# B Entropy on a Small Scale

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1. Mar 5, 2017

### John Morrell

So when my dad first explained the fundamental idea of thermodynamics to me, that entropy never decreases, he pointed out the odd fact that according to basically all other laws of physics, any motion or reaction could be run backwards and be just as valid as it is run forwards. It would break no laws other than the laws of thermodynamics for heat to flow from a cold body to a hot one; on a small scale nothing would necessarily be wrong with that. Having that understanding, I've always understood the laws of thermodynamics to be true only on a macroscopic scale. Given trillions of particles all interacting constantly, the laws of thermodynamics are generally going to be followed and on a large scale will always be true.

My question is, is there some actual fundamental law requiring entropy to increase on any scale? If you get down small enough, isn't it possible for entropy to decrease in a reaction, or does that violate something other than mere probability?

2. Mar 6, 2017

### BvU

Entropy is a statistical variable. On an ever smaller scale it is noisy, so yes it can increase -- temporarily. Probability wins. On a still smaller scale it is meaningless (I think).

You can do the exercise: find out how long it takes you to flip 100 heads in a row. Then think back at the order of magnitude of e.g. molecules involved in a reaction.

3. Mar 6, 2017

### gianeshwar

Validity of entropy I think is due to the fact that there is so far nothing against it.
On a scale of cells of our body where errors keep accumulating and hence we age, is due to second law of thermodynamics.

4. Mar 6, 2017

### gianeshwar

We talk of probability density function in atomic orbitals but I think not about laws of thermodynamics there.

5. Mar 6, 2017

### anorlunda

There are irreversible processes at the quantum level.

But consider an easier case of things that are not reversible even at the macro level. Let's say that you have a sunglass lens that only lets 4% of light through. If you use a mirror to reflect the 4% back through the lens again, you don't get the original 100% out. The 96% absorbtion of the lens is irreversible. So not all things can be run backwards.

6. Mar 6, 2017

### Khashishi

I guess you mean a law other than the second law of thermodynamics. You are looking for something more mechanistic.

In both classical and quantum mechanics, there's a concept of conservation of phase space volume during any process. You can find some info here https://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)
The entropy of a system is related to the phase space volume. One definition of entropy is
$S = k_b \ln \Omega$
where $\Omega$ is the number of microstates in the observed macrostate. Each microstate is represented as a cell in phase space, and a macrostate is represented as a volume in phase space. This volume changes shape over time, but can't shrink. This means this definition of entropy cannot decrease. This fact can be derived from the laws of motion. Well, this derivation implies that entropy can't increase either, so there are several caveats here.

If we know the state is one of the microstates within the macrostate, but we are ignorant of which one, then we forget some details about the system, that means we have to make our macrostate larger, and the entropy increases. But why is it that we can forget details and we cannot "unforget" details to make the macrostate smaller? I don't think this can be answered from the fundamental laws of motion.

7. Mar 16, 2017

### John Morrell

I understand that there are irreversible processes at the macroscopic level, but that seems (to me, at least) to just be a function of probability. It is conceivable that the atoms in the sunglasses could, on a very small scale, and in a very fantastic case, vibrate in such a way as to move a lot of energy into a few atoms which then re-emit the light that was originally absorbed. That obviously doesn't happen, but it could. It's just that that is one micro-state out of trillions of trillions, and the majority of microstates don't involve the thermal energy being concentrated enough for light to be emitted.

But on a small scale, when you have only a couple atoms, it's not so inconceivable that some light that hits one of the atoms is eventually re-emitted back out.

What examples are there of small scale irreversible processes?

8. Mar 16, 2017

### Stephen Tashi

You seem to be conceptualizing a mixture of probabilistic situations and non-probabilistic situations. If you think of the atoms in a particular pair of sun glasses doing anything definite, you have discarded any notion of a non-trivial "probability" being involved. It's like thinking of a "fair coin" that has already been tossed and landed heads.

You are correct that the mathematical theory of probability does not prohibit (or enable) improbable events from actually happening. In fact, the mathematical theory of probability does not say anything at all about whether an event with a probability will or will not actually happen. It is people that apply probability theory to particular problems that make interpretations about whether events will actually happen. Different people may have different opinions. So when you ask if some physical situation can or cannot occur, you are not asking a mathematical question.

9. Mar 16, 2017

### John Morrell

Yeah, that's exactly my question though. My point is that in my understanding entropy only increases because it's overwhelmingly likely to. There are millions of different outcomes possible and many more of them result in an increase in entropy than in a decrease, and so on a macroscopic scale entropy increases. What I'm asking is;
Is there any law that requires entropy to increase in even a small reaction? Is is possible that the entropy in the universe could spontaneously decrease?

10. Mar 16, 2017

### Jilang

Of course theoretically even for the universe, (but the odds would be so staggeringly, overwhelmingly against it it would not happen in the lifetime of the universe!) On a small scale with isolated systems it happens all the time for short timescales.

11. Mar 16, 2017

### John Morrell

Okay, that answers the question I think. It could happen, it just doesn't. So on a small scale it shouldn't really even be that hard for it to happen. To be fair, though, any particular instance of the universe is already staggeringly, overwhelmingly improbable :).

12. Mar 16, 2017

### Stephen Tashi

You have answer the question "Outcomes of what?" It isn't possible to formulate a coherent question about "outcomes", "probability" or the entropy associated with probability unless it is clear what outcomes and probabilities are involved.

If an "outcome" is the state of a gas, specified by a deterministic description of all its atoms then there is no longer any probability to talk about. Hence there is no defined entropy. It doesn't matter whether the specific state of the gas looks "random" or whether all the molecules of the gas are bunched up on one side of a container. If the gas is in some completely described state, then there is no probability left to talk about and there is no way to define entropy.

For example, in another thread, someone proposed to write a detailed simulation of gas that is initially at equilibrium in a chamber. A passage is opened that allows the gas to flow into another chamber. Eventually the gas reaches an equilibrium state occupying both chambers. He proposed to simulate the position and velocity of each atom of the gas as time passes. His question was how to compute the entropy of the gas in his simulation as a function of time. That brought up various problems. 1) Thermodynamic entropy is only defined for gases in an equilibrium state. 2) Thermodynamic entropy is defined for ensembles of systems, not for one particular system. 3) Thermodynamic quantities such as "pressure of a gas" and "volume of a gas" are only defined for gases in an equilibrium state.

What you have to do is find a definition "outcome" such that an outcome can happen and still leave something probabilistic to talk about.

It isn't clear how to define the thermodynamic "entropy of the universe". We might need an "ensemble of universes" to accomplish that definition.

"Entropy" is only a defined quantity in very specific physical situations and the term is used with different meanings. Since you mention chemical reactions, it would be interesting to discuss how entropy is defined in that context. For example, you visualize entropy as a quantity that can vary with time in chemical reactions. Can you point me to a link that describes this happening? I have only found examples of computing the entropy "of a reaction" as a single number.

13. Mar 17, 2017

### John Morrell

Hmm, that's a very good point.

I guess I've been thinking of "outcomes" just as possible states for a system to be in. So for, say, a glass of water, there are numerous different configurations of velocities and positions the water could be in. Well, those configurations are constrained by the total energy in the system, because obviously you can't have the water moving in any way that exceeds the total energy contained in the system.

In life we only ever see the glass of water in thermal equilibrium. But technically, you could have almost all the energy in the glass concentrated in one water molecule moving at ridiculous speeds and the rest of the water and glass near absolute zero. Or you could have the water all spontaneously freeze and the glass around it melt. As far as I understand, there is no absolute law that prohibits this. It's just that the majority of configurations that obey the law of conservation of energy are comprised of the water and glass at thermal equilibrium.

That's the perspective I've been thinking in (whether or not it's correct, I don't know). The universe is like the glass of water; there is only so much energy and mass in it, and they can be configured in any number of ways. Entropy is the measure of how spread out, how close to "thermal equilibrium" the energy and mass are. It's possible for it slowly lose entropy, just very unlikely.

As for chemistry, it seems possible for random molecules in the air to take the thermal energy around them and spontaneously form into clouds of toxic gas; it's just not probable, because that would represent a decrease in entropy, a movement from a large set of micro states into a small set.

14. Mar 17, 2017

### Stephen Tashi

Technically, it isn't the individual molecules of "the" air that occupy "microstates". A given volume of air is in only one microstate. The microstate of this particular volume of air describes the state of each of its molecules. The probabilities associated with a microstate have to do with the probability that a "randomly selected" example of a volume of air (with given values of gross thermodynamic properties such as temperature, pressure, etc) will be found in that particular microstate. However, your general train of thought is clear - i.e. the fact that theory assigns a very small probability to a given microstate implies it is theoretically possible for a some volume of air to be in that microstate.

Your particular interest is thermodynamic-like questions. However, I don't (at the moment) see anything particular in that type of question that distinguishes from the general issues of applying probability theory to any other real world phenomena. The question of whether some as-yet-never-observed physical event is "possible" though improbable can be approached empirically by asking whether our current theory seems to correctly predict the probabilities of those events that have been observed and making some estimate about the probability of observing as-yet-never-observed events with some constraint such as "within your lifetime", "within the age of the universe" etc.

As to the metaphysical question of whether an as-yet-never-observed event is possible, that is complicated. The issue of "possible" vs "actual" is complex. For example, are there "possible" events that never actually happen? Or is the assertion that an event is "possible" equivalent to saying that at some time or place, the event will happen or did happen?