I've followed Leonard Susskinds series on "the theoretical minimum".

In the entropy/reversibility episode in statistical mechanics, he explains (at least one way of) how the apparent paradox of reversibility vs entropy increase is resolved. (See http://theoreticalminimum.com/courses/statistical-mechanics/2013/spring/lecture-7 and http://theoreticalminimum.com/courses/statistical-mechanics/2013/spring/lecture-8)

The explanation goes something like this:

  1. Assume you know that a system is in within a region of phase space $\Delta$, with a "diameter" $\delta$. The reason would be that you are unable to distinguish between points a distance $\delta$ apart. (Assuming classical mechanics, it actually is at a specific point, but we don't know which.)
  2. Let the system evovle. $\Delta(t)$ will spread out sort of fractally, but keep its volume.
  3. Now, since you can't tell points apart if they're within $\delta$, you basically get a "product" between a sphere with diameter $\delta$ and $\Delta(t)$, giving a new region $\Delta^{\star}(t)$, which tends to get larger.
  4. Finally the entropy is just logarithm of the volume of $\Delta^{\star}(t)$.

I get this, and that it will eventually decrease etc. What I don't get is why you need to do the third step.

If I measure the system a second time, it still is in a specific point in phase space, and I should be able to distinguish it to the same precision again? Meaning that the entropy should decrease to the starting value immediately?

EDIT: I do understand the concept of "covering with spheres". My problem with this concept is the following: Somehow we begin to know that the system is in a certain region. I assume we do this by some measurement?

Now, if we can do a measurement with accuracy $\delta$, i.e. with a maximum deviation $\delta$ from the "true" point, at $t=0$, why can't we do it at again at a later time?

EDIT 2: I'm going to try to explain how I think again. Let's choose to view this as a grid of size $\delta$ instead of spheres, it's easier in the following context.

Assume the system is at a specific point in phase space, $x(0)$. By measuring it, we can find it resides in a certain grid element (the one that contains $x(0)$).

The system evolves to a new - specific - point in phase space, $x(t)$. Regardless of our initial measurement (the grid cell) or the evolution of that initial grid cell, a new measurement would also give us a single grid element, namely the one that contains $x(t)$?


The idea of "coarsening" is that we never have sufficient resolution to follow the precise evolution of a volume element of phase space space. It is one way to reconcile the equations of motion (reversible in time) with the apparent irreversibility of nature. The basic argument goes like this: initially we know that the system is somewhere inside a small region of phase space. Because of chaotic evolution, points that are nearby evolve in substantially different ways. The effective volume occupied by the ensemble at some later point is larger than what we started with because of this expansion. Equilibrium is reached when the original blob has spread itself to virtually occupy all accessible space.

This argument essentially describes the evolution of a system in phase space by analogy to diffusion: a blob of ink placed in water eventually diffuses to occupy the entire volume. The initial blob represents a macroscopic region of space that contains all the ink particles. All we can say is all the ink particles are in that blob but we can't tell where each particle is. If at some later time we make a "measurement", as you suggest, we will be mapping the region that contains ink. That region always grows. When it doesn't grow any more we have equilibrium.

Criticism This is one possible way to interpret the second law. The argument requires the ergodic hypothesis, which assumes a distribution of microstates under the action of the laws of mechanics will end up spreading in the entire phase space independently of the details of the initial distribution.

  • $\begingroup$ I think that this interpretation of the second law is observer-centric, in that entropy is only defined in terms of what the observer knows about the system. Then exactly as you say, chaotic time evolution means the observer is more and more uncertain of the final state of the system and thus entropy increases. However, I think OP's question might be more along the lines of why one can't measure a second time with the resolution $\delta$. I think the answer to that question is that one _can measure a second time, with the entropy decreasing accordingly. $\endgroup$ – user196574 Sep 2 at 23:37
  • $\begingroup$ @user196574 If we measure at a later time with resolution $\delta$ we will obtain the new volume occupied by the diffusing blob -- the entire blob. This will not give us a small volume from which to make accurate (low entropy) predictions for the future, The point is that every region of order $\delta$ within the blob contributes to uncertainty, and the more such regions we have (i.e. the larger the blob), the larger the uncertainty. $\endgroup$ – Themis Sep 2 at 23:56
  • $\begingroup$ I'm imagining a system of gas particles characterized by position and momentum. We make a measurement of each gas particle's position and momentum up to some resolution $R$, the uncertainty characterized by entropy $S_0$. Given our uncertainty about the positions and momenta, the chaotic time evolution of the positions and momenta of the particles in the box leads our coarse grained entropy to increase with time, just as you and PSE 390445 state. On a 2nd measurement of the positions and momenta, don't we again shrink to a blob centered around the system's true position and momenta? $\endgroup$ – user196574 Sep 3 at 0:13
  • $\begingroup$ @user196574 Think of the analogy of diffusing ink: we are measuring the space occupied by ink at some time $t$ with resolution $R$ and use this information to predict the blob in the future. Each subregion of size $R$ in that blob diffuses with some uncertainty and produces a region of larger volume. Because the blob contains many regions of size $R$, the entire blob grows larger. (By the way, I don't like this explanation; I am just interpreting what Susskind says in these videos.) $\endgroup$ – Themis Sep 3 at 0:21

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