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I am ever more impressed with Wolfram's ability to obfuscate and self-aggrandize. The second law is very easy to understand when presented as Boltzmann formula

by lisper 4y ago

I am ever more impressed with Wolfram's ability to obfuscate and self-aggrandize. The second law is very easy to understand when presented as Boltzmann formulated it, and Shannon later generalized it: the entropy of a system is the (log of the) number of states that system could possibly be in given a set of values we can actually measure. In a gas, for example, we can measure its volume, temperature, and pressure, but that still leaves a lot of possible actual underlying configurations of the gas particles. The more possible configurations there are, the greater the entropy. And this explains the second law: given a set of measurement values, the more states there are which produce those values, the more likely we are, all else being equal, to observe those values. And so a closed system will evolve by pure chance into a state where the entropy is maximized simply because you are more likely to observe values corresponding to greater numbers of underlying states. That's really all there is to it on a conceptual level.

kurthr 4y ago
Well you're giving the statistical mechanics explanation (or information theory), which some thermodynamicists don't appreciate so much, but... I totally agree. Even in person we would joke that his ego filled an entire room, and that's before he got famous. Thank you for reading it so I didn't have to.
- lisper 4y ago
  > some thermodynamicists don't appreciate so much Why?
  - krastanov 4y ago
    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)
SnowHill9902 4y ago
That’s not the fundamental formulation which does not require statistical mechanics. Anybody saying entropy is very easy to understand doesn’t understand it.
- lisper 4y ago
  > That’s not the fundamental formulation Why not? By whose standard is a different formulation more "fundamental"?
  - krastanov 4y ago
    It is more about the fact that there are two "equally fundamental" explanations, and saying there is only one is a great disservice to the curious mind: the most fascinating and deep insights about entropy come from understanding how the two definitions are equivalent.
IIAOPSW 4y ago
To repeat even more tersely, the fraction of information you have about a system can only go spontaneously down, never up. Knowing less is free, learning is at cost.
- lisper 4y ago
  That is brilliant. Is that your idea, or did you get it from somewhere?
  - IIAOPSW 4y ago
    Mine.
    - lisper 4y ago
      Cool. I would like to use this idea and credit you but you have no identity info in your profile. Shall I just credit this to your handle, IIAOPSW?
      - IIAOPSW 4y ago
        Do me a favor, credit it to Euler.
  - krastanov 4y ago
    It is the notion behind "Completely Positive Trace Preserving" maps. It is used a lot in quantum information and quantum computing. It is present by other names in other situations.
ordu 4y ago
> Today I would have more strongly made the rather Feynmanesque point that if you have a theory that says everything we observe today is an exception to your theory, then the theory you have isn’t terribly useful. I've found this quote amusing and I believe it explains to some extent problems with the Second Law. > The second law is very easy to understand when presented as Boltzmann formulated it When I read prominent scientists I often become impressed by their creative ability to not understand "simple" ideas and to spend years in attempts to understand them. It seems for me that this ability and willingness to think is the main driver of a scientific progress. Einstein for example rejected a very simple idea of time, and reinvented it to create the theory of relativity. How on Earth someone with a brain can not understand time? But scientists can and they find this useful. I think they have special secret training on incomprehension. To apply your mind to a problem you need first to find a problem to attack, and this is the most tricky part of science. If you do not see problems with existing theories you are out of luck, you have nothing to attack. So your first reaction to any idea presented to you better be "I do not understand" then "it is very easy and even obvious".
- lisper 4y ago
  > problems with the Second Law Like what?
- hnaccount_rng 4y ago
  > Einstein for example rejected a very simple idea of time, and reinvented it to create the theory of relativity. But that _really_ isn't what he did! He started from the seemingly obvious notion of linear time. Added the _observation_ of the finite speed of light. And _noticed_ that the "obvious" notion of linear time leads to contradictions (basically saying "at the same time" has different meanings for different observers). And from that he _had_ to go out and construct relativity in order to reconcile these contradictions. Does it matter to anyone? In practice? Probably not. As long as you do not get close to the speed of light (either in velocity or in mass. Yes.. that statement makes sense as is) relativity really is indistinguishable from our notion of linear time. This is _necessary_ for relativity to be a useful theory. But it is _also_ a massive shift in how we think about the world. That's the danger with "intuitive" understandings. They can only ever be based on experiences and those do not lend themselves to extrapolations. The 2nd law of thermodynamics is one of those things. It seems completely obvious if learn about it the first time. Then you go a bit deeper, start picking it a part, thinking about the second and third order consequences of it. Then you argue a bit about whether the probabilistic formulations are worth anything in the first place, then you try to reconcile it with information theory. At this point you get into the whole quantum mess and ask yourself how this "law" can hold in all these circumstance, even if it doesn't seem to have any reason to mean anything anymore.
kgwgk 4y ago
The question of what are the states to count and how is not obvious. “It cannot be emphasized strongly enough that W is not the measure of the set C of all states […] compatible with the external macroscopic constraints; rather, it is the dimension of the subset of those states that can be realized in the greatest number of ways.” [Entropy and the Time Evolution of Macroscopic Systems, Walter T. Grandy Jr.]
- deleted 4y ago
  [deleted]
- lisper 4y ago
  That's interesting, I didn't know that. I always thought that W in k ln(W) was the number of possible states. Thanks for bringing this to my attention! I wonder, though, does this actually matter to anyone but a physicist? Consider the common just-so story about electromagnetic waves: a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. Therefore, once you start to wiggle an electron around, the changing field become self-sustaining and you end up with a wave. This story is wrong because it would lead you to conclude that the E and B fields are 90 degrees out of phase when in fact they are in phase, and understanding why that is requires going into quite a bit more detail. But does that actually matter to anyone other than a physicist or an antenna engineer?
  - kgwgk 4y ago
    > I always thought that W in k ln(W) was the number of possible states. That formula - Boltzmann's entropy - is only applicable when all the microstates are equiprobable. The question of what does it mean in general is subtle. If we have a gas in a container in thermal equilibrium with a heatbath, for example, the energy of the microstates compatible with those constraints is not fixed and their probability depends on the energy. The energy and the pressure fluctuate. > I wonder, though, does this actually matter to anyone but a physicist? The people most concerned with the question of entropy on a conceptual level tend to be physicists - and philosophers. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43415/11098_2004_Article_207174.pdf https://deepblue.lib.umich.edu/bitstream/handle/2027.42/4341...
effie 4y ago
> The more possible configurations there are, the greater the entropy. And this explains the second law: given a set of measurement values, the more states there are which produce those values, the more likely we are, all else being equal, to observe those values. And so a closed system will evolve by pure chance into a state where the entropy is maximized simply because you are more likely to observe values corresponding to greater numbers of underlying states. This is an argument for observing high entropy macrostates at any time, assuming all microstates are equally probable. But not for evolution in time that makes entropy higher in the future, or at least non-smaller in the future. You assume the closed system evolves its microstate in time "by pure chance", i.e. any microstate is equally probable as the next microstate, which ignores many constraints there are due to physics laws like conservation of phase volume or energy. This probabilistic argument could be applied to past just as well as the future. But observations show entropy of an isolated system is lower in the past than in the present. So your argument does not really derive all aspects of 2nd law.
- lisper 4y ago
  > But not for evolution in time that makes entropy higher in the future, or at least non-smaller in the future. Well, obviously the system is subject to the constraints of its dynamics. It's not just picking a new microstate at random at each instant in time. It can only evolve to states that are accessible from its current state according to its dynamics. But given that set of accessible microstates, you are more likely to get a macrostate corresponding to a larger number of microstates than a smaller number. The reason entropy was lower in the past is that the universe started out in a low-entropy state. The reason for that is still a mystery.
  - effie 4y ago
    > But given that set of accessible microstates, you are more likely to get a macrostate corresponding to a larger number of microstates than a smaller number. This is due to mixing in phase space, which is plausible for many particle systems. However, it works in both directions of the parameter t. The only way to derive 2nd law including its time asymmetry (which is the major point of 2nd law) is to restrict considerations to those microscopic processes that are compatible with macroscopic processes that systematically lead to the same final state. This means we consider only subset of all microscopically possible processes. E.g. Jaynes selects those macroscopic processes that bring one macrostate A, via some irreversible evolution, to another macrostate B, repeatedly. Then, assuming this reliability of the resulting state, he derives non-decrease of entropy from maximum entropy principle and Hamiltonian statistical physics. If we didn't assume this restriction and allowed all mechanically possible processes, entropy could both increase and decrease. > The reason entropy was lower in the past is that the universe started out in a low-entropy state. The reason for that is still a mystery. This often repeated idea is in vein with Clausius' ostentatious statements on energy and entropy of the Universe, which have been out of place and plagued discussions of thermodynamics with needless mystical assumptions ever since he wrote them. There is no evidence for them, as there are no reliable measurements of entropy of the Universe. There is no need for them, as there are no good reasons to believe Universe has defined entropy at all. Thermodynamics, including 2nd law, is about finite sized systems free of strong gravity effects. It does not extrapolate easily beyond compact systems like planets/stars to ever bigger systems. There is no Universe in it anywhere. 2nd law that we see in daily life or a lab is not a statement about entropy of Universe being lower in the past, nor does it hang on such an assumption. It is (in one of its variants) the statement that a finite isolated system can change its macrostate to some other final macrostate, but this final macrostate is of higher or same entropy, so in real process entropy can't decrease.

by lisper 4y ago

kurthr 4y ago

Well you're giving the statistical mechanics explanation (or information theory), which some thermodynamicists don't appreciate so much, but... I totally agree. Even in person we would joke that his ego filled an entire room, and that's before he got famous. Thank you for reading it so I didn't have to.

lisper 4y ago
> some thermodynamicists don't appreciate so much Why?
- krastanov 4y ago
  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)

SnowHill9902 4y ago

That’s not the fundamental formulation which does not require statistical mechanics. Anybody saying entropy is very easy to understand doesn’t understand it.

lisper 4y ago
> That’s not the fundamental formulation Why not? By whose standard is a different formulation more "fundamental"?
- krastanov 4y ago
  It is more about the fact that there are two "equally fundamental" explanations, and saying there is only one is a great disservice to the curious mind: the most fascinating and deep insights about entropy come from understanding how the two definitions are equivalent.

IIAOPSW 4y ago

To repeat even more tersely, the fraction of information you have about a system can only go spontaneously down, never up. Knowing less is free, learning is at cost.

lisper 4y ago
That is brilliant. Is that your idea, or did you get it from somewhere?
- IIAOPSW 4y ago
  Mine.
  - lisper 4y ago
    Cool. I would like to use this idea and credit you but you have no identity info in your profile. Shall I just credit this to your handle, IIAOPSW?
    - IIAOPSW 4y ago
      Do me a favor, credit it to Euler.
- krastanov 4y ago
  It is the notion behind "Completely Positive Trace Preserving" maps. It is used a lot in quantum information and quantum computing. It is present by other names in other situations.

ordu 4y ago

> Today I would have more strongly made the rather Feynmanesque point that if you have a theory that says everything we observe today is an exception to your theory, then the theory you have isn’t terribly useful. I've found this quote amusing and I believe it explains to some extent problems with the Second Law. > The second law is very easy to understand when presented as Boltzmann formulated it When I read prominent scientists I often become impressed by their creative ability to not understand "simple" ideas and to spend years in attempts to understand them. It seems for me that this ability and willingness to think is the main driver of a scientific progress. Einstein for example rejected a very simple idea of time, and reinvented it to create the theory of relativity. How on Earth someone with a brain can not understand time? But scientists can and they find this useful. I think they have special secret training on incomprehension. To apply your mind to a problem you need first to find a problem to attack, and this is the most tricky part of science. If you do not see problems with existing theories you are out of luck, you have nothing to attack. So your first reaction to any idea presented to you better be "I do not understand" then "it is very easy and even obvious".

lisper 4y ago
> problems with the Second Law Like what?
hnaccount_rng 4y ago
> Einstein for example rejected a very simple idea of time, and reinvented it to create the theory of relativity. But that _really_ isn't what he did! He started from the seemingly obvious notion of linear time. Added the _observation_ of the finite speed of light. And _noticed_ that the "obvious" notion of linear time leads to contradictions (basically saying "at the same time" has different meanings for different observers). And from that he _had_ to go out and construct relativity in order to reconcile these contradictions. Does it matter to anyone? In practice? Probably not. As long as you do not get close to the speed of light (either in velocity or in mass. Yes.. that statement makes sense as is) relativity really is indistinguishable from our notion of linear time. This is _necessary_ for relativity to be a useful theory. But it is _also_ a massive shift in how we think about the world. That's the danger with "intuitive" understandings. They can only ever be based on experiences and those do not lend themselves to extrapolations. The 2nd law of thermodynamics is one of those things. It seems completely obvious if learn about it the first time. Then you go a bit deeper, start picking it a part, thinking about the second and third order consequences of it. Then you argue a bit about whether the probabilistic formulations are worth anything in the first place, then you try to reconcile it with information theory. At this point you get into the whole quantum mess and ask yourself how this "law" can hold in all these circumstance, even if it doesn't seem to have any reason to mean anything anymore.

kgwgk 4y ago

The question of what are the states to count and how is not obvious. “It cannot be emphasized strongly enough that W is not the measure of the set C of all states […] compatible with the external macroscopic constraints; rather, it is the dimension of the subset of those states that can be realized in the greatest number of ways.” [Entropy and the Time Evolution of Macroscopic Systems, Walter T. Grandy Jr.]

deleted 4y ago
[deleted]
lisper 4y ago
That's interesting, I didn't know that. I always thought that W in k ln(W) was the number of possible states. Thanks for bringing this to my attention! I wonder, though, does this actually matter to anyone but a physicist? Consider the common just-so story about electromagnetic waves: a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. Therefore, once you start to wiggle an electron around, the changing field become self-sustaining and you end up with a wave. This story is wrong because it would lead you to conclude that the E and B fields are 90 degrees out of phase when in fact they are in phase, and understanding why that is requires going into quite a bit more detail. But does that actually matter to anyone other than a physicist or an antenna engineer?
- kgwgk 4y ago
  > I always thought that W in k ln(W) was the number of possible states. That formula - Boltzmann's entropy - is only applicable when all the microstates are equiprobable. The question of what does it mean in general is subtle. If we have a gas in a container in thermal equilibrium with a heatbath, for example, the energy of the microstates compatible with those constraints is not fixed and their probability depends on the energy. The energy and the pressure fluctuate. > I wonder, though, does this actually matter to anyone but a physicist? The people most concerned with the question of entropy on a conceptual level tend to be physicists - and philosophers. https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43415/11098_2004_Article_207174.pdf https://deepblue.lib.umich.edu/bitstream/handle/2027.42/4341...

effie 4y ago

> The more possible configurations there are, the greater the entropy. And this explains the second law: given a set of measurement values, the more states there are which produce those values, the more likely we are, all else being equal, to observe those values. And so a closed system will evolve by pure chance into a state where the entropy is maximized simply because you are more likely to observe values corresponding to greater numbers of underlying states. This is an argument for observing high entropy macrostates at any time, assuming all microstates are equally probable. But not for evolution in time that makes entropy higher in the future, or at least non-smaller in the future. You assume the closed system evolves its microstate in time "by pure chance", i.e. any microstate is equally probable as the next microstate, which ignores many constraints there are due to physics laws like conservation of phase volume or energy. This probabilistic argument could be applied to past just as well as the future. But observations show entropy of an isolated system is lower in the past than in the present. So your argument does not really derive all aspects of 2nd law.

lisper 4y ago
> But not for evolution in time that makes entropy higher in the future, or at least non-smaller in the future. Well, obviously the system is subject to the constraints of its dynamics. It's not just picking a new microstate at random at each instant in time. It can only evolve to states that are accessible from its current state according to its dynamics. But given that set of accessible microstates, you are more likely to get a macrostate corresponding to a larger number of microstates than a smaller number. The reason entropy was lower in the past is that the universe started out in a low-entropy state. The reason for that is still a mystery.
- effie 4y ago
  > But given that set of accessible microstates, you are more likely to get a macrostate corresponding to a larger number of microstates than a smaller number. This is due to mixing in phase space, which is plausible for many particle systems. However, it works in both directions of the parameter t. The only way to derive 2nd law including its time asymmetry (which is the major point of 2nd law) is to restrict considerations to those microscopic processes that are compatible with macroscopic processes that systematically lead to the same final state. This means we consider only subset of all microscopically possible processes. E.g. Jaynes selects those macroscopic processes that bring one macrostate A, via some irreversible evolution, to another macrostate B, repeatedly. Then, assuming this reliability of the resulting state, he derives non-decrease of entropy from maximum entropy principle and Hamiltonian statistical physics. If we didn't assume this restriction and allowed all mechanically possible processes, entropy could both increase and decrease. > The reason entropy was lower in the past is that the universe started out in a low-entropy state. The reason for that is still a mystery. This often repeated idea is in vein with Clausius' ostentatious statements on energy and entropy of the Universe, which have been out of place and plagued discussions of thermodynamics with needless mystical assumptions ever since he wrote them. There is no evidence for them, as there are no reliable measurements of entropy of the Universe. There is no need for them, as there are no good reasons to believe Universe has defined entropy at all. Thermodynamics, including 2nd law, is about finite sized systems free of strong gravity effects. It does not extrapolate easily beyond compact systems like planets/stars to ever bigger systems. There is no Universe in it anywhere. 2nd law that we see in daily life or a lab is not a statement about entropy of Universe being lower in the past, nor does it hang on such an assumption. It is (in one of its variants) the statement that a finite isolated system can change its macrostate to some other final macrostate, but this final macrostate is of higher or same entropy, so in real process entropy can't decrease.