all, the thermodynamic project. You can convert heat into motion, and you can convert motion into heat, but the latter is much, much easier than the former. Why? The statistical answer is: because heat is motion. It's a particular kind of motion. But if you deny yourself a referent to the things that are moving in the case of heat (atoms) then you deny yourself access to this simple straightforward explanation of the fundamental question of thermodynamics. > "Understanding" entropy is about understanding why these vastly different definitions (thermodynamic and statistical) are equivalent I agree, but I think it's much easier to start with the statistical definition and show how the thermodynamic one follows (because that is straightforward) than to start with the thermodynamic definition and try to extend it to non-thermodynamical systems. I've never seen that done successfully. There's a reason that the atomic theory is a thing.