If the laws of physics work the same forward and backward in time, why does entropy grow in one direction?

  • $\begingroup$ Perhaps we are a little caught up in the thinking of the last century and the conditions of the earth. "The second law of thermodynamics states that the entropy of an isolated system never decreases with time". Take the example of mass aggregation into stars and black holes. Exploding stars allow heavy elements from the aggregation of hydrogen to enter the world. The process from hydrogen to heavy elements increases entropy. $\endgroup$ Oct 22 '20 at 3:50

The question you ask is one of the major open philosophical questions in science today. Why does time appear to move "forward?" Many great minds like Feynman have explored the question (and its dual "what if time doesn't just move forward?") As such, no answer can be completely satisfactory, but some statements can be made and they will hopefully help.

As we see in JulianDeV's answer, entropy and its inevitable statistical increase are baked into the problem statement. They arise when dealing with systems where one can theoretically predict the past and future state of the system perfectly, given enough information, but where "enough" information is unacquirable. For example, the velocity of particles cannot be known perfectly. The best we can do is state some bounds on the velocity.

When you fill in the missing unknowns with randomness, in the truest mathematical sense of the word, your equations exhibit entropy. The measurable "states" and the unmeasurable multitude of "microstates," which appear the same upon measurement but are fundamentally different in the phase space of the systems, inherently lead to entropic like behavior. And in the direction of evolution of the system, we see entropy increasing (on average, as JulianDeV points out). So long as you wish to use random variables to fill in for that which you do not know, this entropic behaviors will emerge as unknown compounds upon unknown.

The great philosophical question is why are these models typically applied to systems with time going in one direction. Is there a fundamental concept of "forward" to the universe, or is it a side effect of the way the H. sapiens mind thinks about time?

And that is where the satisfactory answers must end. When we step beyond "what models exhibit entropic behaviors?" and into "what aspects of reality should be modelable this way?", we quickly get into the fundamental philosophical meaning of time. For an introduction into that question, I highly recommend the SEP article on Time.

  • $\begingroup$ I am tempted to think of time as a dimension just like space as relativity treats it. But space is symmetrical and it just seems really strange that time would have this probabilistic mechanism that makes it asymmetrical. I will read your recommended article. $\endgroup$
    – Jordan
    Oct 21 '20 at 20:40
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    $\begingroup$ @Jordan Time typically thought of as a dimension, but as a temporal dimension rather than a spatial one. Even in relativity, time is different than a spatial dimension (even if its part of one big 4-d vector). There are different symmetries for temporal and spatial dimensions (which becomes a big deal once you get to Nother's theorem) $\endgroup$
    – Cort Ammon
    Oct 21 '20 at 20:49

It is actually not, the total entropy can in fact decrease, but it's just highly unlikely.

I will make an analogy to the statistical interpretation of entropy as counting the amount of microstates $\Omega$ according to some configuration in phase space.
$S = k \ln \Omega$
Consider a uniform gas of indistinguishable particles in a box. Each particle has a position in 3D space and a momentum vector, together they form the $6N$-dimensional phase space where each point labels a certain configuration:
We will now compare 2 particular states:

  1. The gas is uniformly distributed over the entire box
  2. All gas particles suddenly go to one corner of the box and reside there

Clearly, the first state can be realised in many more ways than the second. Therefore, $\Omega_1 \ggg \Omega_2$ and hence $S_1 \ggg S_2$. There is a higher probability to find the gas uniformly distributed than in one corner.
However, does this mean that the second case is unphysical as it would lower the entropy?
No, it is just extremely unlikely! The second law of thermodynamics is not absolute, it's just that processes where the total entropy decreases are highly improbable.

For other examples: look up negative temperatures, the entropy decreases when adding heat.

  • $\begingroup$ Yes. I get the statistical interpretation of entropy. I get that a high entropy state is far more likely than a low entropy state. This is intuitive for me at this point. I see it every morning when I stir milk into my coffee. But if you think of the universe moving backwards in time. All of the physics would still be the same, but entropy would be globally reducing. Why is the probability one directional while the physics is two? $\endgroup$
    – Jordan
    Oct 21 '20 at 20:34
  • $\begingroup$ I am not sure about this answer. The entropy is given by the amount of knowledge you have about the system. If you dont know anything about the particle position the entropy will be S1 in your example. If you know all particles are in a corner the entropy will be S2. But the entropy is fixed by what you know it will not decrease like this. Because following your reasoning we could also say that the system is at all time in an exact position so the entropy would always be 0. $\endgroup$
    – StarBucK
    Oct 21 '20 at 21:49
  • $\begingroup$ I thought entropy was related to knowledge only insofar as probability was related to knowledge. But given quantum mechanics the probability is inherent to the fundamental physics of the system. Somethings can not be known. Am I missing something? $\endgroup$
    – Jordan
    Oct 22 '20 at 13:49

Once you define entropy, such as Cort Ammon descibes, add completely symmetric laws of physics (which is the case, as you say), you still need some kind of initial or prior condition to deduce a single emergent arrow of time.

The prior condition is the big bang, and it was "large" enough and lower entropy than today. The laws of physics allow for trajectories in either entropic direction. But if you already have a relatively low entropy state, the most common trajectories will be toward a higher overall entropy state. And thus from one snapshot to the next, all the way to the big bang, entropy has increased overall. And thus a single emergent arrow of time, the second law.

There are other formulations and assumptions that can lead to multiple arrows of time, like shape dynamics of Julian Barbour. They do not assume a low entropy big bang, and one trade off is having multiple arrows of time.

There is much more to the story but I belive this is the bare logical arguement.


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