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I am puzzled with thought experiment that resembles/is version of Bolzmann's brain-hyphothesis. I could explain it in following way:

Let’s assume that we have isolated system full of some stuff, let’s say room full of air. We have some entropy meter in that room and we can also observe what happens in the room.

As well known, air particles in room are bumping on each other. Sometimes, very very rarely, all particles might be packed in the other half of the room and other half would be empty. On that time instant, entropy in room decreases rapidly and there is some usable energy in the room. Let’s take a leap of faith and assume that on that instant life would pop up in that room.

Here comes the catch that confuses me: Direction of time is always thought to be pointing towards state of higher entropy. Then, if life would pop up on downward slope of measured entropy curve (when entropy starts to drop), life inside the room would assume that arrow of time is opposite than ours (Direction to higher entropy inside the room is opposite to our arrow of time). If life would pop up on upward slope of measured entropy curve, life inside the room would have same direction of time than we would.

Am I missing something here? It seems counter intuitive that time in some insulated system at times might have different direction than system surrounding it.

It's getting even more confusing when trying to think how speed of time in such room would compare to ours. Maybe speed of time in that room would be somewhat relational to slope? If slope is steep, maybe time for beings in the room would be fast?

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    $\begingroup$ "Direction of time is always thought to be pointing towards state of higher entropy." - No. $\endgroup$ – ACuriousMind Dec 23 '15 at 20:21
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    $\begingroup$ If you can observe what happens in the room then it's not fully isolated. Your observation that one can't define a direction of time in an equilibrium situation is correct, but your leap of faith is misguided, though. Life requires non-equilibrium, so even if, by chance, a Boltzmann brain or an amoeba pops up, it won't live for long because brains and amoeba in isolated environments die. The only way to evade that problem is to make the room so large that it contains the equivalent of the entire environment, which means it would be equivalent to the universe. No gain, there. $\endgroup$ – CuriousOne Dec 23 '15 at 20:21
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    $\begingroup$ So why not extend it to the universe? Suppose there was a massive fluctuation to a low entropy state for some unknown reason. The op is saying that if life sprung up on one side of the minimum they would experience time in a different direction than the if it was the other side. I think that's valid. $\endgroup$ – octonion Dec 23 '15 at 23:01
  • $\begingroup$ I agree with octonion, regarding the last question: the "speed" of time (whatever it means) will not change because the laws of physics are not invariant to changes in timescale $\endgroup$ – user83548 Dec 24 '15 at 0:49
  • $\begingroup$ Bruce: Your answer "laws of physics are not invariant in changes to timescale" sounds interesting. Is this related to Noerthers theorem? Intuitively I guessed that the if changes in entropy inside the room would happen quickly/slowly, that would effect "speed" of time beings inside the room would experience. But propapbly this intuition is wrong then $\endgroup$ – Juho Salminen Dec 26 '15 at 9:14
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Let’s assume that we have isolated system full of some stuff, let’s say room full of air. We have some entropy meter in that room and we can also observe what happens in the room.

Already your instruments reduce the isolation

As well known, air particles in room are bumping on each other. Sometimes, very very rarely, all particles might be packed in the other half of the room and other half would be empty. On that time instant, entropy in room decreases rapidly

No, entropy does not decrease because it is a statistical phenomenon and you are describing one instant of it., a microstate.

and there is some usable energy in the room. Let’s take a leap of faith and assume that on that instant life would pop up in that room.

If life popped up, life works against entropy within a larger environment, it cannot be isolated. The decrease in entropy ( increase in order of the atoms and molecules of life) is at the expense of increase of the total entropy, not to forget all those photons from all those interactions that keep life alive.

Here comes the catch that confuses me: Direction of time is always thought to be pointing towards state of higher entropy. Then, if life would pop up on downward slope of measured entropy curve (when entropy starts to drop), life inside the room would assume that arrow of time is opposite than ours (Direction to higher entropy inside the room is opposite to our arrow of time). If life would pop up on upward slope of measured entropy curve, life inside the room would have same direction of time than we would.

Life does not exist unless it can interact with a larger environment since by construction it is an engine that decreases entropy locally.

So all the rest of speculations of different times are based in misconception of microstates as entropy defining states. They are just one instant in the probability distribution that will define entropy and a possible arrow of time.

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  • $\begingroup$ en.wikipedia.org/wiki/Fluctuation_theorem $\endgroup$ – octonion Dec 23 '15 at 22:57
  • $\begingroup$ @octonion from the description in your link, this theorem applies to non equilibrium systems, not isolated ones . $\endgroup$ – anna v Dec 24 '15 at 6:32
  • $\begingroup$ Thanks for your answers! I can see that my question was somewhat unclear. As in Size of the room or it's complete isolation from the world around it is not important here. $\endgroup$ – Juho Salminen Dec 26 '15 at 8:56
  • $\begingroup$ Thanks for your answers! I think my question was somewhat unclearly written. As in original Bolzmann's brain-hyphothesis, size of the room or it's complete isolation from the world around should be not that important. Neither is the stuff that is inside the room. It can be whatever hyphothetical stuff that can be in nonequilibrium state for a while because of statistical reasons. If room/universe is in low entropy state, it does not go to high entropy state immediately, but it takes some time. This time should be enough to produce some life in it. $\endgroup$ – Juho Salminen Dec 26 '15 at 9:07
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This is actually a really profound question. If I understand Boltzmann's answer to this question, it is that indeed, if you were on the downward slope of the fluctuation, you would see time running backwards. However, were you a creature that came into being during that process, it would seem natural to you that the "future" causes the "past". Since we are creatures that seemingly exist on the upward trajectory towards higher entropy, we see time moving from "past" to "future". For the case of the isolated room, I think you would be right in that there would be a sense in which time would be running backwards. You can imagine the case of a movie of a pane of glass being shattered playing in reverse - that is essentially equivalent to your case where the gas goes from being uniformly distributed to suddenly fluctuating into one side of the room. Time would seem to run backwards.

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Your question is interesting, but it is ill-posed because it contains some wrong definitions and some circular elements.

First of all, the thermodynamic entropy of a system depends on the description of the system. For example, if you describe air as a gas, it has one thermodynamic entropy; if you describe it as a mixture of gases, it has a different entropy – both in numerical value and in functional form. This fact is discussed, for example, by Grad: The many faces of entropy, Comm. Pure Appl. Math. 14 (1961) 323 https://doi.org/10.1002/cpa.3160140312.

If you use a particle-like description of air, thermodynamic entropy and temperature vanish altogether from your description. The system is fully described by the position and momentum of each particle, and possibly by other internal degrees of freedom of each particle. With these variables and the expression of internal and wall forces you can determine the evolution of the system. Nothing else is needed.

You can introduce a statistical entropy and a statistical temperature in your description, but you only do so when you are uncertain about the initial state of the system or of the specific functional form of the forces involved. So these quantities are not physical properties of the system: they are quantities connected with the particular probabilistic description you are using – either out of necessity or as a whim.

The key word in the above considerations is "your description", which appeared over and over again. It shows that the notion of time, with a specific direction, was already present all along. The moment you describe a system in a physical way, you are implicitly introducing a time variable. It's a primitive, in the formal-logical sense of this term. This logically primitive variable has specific properties implicit in its use and definition. Of course you can decide to demote "time" from its role of primitive to the role of a derived quantity, but then you have to specify some other primitive notion in its stead, and do so in a non-circular way.

The question of the relation between entropy (which one, thermodynamic or statistical? and based on which physical description?) and time (what is time? do we have in mind its relation with the notion of "memory"? But how is "memory" defined? what is its physics?) is far from settled. Discussions about it, to me personally, always seem to contain circularities and ambiguous terms, whose meaning silently change during the course of the arguments.

There is also the question of "life". What is "life"? how does it "have a sense of time"? A bacterium is a life form, for example. Suppose one appears in the room. How is this bacterium sensing time? But a bacterium is also just a physical system. So you've only introduced another physical system in the room. Aren't we back at the beginning then, with a room containing a physical system? Maybe the air already had a "sense of time" by itself then?

Here are some very interesting readings on these topics:

And also these:

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