What causes entropy?

[pgwenthold]

Sure. Energy is a function of the distribution of states; see the Bohr model of the hydrogen atom for an example.

It's just very difficult to work with systems larger than a single hydrogen atom, which is one reason that energy is usually treated as a primitive.

Actually, think the better answer is, "yeah, that's what temperature is."

Temperature is based completely on the probability distribution of energies (that being Maxwell-Boltzman)

There is a lot of bizarre thinking in this thread. Entropy, like any thermodynamic function, is a macroscopic quantity that describes a sufficiently large population (where "sufficient" means it can be described by using population statistics). Talking about the entropy of a single particle makes no more sense than talking about the temperature of a single particle.

A single particle might have a partition function, just as it might have an internal energy, but to describe them in terms of entropy and temperature, you need a macroscopic sample.

 

[rocketdodger]

Actually, think the better answer is, "yeah, that's what temperature is."

Temperature is based completely on the probability distribution of energies (that being Maxwell-Boltzman)

There is a lot of bizarre thinking in this thread. Entropy, like any thermodynamic function, is a macroscopic quantity that describes a sufficiently large population (where "sufficient" means it can be described by using population statistics). Talking about the entropy of a single particle makes no more sense than talking about the temperature of a single particle.

A single particle might have a partition function, just as it might have an internal energy, but to describe them in terms of entropy and temperature, you need a macroscopic sample.

It seems like you could, though -- If entropy is a measurement of the number of states with a given energy, and a single particle can have energy, then a single particle should be able to have entropy as well.

 

[sol invictus]

Is that really lowering the entropy of the sandwich?

Yes.

It seems like there would be more low energy states than high ones, so a colder sandwich would in fact have higher entropy. Is this incorrect?

Yes, it's incorrect. There are almost always more states at high energy. There are exceptions, but they are rare (and in fact in those cases the temperature would go through infinity and then become negative as you increase the energy).

 

[sol invictus]

There is a lot of bizarre thinking in this thread. Entropy, like any thermodynamic function, is a macroscopic quantity that describes a sufficiently large population (where "sufficient" means it can be described by using population statistics). Talking about the entropy of a single particle makes no more sense than talking about the temperature of a single particle.

A single particle might have a partition function, just as it might have an internal energy, but to describe them in terms of entropy and temperature, you need a macroscopic sample.

Energy and entropy have statistical mechanical definitions that apply perfectly well to any number of particles. It's just that the laws of thermodynamics only follow from those microscopic definitions when the number of particles is large (the example I gave early in the thread shows quite clearly how that works in the case of the 2nd law of thermodynamics).

For example, in quantum mechanics the entropy of a single particle state will be zero if and only if it is pure. Clearly it has an energy as well.

Can it have a temperature? That's a slightly more subtle question. Certainly one can put a single particle into the thermal state that corresponds to the canonical ensemble at some temperature T, and that single particle state can exist independently of any thermal reservoir or heat bath. But I agree it's a little odd to ascribe a temperature to a single particle; I'd prefer to be specific and simply say it's in a thermal state with temperature T or something to that effect.

 

[sol invictus]

It seems like you could, though -- If entropy is a measurement of the number of states with a given energy, and a single particle can have energy, then a single particle should be able to have entropy as well.

Right. (And bear in mind that it isn't necessarily the number of states at an exactly determined energy - it can be the number of states with energy below a given energy, or in some energy range, or at a given temperature. That depends on the ensemble and precisely which definition of entropy you're using.)

By the way, the definition of entropy is more general even than this. It lies at the basis of Shannon's formulation of information theory. It tells you how much information can be stored or transmitted in a given system or channel.

 

[blobru]

It seems like you could, though -- If entropy is a measurement of the number of states with a given energy, and a single particle can have energy, then a single particle should be able to have entropy as well.

Though a single particle, considered as an isolated system, is automatically in equilibrium with itself (by Newton's First Law of Motion: whether at rest or in motion, whatever "energy state" it is in won't change until acted on by an external force). So as a system by itself, its entropy can't change, and isn't a meaningful measure (as I understand it, at least -- though I suppose it's different in QM's probabilistic treatment of "particles").

 

[pgwenthold]

It seems like you could, though -- If entropy is a measurement of the number of states with a given energy, and a single particle can have energy, then a single particle should be able to have entropy as well.

No, a single particle has a partition function.

 

[pgwenthold]

I'd prefer to be specific and simply say it's in a thermal state with temperature T or something to that effect.

Huh?

If you are going to be "specific" why not just say that it has energy X?

Then again, what do you even mean a "thermal state" of an individual particle? "Thermal" refers to the distribution of energies within the sample.

ANY state of a particle can be a "thermal" state. It's just that some of the states are more common than others.

 

[sol invictus]

Though a single particle, considered as an isolated system, is automatically in equilibrium with itself (by Newton's First Law of Motion: whether at rest or in motion, whatever "energy state" it is in won't change until acted on by an external force). So as a system by itself, its entropy can't change, and isn't a meaningful measure (as I understand it, at least -- though I suppose it's different in QM's probabilistic treatment of "particles").

The state does in general have a time dependence. It's true that the expectation value of the energy will not change without some external influence.

No, a single particle has a partition function.

rocketdodger was correct - a single particle is described by a state, to be specific a single-particle state, and that state might be pure or mixed. If it's mixed it has a non-zero entropy that arises from the fact that the mixed state includes more than one pure state. That can happen even when the energy is definite, if there is degeneracy (i.e. more than one state with the same energy).

Huh?

If you are going to be "specific" why not just say that it has energy X?

Because it doesn't. In my example it doesn't have a definite energy. It has a definite temperature.

Then again, what do you even mean a "thermal state" of an individual particle?

[latex]\rho = e^{-\beta H}/Z[/latex]. H is the Hamiltonian operator acting on that single particle, beta=1/T, and Z is the partition function (which just normalizes that matrix).

ANY state of a particle can be a "thermal" state. It's just that some of the states are more common than others.

Perhaps you're talking about classical mechanics? If so, it's no wonder you're having trouble following this. There isn't really such a thing as classical statistical mechanics.

 

[Dr. Keith]

Actually, think the better answer is, "yeah, that's what temperature is."

It has been many years since I had a conversational and computational grasp on entropy, but when I did it was through the study of steam engines.

It was a very good way to hammer this through the thick skulls of promisingly smart kids who all thought they could build a perpetual motion machine, even if they didn't say so out loud, ie first year mechanical engineering students.

I really need to build one of these beautiful things one day. Steam engines are so incredibly cool. Well, except where they are hot. Hotter?

 

[soylent]

Can someone explain how energy and entropy and this whole probability idea are related?

Imagine you have a water-tight box with two compartments. In both compartments are dye solutions, but in one compartment the concentration is twice as high. The dye molecules bounce around randomly in the solution; if you punch a hole in the wall between the two compartments some dye molecules will diffuse over from the compartment with the weaker solution to the stronger solution and some vice versa. Since there are many more dye molecules in the stronger dye solution it is statistically more likely for a dye molecule to cross from the stronger solution to the weaker solution. Stated differently, there are many more microstates in which dye molecules are evenly distributed between the two containers than in which they are unevenly distributed between the two containers and if you allow mixing to occur the system will tend towards a homogenous mix.

Imagine a similar situation in which you have two reservoirs, one hot and one cold. This is a very similar situation to the dye. The initial state is low entropy because if you connected the reservoirs with a heat-pipe to allow "mixing" there would be a net-flow of heat from the hot reservoir to the cold reservoir until they are the same temperature. Like with the dye there are many more microstates in which both reservoirs have the same temperature than there are microstates in which both reservoirs differ in temperature.

Because heat will flow spontaneously from the hot reservoir to the cold you could build a heat engine that will allow heat to flow from the hot reservoir to the cold reservoir and convert some of it to mechanical work. There is free energy in the thermodynamic sense.

If instead the temperature of both reservoirs was the same the total amount of energy in the system would be the same, but there is no free energy, no capacity to do work.

When you use energy, you don't destroy any of it, you simply degrade it to a high-entropy form of energy(like low-grade heat) that is no longer capable of doing what you want it to do. What you're using up is free energy, or the energy available to do work.

In order to decrease the entropy somewhere, you need to increase the entropy elsewhere by at least an equivalent amount. In order for plants to create the low entropy chemical energy they contain and the oxygen they emit they have to take low entropy sunshine and degrade it to high entropy thermal energy. In order to desalinate sea water into brine and fresh water you have to decrease the entropy of mixing, to do that you have to increase entropy elsewhere(such as by taking low entropy chemical energy like methane and degrading it to high entropy exhaust while extracting some mechanical energy to run an RO-plant).

 

[blobru]

The state does in general have a time dependence. It's true that the expectation value of the energy will not change without some external influence.

Okay... that makes sense. Thanks.

 

[RussDill]

I dunno, the time invariance thing is interesting from the question of why entropy was so low at a given point in time. But the universe definitely has a time dependence other than the entropy just being lower. You can always tell which way is the future, and which way is the past even if we reach heat death, check to see if the expansion of the universe is accelerating.

It would be interesting to see what an opposite universe would appear as, start way in the past with a huge universe with particles/energies spaced in a precise lattice. The universe is contracting at an incredible rate, barely being slowed by dark energy. As soon as the universe gets small enough, quantum instabilities cause particles to combine and gas clouds to form, eventually forming stars, planets, etc. Dark energy eventually slows the contraction of the universe to a much lower rate. Perhaps the universe will expand again? However, as the universe gets smaller, the force of dark energy reduces, and the contraction of the universe accelerates as gravity takes over, eventually ending in a big crunch.

 

[sol invictus]

I dunno, the time invariance thing is interesting from the question of why entropy was so low at a given point in time. But the universe definitely has a time dependence other than the entropy just being lower.

That's not at all clear. I'm not sure how you even separate the two.

You can always tell which way is the future, and which way is the past even if we reach heat death, check to see if the expansion of the universe is accelerating.

Accelerating expansion actually doesn't distinguish the future from the past once you reach "heat death". The universe's spacetime asymptotes to de Sitter. de Sitter can be thought of as eternally accelerating, but can equally well be thought of as eternally decelerating - in fact, it's really static. Moreover, we can't see the whole universe. It's possible the part we inhabit is expanding (and time is running "forward"), while somewhere else it's running the other way.

It would be interesting to see what an opposite universe would appear as, start way in the past with a huge universe with particles/energies spaced in a precise lattice. The universe is contracting at an incredible rate, barely being slowed by dark energy. As soon as the universe gets small enough, quantum instabilities cause particles to combine and gas clouds to form, eventually forming stars, planets, etc. Dark energy eventually slows the contraction of the universe to a much lower rate. Perhaps the universe will expand again? However, as the universe gets smaller, the force of dark energy reduces, and the contraction of the universe accelerates as gravity takes over, eventually ending in a big crunch.

That's a kind of opposite, but it's not the kind we were discussing. Consider something else instead - take our universe and simply reverse the direction of time (i.e. think of our universe as a movie, and play it backwards).

Now from our point of view, time is running backwards in the movie. As we watch people rise from death, get younger and younger, then pop back into the womb. Omelettes uncook themselves, divide into ingredients, the eggs unbreak. The universe shrinks and heats up.

But it's important to realize that this backwards universe is a perfectly valid solution to the laws of physics. Given the appropriate initial conditions, it's what would happen. And if it did happen, the people living in it would think of the direction of time we call the past as their future. They would have to, because their lives and brains are identical to ours, except running in reverse.

So the direction of time we call "future" is determined only by the local conditions in our part of the universe. To make a very rough analogy, if you had lived you whole life in the USA you might think there's a rule of automotive engineering that requires steering wheels to be on the right. A trip to the UK would disabuse you of that notion. Similarly, there's no law of physics that requires time to run in the direction we think it does.

 

[dafydd]

Time.

 

[dlorde]

Now from our point of view, time is running backwards in the movie. As we watch people rise from death, get younger and younger, then pop back into the womb. Omelettes uncook themselves, divide into ingredients, the eggs unbreak. The universe shrinks and heats up.

But it's important to realize that this backwards universe is a perfectly valid solution to the laws of physics. Given the appropriate initial conditions, it's what would happen. And if it did happen, the people living in it would think of the direction of time we call the past as their future. They would have to, because their lives and brains are identical to ours, except running in reverse.

Now I think you have to be really careful about what people in the time-reverse universe are thinking... if their lives are identical to ours but running backwards in time, they will 'unthinking' thoughts and unmaking memories of their future (to them) lives - a bit weird from an external point of view, but for them, life will surely be identical to ours - the same memories, the same events. It's certainly arguable whether they'd be aware of the reverse timeline.

If this model seems uncomfortable, the alternative seems even more uncomfortable- the idea that the entire universe runs time-reverse, but human thought is exempt, being able to think about the unscrambling of eggs rather than unthinking the scrambling of eggs... unless you prefer a dualist view, where conscious awareness is some kind of metaphysical interpreter outside spacetime, but accompanying the physical entity in following the arrow of time forwards or backwards.

This leads to thoughts of a Parminidean 4D block universe, where everything has, in a sense, 'already happened' (the timeline is established), so the external dualist awareness can traverse in either direction. Can't say I like it.

 

[sol invictus]

Now I think you have to be really careful about what people in the time-reverse universe are thinking... if their lives are identical to ours but running backwards in time, they will 'unthinking' thoughts and unmaking memories of their future (to them) lives - a bit weird from an external point of view, but for them, life will surely be identical to ours - the same memories, the same events. It's certainly arguable whether they'd be aware of the reverse timeline.

It's not arguable: they couldn't possibly be aware of it. Our past is their future. For them, memories form normally, everything proceeds just as it does for us - except backwards, from our point of view (and we live backwards from theirs).

Their experience is subjectively identical to ours, which is the point - it seems that there isn't a fundamental arrow of time, only an environmental one determined by the local conditions.

If this is confusing, imagine freezing time, picking up the entire observable part of the universe, making an exact copy, and rotating the copy by 180 degrees. Now put the copy and the original down far away from each other, and let time go again.

Can the inhabitants tell whether they're in the copy or the original? Nope - everything seems exactly the same for both.

If this model seems uncomfortable, the alternative seems even more uncomfortable- the idea that the entire universe runs time-reverse, but human thought is exempt, being able to think about the unscrambling of eggs rather than unthinking the scrambling of eggs... unless you prefer a dualist view, where conscious awareness is some kind of metaphysical interpreter outside spacetime, but accompanying the physical entity in following the arrow of time forwards or backwards.

This leads to thoughts of a Parminidean 4D block universe, where everything has, in a sense, 'already happened' (the timeline is established), so the external dualist awareness can traverse in either direction. Can't say I like it.

You've lost me a bit there. Unless you're religious there's nothing special about human thought; it's just another physical process, like milk mixing with coffee, stars burning, etc.

 

[dlorde]

Okay - looks like we're on the same page after all. I misinterpreted what you meant when you said

the people living in it would think of the direction of time we call the past as their future.

It's a little ambiguous whose future whom is viewing as which - if you follow me. The people living in the reversed timeline have a future that we call the past, but they (subjectively) don't think about it as that, their experience is identical to ours.