Thermodynamic and the Big Crunch

[Enola Straight]

One of the possible Ends of the Universe involves gravity ultimately overcoming the outward expansion of space/time/matter/energy since the Big Bang and EVERYTHING contracts to a geometric point.

Doesn't that violate "the universe as a whole decays from higher order to entropy"?

[SpottedKitty]

One of the possible Ends of the Universe

Less possible than used to be thought consistent with theory — the universe is still expanding, all right, but observations now suggest the expansion isn't slowing down, it's speeding up. Look up "dark energy", and don't worry too much if you can't get a grip on the concept, I don't think anyone really understands it yet. More data is needed; there are still theories of really long-term Deep Time in which the universe can cycle between Bang and Crunch, but it's looking less likely.

[Terralthra]

Until and unless we can adequately explain why dark energy is causing spacetime expansion, there's no useful way to model the eventual fate of the universe. If the trend continues as it has, the expansion will continue to accelerate and the universe will never collapse into a Big Crunch...but we have no idea if that will happen, since we don't know why the expansion is accelerating.

[Starglider]

Doesn't that violate "the universe as a whole decays from higher order to entropy"?

Yes, and... ?

The 'laws' of thermodynamics are statistical consequences of particle scale physics operating over huge numbers of particles (in fact this can be said of most classical physics). There are edge cases where the fundamental laws give rise to different aggregate behaviour. This doesn't make any of it less valid

[His Divine Shadow]

A big crunch I think sounds more optimistic, that way one could imagine all that matter exploding in a new big bang again, repeating the process over and over again. Universal heat death is much more depressing a fate.

[Simon_Jester]

One of the possible Ends of the Universe involves gravity ultimately overcoming the outward expansion of space/time/matter/energy since the Big Bang and EVERYTHING contracts to a geometric point.

Doesn't that violate "the universe as a whole decays from higher order to entropy"?

Entropy is one of the major drivers of change in physical systems; the other is energy. A system will tend towards increasing entropy as long as this can happen without requiring an expenditure of energy to sustain that increase in entropy.

So when you break something, it tends to stay broken, because the 'broken' state is energetically favored over the 'unbroken' state. If you try to break a system where the 'broken' state has higher energy content, it tends to bounce back to where it was. Low energy chemical compounds do not, as a rule, 'decay' into higher energy ones, for instance.

As a thought experiment of this nature, imagine a room containing rocks, in free fall. Rocks are distributed randomly throughout the room, floating in midair in various locations. This is a high-entropy state, because there are a greater number of possible states of the system than if all the rocks were piled in a corner, or bumping against the ceiling.

Now imagine the same room with gravity. All the rocks are on the floor. This reduces the number of possible states of the system (you can't specify a possible configuration of the rocks where one rock is on the ceiling). And yet it happens; this is the natural 'ground' state of a room full of rocks on Earth. Even though a higher-entropy state can be imagined, such a state cannot exist unless extra energy is pumped into the system, and will not remain stable unless some new force is introduced to counterbalance gravity.
______________

So while entropy is a good concept for describing the degree to which systems are ordered (more order means fewer possible configurations of the system, which in turn means lower entropy)...

...Ultimately they do not counteract the fact that physical forces still cause objects to move.

And the Big Crunch is precisely such a thing- the image of the entire universe being pulled back together into a geometric point by the immense power of gravity caused by all the matter within that universe.

Of course, it's probably moot since most evidence indicates that we don't live in a universe likely to Crunch...

[jwl]

Entropy is conserved so long as there is an increasing range of microstates. All it means is that the big crunch won't look like a time-reversal of the big bang. It's entirely possible to compress gases whilst keeping to the second law as well.

As a thought experiment of this nature, imagine a room containing rocks, in free fall. Rocks are distributed randomly throughout the room, floating in midair in various locations. This is a high-entropy state, because there are a greater number of possible states of the system than if all the rocks were piled in a corner, or bumping against the ceiling.

Now imagine the same room with gravity. All the rocks are on the floor. This reduces the number of possible states of the system (you can't specify a possible configuration of the rocks where one rock is on the ceiling). And yet it happens; this is the natural 'ground' state of a room full of rocks on Earth. Even though a higher-entropy state can be imagined, such a state cannot exist unless extra energy is pumped into the system, and will not remain stable unless some new force is introduced to counterbalance gravity.
______________

So while entropy is a good concept for describing the degree to which systems are ordered (more order means fewer possible configurations of the system, which in turn means lower entropy)...

...Ultimately they do not counteract the fact that physical forces still cause objects to move.

This is still an increase in entropy because the ordered potential energy is converted to disordered heat energy when the rocks hit the floor. There are no known violations of the second law of thermodynamics, unless you count extremely temporary, local, decreases in entropy that you can't extract work from.

Not sure if you meant this anyway though.

[Simon_Jester]

What I meant is that a system that superficially appears to be more ordered can be created out of a disordered system by application of an outside force. Such as gravity.

Also that when a force has been applied to a system literally all along, rather than being introduced suddenly, that will affect the initial entropy of the system. For example, rocks aren't uniformly spaced throughout the Earth's atmosphere because gravity has always been acting on those rocks- they never had a chance to float up to some great height and adopt a higher-entropy configuration.

The Second Law of Thermodynamics does not state that all systems will automatically adopt the highest-entropy state conceivable. It states that they will adopt the highest-entropy state consistent with other physical laws and the outside forces acting on them.

[SpottedKitty]

<nod> Something to keep in mind in any discussion of entropy and Second Law conservation — it only makes sense if it's applied to completely closed systems. If there's any external source of energy, or a way the energy can leak out to somewhere else, then it isn't a completely closed system and it's possible to appear as if the entropy level is doing something it shouldn't.

[Simon_Jester]

Also note that any outside force acting on the system can have this kind of effect, because it creates situations where some configurations are more... energetically favored... than others. So that the system will automatically assemble itself into the configuration of lowest possible energy, radiating away the excess.

Of course, the radiated-away energy then carries with it extra thermodynamic entropy.

[Kuroneko]

In a Big Crunch scenario, the smooth FRW cosmology is an over-idealization. A collapse would be a very inhomogeneous process. For example, lots of black holes should form here and there, and black holes maximize entropy for their size. Therefore, it is inappropriate to think of the Big Crunch singularity as a low-entropy state: not only is there no proper 'state' for singularity with respect to currently-accepted physics, but 'realistic' Big Crunches would approach the singularity while increasing entropy as usual.

Additionally, the description of a singularity as a 'geometric point' is an over-simplification. Geometrically, this kind of singularity is not a point on the spacetime manifold, so 'geometric point' is just a crude analogy (sometimes highly inaccurate, because in many situations the singularity is not anything like a point).

Until and unless we can adequately explain why dark energy is causing spacetime expansion, there's no useful way to model the eventual fate of the universe.

We wouldn't need an explanation of the origin for a model per se. There is a good theoretical reason why any vacuum energy density has to take the form of a cosmological constant: it has only possible stress-energy tensor which is the same in all local inertial frames. Thus any vacuum energy (which is an ordinary feature of QFTs) would have to take that form. So it's expected that the biggest mystery is why vacuum energy is so tiny--it is almost zero.

What I meant is that a system that superficially appears to be more ordered can be created out of a disordered system by application of an outside force. Such as gravity.

Yes, but the notion of 'disorder' is itself quite superficial in discussions of entropy and should be avoided. The simplest go-to example is a monoatomic gas in a cubical box, so let's imagine identically-sized cubical boxes filled with some particular noble gas, one box for each element, at the same temperature.

Naively, one might think that the 'disorder' only depends on the possible arrangement at an instant in time, and so since we have the same amounts distributed uniformly throughout equal volumes, they should have the same entropy. This is mistaken, as entropy goes up with temperature. Alternatively, one might think that the since lighter noble gases are moving faster than their heavier ones at identical temperatures, they are more 'disorderly' and so the lighter gases are more entropic. However, this is also mistaken, as actually the standard entropies of the noble gases increase in the same order as their masses (He<Ne<Ar<Kr<Xe<Rn). The intuitive notions of 'disorder' have led us wrongly, here and in many other situations besides.

One can't say that entropy is disorder without using 'disorder' in some highly peculiar sense quite different from any of its usual meanings, at which point one might as well just use a different word without trying to reduce it to a more common meaning--and the current best word for entropy is 'entropy'. Carrying this baggage of common-sense meanings of 'disorder' that muddles understanding and leads many people to confusion.

Also that when a force has been applied to a system literally all along, rather than being introduced suddenly, that will affect the initial entropy of the system. For example, rocks aren't uniformly spaced throughout the Earth's atmosphere because gravity has always been acting on those rocks- they never had a chance to float up to some great height and adopt a higher-entropy configuration.

I don't think it's appropriate to think of configurations as higher or lower entropy. An important part of it was under-emphasized in your earlier post: entropy is not a function of the particular configuration, but rather of your knowledge. Entropy is a measure of the information you don't have. If you have your room with rocks, with our without gravity, and you look into it and see each of the rocks, then at that point, it makes no difference whether they're on the floor, floating spread about the room, or in a corner. You know how they're arranged and how they're moving, so there is zero entropy.

Suppose you haven't looked into the room yet (or cannot do so), and the only thing you know about it is that the rocks have the minimal energy, which is achieved when the rocks neither move nor rotate. In the case of no gravity, they could be anywhere in the room, but if there's gravity, then the inclusion of gravitational potential energy means that they have to be on the floor. In either case, though, the highest-entropy probability distribution is a uniform one. If you assign any other probability distribution, then you're assuming more information than just the total energy.

On the other hand, suppose that instead of particular definite energy, you know a definite temperature. This is a very different condition, because then the energy can be subject to fluctuations, although its expected value is still defined. In this case, the maximum-entropy distribution is not uniform.

That the Gibbs (thermodynamic) entropy and the Shannon (information) entropy are identical up to dimensional factor in terms of probabilities of microstates is not an accident. The intuitive reason for the second law is that information is conserved but is lost to the environment. For example, even if you knew everything about a macroscopic amount of gas in a box at some instant in time, when it's in thermal contact with its environment, you will not be able to keep precise track of those interactions, and so your knowledge will quickly become less and less. Entropy can be understood as the amount of such 'missing' information. But of course, practically you wouldn't know anything not available to macroscopic measurements in the first place, and so thermodynamics is primarily concerned with macroscopically measurable quantities.

[Simon_Jester]

We wouldn't need an explanation of the origin for a model per se. There is a good theoretical reason why any vacuum energy density has to take the form of a cosmological constant: it has only possible stress-energy tensor which is the same in all local inertial frames. Thus any vacuum energy (which is an ordinary feature of QFTs) would have to take that form. So it's expected that the biggest mystery is why vacuum energy is so tiny--it is almost zero.

To expand upon this, if I am not mistaken-

If there was exactly zero vacuum energy, this would be less provocative a mystery than for there to be 'some but only a tiny amount,' because all the models that predict it should exist at all predict that it should be extremely large, correct?

Yes, but the notion of 'disorder' is itself quite superficial in discussions of entropy and should be avoided. The simplest go-to example is a monoatomic gas in a cubical box, so let's imagine identically-sized cubical boxes filled with some particular noble gas, one box for each element, at the same temperature.

Naively, one might think that the 'disorder' only depends on the possible arrangement at an instant in time, and so since we have the same amounts distributed uniformly throughout equal volumes, they should have the same entropy. This is mistaken, as entropy goes up with temperature. Alternatively, one might think that the since lighter noble gases are moving faster than their heavier ones at identical temperatures, they are more 'disorderly' and so the lighter gases are more entropic. However, this is also mistaken, as actually the standard entropies of the noble gases increase in the same order as their masses (He<Ne<Ar<Kr<Xe<Rn). The intuitive notions of 'disorder' have led us wrongly, here and in many other situations besides.

Begging your pardon; had I been precise, I would have observed that a system where rocks are distributed randomly throughout a three dimensional volume has more possible microstates than a system where they are distributed randomly throughout two dimensions but are constrained to lie on the floor.

To extend that explanation for everyone else's benefit, the entropy of a system increases as the number of possible microstates (individual specific ways for the system to be in that state) increases. Thus, having all the helium atoms neatly arranged on one side of a room, and all the neon atoms on the other, is lower-entropy than having a mixture of both... because the latter arrangement allows each individual atom more space it can potentially occupy. Thus, it doubles the number of possible microstates of the system once for the first atom, once for the second atom.... and so on for all atoms of both gases in the room.

Now, clearly, constraining particles to be located in two dimensions will sharply reduce the number of available microstates. If this comes with an increase in the thermal energy of individual atoms, or the like, then overall entropy of the system can certainly increase!

But my original point was that the trend toward increasing entropy doesn't automatically cause the rocks to levitate just because that would be a higher-entropy state. Entropy is a description of a system in terms of information and the number of possibilities there are available to be occupied; it cannot supply energy where none was available before, nor can it nullify the operation of a force acting upon a mass.

Sometimes people can conceptualize the Second Law of Thermodynamics as a trend towards maximal entropy. I was trying to point out that this is only true if we add the phrase "maximal entropy subject to other constraints," because the presence or absence of other constraints and forces define the context for how we compute the entropy of the system, and impose other constraints through processes like energy conservation.

Fair enough?

I don't think it's appropriate to think of configurations as higher or lower entropy. An important part of it was under-emphasized in your earlier post: entropy is not a function of the particular configuration, but rather of your knowledge. Entropy is a measure of the information you don't have. If you have your room with rocks, with our without gravity, and you look into it and see each of the rocks, then at that point, it makes no difference whether they're on the floor, floating spread about the room, or in a corner. You know how they're arranged and how they're moving, so there is zero entropy.

Ah. I was thinking of the stat-mech approach of describing the entropy of a system in terms of the number of possible microstates that match a given description. Rocks happen to be visible where, say, air molecules are not, but this is largely incidental to the question of whether there are more microstates where the rocks are distributed randomly throughout a 3D volume than where they are distributed randomly throughout a 2D area.

[jwl]

But it isn't a lower entropy state though, that's the thing. The fact is, rocks falling to the floor generates heat, and heat is what entropy is primarily concerned with, not arrangements of rocks. There's a reason the second law is often expressed in terms of heat flowing from hot to cold.

To show this further, imagine you swapped those rocks with bouncy balls. They would remain in the air for a significant amount of time, (partially) because heat isn't being generated to vent away the entropy.

[Kuroneko]

If there was exactly zero vacuum energy, this would be less provocative a mystery than for there to be 'some but only a tiny amount,' because all the models that predict it should exist at all predict that it should be extremely large, correct?

It would be less provocative a mystery, yes, in that physicists are quite used to extremely large (even formally infinite) quantities being cancelled by opposite contributions from other things. Having it zero would suggest that the extremely large predicted energy density is cancelled by to some unknown and unbroken symmetry, which would at least be 'like' the things physicists have encountered many times before. Having it be very large but smaller than the enormous predictions would be fine too, as physicists have dealt with broken symmetries before. Having it be almost zero, though, means that either the physics is ludicrously fine-tuned or something must be very, very different indeed.

For comparison, see the 'hierarchy problem' of why the Higgs boson is so light. One proposed explanation is that the interactions that would drive its mass higher are cancelled by their supersymmetric partners, leaving the Higgs much lighter than the Planck mass, but still heavy because the symmetry is broken at low energies. That's not an accepted theory, but the reason it's there is that that mode of explanation has worked several times before regarding other things.

Incidentally, supersymmetry would knock down vacuum energy density by about five dozen orders of magnitude from what's expected by the standard model alone. It's too bad that it would still be too large even then, by about as many orders of magnitude.

Fair enough?

Ah, I was actually talking about a box of only helium, a box of only neon, and so forth. If you have the same amount of gas in each box with the same volume and the same temperature, then the lighter noble gases are faster, and one might expect them to be more entropic because they look more 'disorderly' than the heavier noble gases. But actually, the faster-moving gases are less entropic than their heavier brethren.

It's actually fairly easy to motivate the correct order in a quantum-mechanical context: one would look at the energy levels of a particle in a box problem and how the spacing changes with respect to the mass of the particle, and what this implies about the density of states. But one doesn't really need QM here: the Sackur-Tetrode equation predicting the entropy of a monoatomic gas is product of classical thermodynamics.

One essential feature that's shared with QM, though, is that the particles are indistinguishable--in QM, fundamentally so, but classically, we're only concerned about indistinguishability through macroscopic measurements. This was recognized long before quantum mechanics, and it's the reason why in a simple situation a microstate is a phase space region of volume hnC, where C is an overcounting correction factor, and h is a parameter having dimensions of length·momentum (same as Planck's constant, but classically, it is a book-keeping parameter related to the fact that classically, only entropy differences are relevant, so entropy can be arbitrarily shifted by any constant amount if done everywhere consistently).

See also the Gibbs mixing paradox, which is quite relevant to your post. However, I do agree with your original point.

Ah. I was thinking of the stat-mech approach of describing the entropy of a system in terms of the number of possible microstates that match a given description. Rocks happen to be visible where, say, air molecules are not, but this is largely incidental to the question of whether there are more microstates where the rocks are distributed randomly throughout a 3D volume than where they are distributed randomly throughout a 2D area.

If your constraint is an exact energy, it is that--a microcanonical ensemble with a uniform distribution over the possible microstates, so you can get entropy just by counting them. But in general,stat-mech has the Gibbs entropy (which was actually first done in a statistical context by Boltzmann rather than Gibbs, but more obscurely), which only reduces to the logarithm of the number of microstates whenever the relevant microstates are equally likely.

But it isn't a lower entropy state though, that's the thing. The fact is, rocks falling to the floor generates heat, and heat is what entropy is primarily concerned with, not arrangements of rocks. There's a reason the second law is often expressed in terms of heat flowing from hot to cold.

Entropy in classical thermodynamics is concerned with information inaccessible to macroscopic measurements. Since positions and speeds of rocks are not unavailable to macroscopic measurements, yes, you are right in that classical thermodynamics is not concerned with arrangements of rocks.

However, this is merely a contextual restriction. The Gibbs thermodynamic entropy for a given distribution over the microstates is S = -kB Sum{ pi log pi }, which is simply the the Shannon information entropy of the probability distribution scaled by Boltzmann's constant. It is also generalized to quantum probabilities by the von Neumann entropy σ = -Tr(ρ log ρ), in which the density matrix ρ is a quantum-probabilistic generalization of a probability distribution and doesn't actually care whether it's describing a physical system, much less whether it's a thermodynamic one. Therefore, it is completely legitimate to talk about the entropy of a probability distribution of rocks, or indeed any probability distribution whatsoever.

But again, I wish to emphasize that entropy is not a function of an arrangement or a configuration, but rather of the probability distribution over such things describing your knowledge of the situation.

To show this further, imagine you swapped those rocks with bouncy balls. They would remain in the air for a significant amount of time, (partially) because heat isn't being generated to vent away the entropy.

I think it more naturally leads to the opposite point. The difference is that you can typically tell when a rock hits the wall of the room, but you can't tell when a molecule of gas hits a wall of its container. But if the walls of the room are such that you can't tell from the outside when a ball bounces off, there is no conceptual difference. If the bouncy balls are sufficiently bouncy, then this system can keep a well-defined energy level distributed among the balls, in which case, the most-entropic probability distribution of bouncy balls under the energy constraint is the standard microcanonical ensemble of thermodynamics. It's exactly the same.

As a side note, I think understanding entropy of any kind as fundamentally about information makes Landauer's principle quite obvious (generalized to more than just energy), in addition to the conceptual coherency of having a common probabilistic basis.

[jwl]

But it isn't a lower entropy state though, that's the thing. The fact is, rocks falling to the floor generates heat, and heat is what entropy is primarily concerned with, not arrangements of rocks. There's a reason the second law is often expressed in terms of heat flowing from hot to cold.

Entropy in classical thermodynamics is concerned with information inaccessible to macroscopic measurements. Since positions and speeds of rocks are not unavailable to macroscopic measurements, yes, you are right in that classical thermodynamics is not concerned with arrangements of rocks.

However, this is merely a contextual restriction. The Gibbs thermodynamic entropy for a given distribution over the microstates is S = -kB Sum{ pi log pi }, which is simply the the Shannon information entropy of the probability distribution scaled by Boltzmann's constant. It is also generalized to quantum probabilities by the von Neumann entropy σ = -Tr(ρ log ρ), in which the density matrix ρ is a quantum-probabilistic generalization of a probability distribution and doesn't actually care whether it's describing a physical system, much less whether it's a thermodynamic one. Therefore, it is completely legitimate to talk about the entropy of a probability distribution of rocks, or indeed any probability distribution whatsoever.

But again, I wish to emphasize that entropy is not a function of an arrangement or a configuration, but rather of the probability distribution over such things describing your knowledge of the situation.

Hence primarily. My point is that the predominant term of entropy in macroscopic situations is going to be the heat and energy distribution, so it shouldn't be at all surprising that the "entropy" of rocks goes up if you exclude heat and energy distribution, because you are then excluding most of what entropy describes. The second law of thermodynamics is still "total entropy does not decrease" not "entropy does not decrease when physically allowed". Or rather, the second is true, but entropy not decreasing is always physically allowed.
Entropy

To show this further, imagine you swapped those rocks with bouncy balls. They would remain in the air for a significant amount of time, (partially) because heat isn't being generated to vent away the entropy.

I think it more naturally leads to the opposite point. The difference is that you can typically tell when a rock hits the wall of the room, but you can't tell when a molecule of gas hits a wall of its container. But if the walls of the room are such that you can't tell from the outside when a ball bounces off, there is no conceptual difference. If the bouncy balls are sufficiently bouncy, then this system can keep a well-defined energy level distributed among the balls, in which case, the most-entropic probability distribution of bouncy balls under the energy constraint is the standard microcanonical ensemble of thermodynamics. It's exactly the same.

Indeed. And in that limit, you can clearly see that the "entropy loss" is due to heat creation, because the only way to eliminate it is to make the balls perfectly bouncy, and if they're perfectly bouncy they are never going to end up on the floor.

[Kuroneko]

Hence primarily. My point is that the predominant term of entropy in macroscopic situations is going to be the heat and energy distribution, so it shouldn't be at all surprising that the "entropy" of rocks goes up if you exclude heat and energy distribution, because you are then excluding most of what entropy describes.

Suppose we have a monoatomic gas of N silver atoms, the ground state of which have single unpaired s-shell electron. Because silver atoms are relatively heavy, the magnetic moment depends overwhelmingly on the spin of this outer electron. The spin entropy is thus S = kB log 2N. For now, it is a constant term, and so is not thermodynamically relevant, and one can ignore with impunity. But it is nevertheless a real, physical entropy, which one can see by turning on a external constant magnetic field. Then the magnetic moment of the silver atoms have a potential energy -μ·B, and the energy levels are split by the Zeeman effect. The spin entropy will be dependent on energy, and so will be thermodynamically relevant because temperature is 1/T = ∂S/∂E.

It would be inappropriate to put this entropy in scare quotes, regardless of whether there is a magnetic field or not.

Now, with Simon_Jester's rock example, rocks can be up or down (albeit in a more complicated manner than electron spin), and the the initially degenerate energy level of their up-or-downness (sans gravity, gravitational potential energy is independent of position) is split by the gravitational field. I don't know in what sense you mean that it neglected energy distribution, when the example was explicitly about energy distribution.

The second law of thermodynamics is still "total entropy does not decrease" not "entropy does not decrease when physically allowed". Or rather, the second is true, but entropy not decreasing is always physically allowed.
Entropy

Sorry, but I don't really understand what it is that you're objecting to. You say the situation is different because when rocks fall, the gravitational potential energy will be lost as heat. I agree, but I don't see how that contradicts what Simon_Jester said. Rather, to me it seems apparent that it's directly implied by what he said. So perhaps I'm misunderstanding one of you, possibly both.

Indeed. And in that limit, you can clearly see that the "entropy loss" is due to heat creation, because the only way to eliminate it is to make the balls perfectly bouncy, and if they're perfectly bouncy they are never going to end up on the floor.

We don't need to eliminate it; we could just take the analogy in the opposite direction, too. If we put the room in contact with a thermal bath, we don't need to posit that the rocks are bouncy balls; what we need is for them to be effectively indestructible and the walls be such that we can't tell when the rocks bang into them. It's fine for them to act like real rocks in all other respects, including dissipative forces, etc.

The rock gas would differ from more ordinary gas particles in that typical gas particles have only a few internal degrees of freedom--which shows up in the specific heat of the gas--whereas the rocks would have an immense number of internal degrees of freedom because they're macroscopic. But I don't think this is a critical conceptual difference for the purposes here. We need we would need ludicrously high temperatures for the thermal bath in order for rock up-and-downness degrees of freedom to thermodynamically participate in any meaningful manner. Not necessarily so high as to have them bouncing into walls, though.

If we're talking about practicality, then yes, absolutely, an indestructible rock gas is a silly thing, and (for example) a monoatomic silver gas would illustrate much the same lessons without being physically unreasonable. On the other hand, I don't see what's objectionable about it for the for the purposes of illustrating the concept of entropy.