# Is a world with constant/decreasing entropy theoretically impossible?

I'm not 110% sure exactly what I mean by this question. It was sparked by a friend who said he wished the law of entropy were reversed, so he wouldn't have to worry about cleaning the bathroom.

Basically, we can imagine many changes to the laws of physics - you could scrap all of electromagnetism, gravity could be an inverse cubed law, even the first law of thermodynamics could hypothetically be broken - we've all imagined perpetual motion machines at one time or another.

However, the second law of thermodynamics seems somehow more 'emergent'. It just springs out of the nature of our universe - the effectively random movement of physical objects over time. Provided you have a Universe whose state is changing over time according to some set of laws, it seems like the second law must be upheld, things must gradually settle down into the state of greatest disorder.

What I'm particularly wondering is if you can prove in any sense (perhaps using methods from statistical mechanics)? Or is it possible to construct a set of laws (preferably similar to our own) which would give us a universe which could break the second law.

Thanks

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 An interesting answer to this question can be found in the book 'Time's Arrow and Archimedies' Point' by Huw Price (amazon.com/Times-Arrow-Archimedes-Point-Directions/dp/…). He points out that if you took our universe and 'ran the tape in reverse' then everything that happened on the microscopic level would be compatible with the laws of physics as we know them*. The universe in which entropy always decreases could just as well be our own universe - we'd have no way to tell the difference. (*you'd also have to reverse charge and chirality to make it work properly.) – Nathaniel Feb 2 '12 at 17:54 I've heard this before, and I guess you're technically right - if we could 'create' a universe in the exact state as our universe and then reverse the motion of all the atoms/charges/forces, etc, then it should run according to relatively familiar laws, except that entropy would constantly be decreasing. Admittedly this is extremely Laplacian, and I'm not sure if quantum mechanics has anything to say on the matter... – tom Feb 2 '12 at 23:12 Everything in quantum-mechanics-as-we-know-it respects time-reversal symmetry (technically CPT symmetry), except for the process of observation, in which the wave function "collapses" into a randomly chosen eigenstate. However, to my mind the asymmetry of this is probably the same as the asymmetry involved when you learn that a hidden ball is in my right hand rather than my left. It's not the ball that changes, it's your state of knowledge. There are arguments against this view based on Bell's theorem etc., but space is too limited to discuss them. – Nathaniel Feb 3 '12 at 10:51

The short answer is that such a universe cannot be envisaged, not with relevance to our known physics.

Entropy as defined in statistical thermodynamics is proportional to the logarithm of the number of microstates of the closed system, the universe in your question. You would have to devise a universe where the number of microstates diminishes with time. The great multiplier in our universe is the photon, which is emitted at every chance it gets, and thus increases the number of microstates, i.e. electromagnetic interactions. A universe without electromagnetism would not have atoms.

It is worth noting that all biological systems decrease entropy, as does the crystallization of materials, but this is possible because the systems are open and the energy exchanges create a large number of microstates thus obeying in the closed system the entropy constraint.

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You may or may not be satisfied with this answer, but I found the question fun, and so I gave it a shot.

Take the classic bowls and balls multiplicity set-up. We have four balls, labeled A,B,C,D and two bowls to put them in. Normally, two balls per bowl is the macrostate with the highest multiplicity, and so for the sake of argument let's say that's where you start the system off at.

Now, each microstate has a particular probability of occurring, and that probability was the item I decided to pay attention to for this exercise. The Complex Klein-Gordon lagrangian allows for negative probability densities (from the current 4-vector $j_\alpha =\psi^{*}\overleftrightarrow{\partial_{\alpha}}\psi$), which is one of the reasons it went out of style for electrons. But it also maintains conservation of probability, $\partial_\alpha j^\alpha =0$.

So, consider the bowl example, except let three of the microstates corresponding to the "two balls per bowl" macrostate have negative probability of occurring (for whatever reason $j_0$ for these states leads to negative probability). Then when one sums up the probabilities, the "two balls per bowl" macrostate still has the highest multiplicity, but does not have the greatest probability of occurring, so if the system started out in the "two balls per bowl" state, it is likely that it would evolve into a different state (most likely the "three balls in one bowl" state). Of course, another microstate would have to gain probability to pick up the slack to conserve it.

I don't know if that tickles your fancy but its the first thing that came to mind when I read the question. Someone might have a slightly more reasonable answer.

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The microscopic laws are reversible in time (if you also change chirality and the sign of all charges). Thus one cannot prove what you'd like to prove.

Statistical mechanics, which is the discipline in which one derives the second law from microphysics, always makes one or the other assumption that induces the direction of time actually observed in our universe: That entropy increases (unless the whole world is in equilibrium, which it currently isn't).

However, you could run the whole universe backward, and it would satisfy precisely the same microscopic laws (if you also change chirality and the sign of all charges). But entropy would decrease rather than increase.

I don't think your friend would like to live in such a world.

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