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It's often said that the entropy of universe is increasing and that the universe's energy is becoming more evenly distributed. But intuitively, we say also that entropy is a measure of the disorder.

Isn't it a contradiction to say that the universe is becoming both more evenly distributed and more disordered?

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    $\begingroup$ On a macroscopic scale, entropy as a measure of increasing disorder of the universe has started falling out of favor to entropy as a measure of increasing uniformity of the distribution of energy in the universe. The two are consistent with one another. The former is "what does that even mean?" The latter is "so that's what entropy is!" $\endgroup$ – David Hammen May 18 '18 at 1:34
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This is a common source of confusion because many folks figure that a uniform distribution would be very ordered.

The thing's that things aren't becoming uniformly distributed; that's just a macroscopic view! What's actually happening is that the macroscopic view is becoming increasingly less informative, i.e. uncorrelated with the information of the system. By the time complete macroscopic uniformity is reached, the observer has completely lost their ability to guess what actually exists at any given point in that space.

For example, consider a glass of water with a block of salt to be dropped into it. Once the salt is dropped into the water, it'll dissolve and evenly spread around. That seems more ordered and uniform, huh? But, it's not!

  • Before mixing, you could say if there was a water particle or salt particle at some randomly selected point in the glass.

  • As mixing occurs, the amount of information you have about relative probabilities of whether there's a salt particle or water particle at any given point become less certain, though you can still make a better-than-no-information guess at the odds.

  • After complete mixing, you have no idea beyond saying that there's a single probability throughout the glass equal to the portion of the particles of the selected type.

This is, the structure (order) that existed before mixing has now been broken down into random chaos. And that's pure entropy.

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  • $\begingroup$ thanks for your correction of traduction and your answer ... it seems you do the analogy with the theory of information and $S= - log I$ with information $I$, don't you ? $\endgroup$ – youpilat13 May 18 '18 at 0:54
  • $\begingroup$ @youpilat13 Sure, though the more general form,$$S~=~-\sum_{{\forall}i}{p_i \ln{\left( p_i \right)}} \,, $$tends to be more appropriate in the general case since the reduced version requires equal probabilities (which makes stuff like the water-and-salt example confusing, since equal portions of salt won't typically dissolve in water - at least not for sodium chloride). $\endgroup$ – Nat May 18 '18 at 0:56
  • $\begingroup$ ok, it's easier to understand with the notion of information than with classical possible combinations of arranging a system, I have to study more the relation between both to grasp subtilities. thanks $\endgroup$ – youpilat13 May 18 '18 at 0:59
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    $\begingroup$ @youpilat13 Definitely - and just a head's up, one of the biggest misconceptions folks have about entropy is believing that it's a property of a system, but it's not. Instead, entropy is a property of a model, such that it's model-subjective. It's initially difficult to understand how a value that was originally empirically measured could possibly be observer-subjective even in classical contexts, but once you get that, I think that the rest tends to fall in place. $\endgroup$ – Nat May 18 '18 at 1:02
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    $\begingroup$ [...] For all points For all points $\left(x_i,y_i,z_i\right)$, you pretty much just know that the odds of there being a water particle in that location are about $\frac{1000}{1000+5}$ (ignoring all sorts of conversion factors, activities, ionic strength, etc., for simplicity) while the odds of there being a salt particle are $\frac{5}{1000+5}$. If you calculate the entropy of this state,$$S~=~-\sum_{{\forall}i}{p_i \ln{\left(p_i\right)}}\,,$$it should be maximized here. $\endgroup$ – Nat May 19 '18 at 1:55
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No, it's not a contradiction. Entropy is not a measure of disorder. The entropy $S$ is a measure of the number of ways to arrange a system,

$$ S = k\ln\Omega. $$

The energy is becoming more "evenly spread" because the entropy has to increase, for this is the most probable macrostate.

There are more ways to arrange evenly-distributed energy than there are to arrange the energy if all of it is concentrated in a single place.

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  • $\begingroup$ @zhutcgens1 So, where does this definition of "measure of disorder" come from ? it is just a microscopic point of view ? $\endgroup$ – youpilat13 May 18 '18 at 0:50
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    $\begingroup$ I would strongly contest that "measure of disorder" is a potential definition of entropy. Its relationship comes from multiplicity $\Omega$; sometimes, disorder and multiplicity can have similar meanings. Like, if you have a messy bedroom, there are more ways to arrange your things because they are disordered. But extending this as "definition" is too far. $\endgroup$ – Zack Hutchens May 18 '18 at 0:55
  • $\begingroup$ This youtube, MIT course, confirms that entropy is the measure of number of ways to arrange a system. Very interesting and great graphics. youtu.be/870y6GUKbwc $\endgroup$ – kamran May 18 '18 at 4:01

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