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The arrow of time is often associated with the fact that entropy always increases. On the other side that should mean, if entropy decreases time should run backwards. But inside a refrigerator we have that situation. Entropy inside a refrigerator decreases (at least while cooling down). However when looking into the refrigerator while cooling down, time doesn't seem to run backwards. Things fall down and not upwards, broken things don't reassemble themselves, etc. .

I do understand that a refrigerator is not a closed system and the second law of thermodynamics doesn't apply. But should that also mean, that no arrow of time can be defined for an open system like a refrigerator? Or must we conclude that the connection between entropy and time is an illusion? If we cannot use entropy to define an arrow of time inside an open system, what is it that makes sure that time doesn't run backwards inside an refrigerator?

UPDATE: I found a recent article Siegel: Where does our arrow of time come from? where the author states basically the same idea in different words:

...if all you did was live in a pocket of the Universe that saw its entropy decrease — time would still run forward for you. The thermodynamic arrow of time does not determine the direction in which we perceive time’s passage. So where does the arrow of time that correlates with our perception come from?

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    $\begingroup$ Entropy isn't Time. Global entropy progression is just an indicator of time's passage. $\endgroup$ – RBarryYoung Oct 29 '16 at 20:59
  • $\begingroup$ To measure and verify that time is going in the same direction inside and outside, we need to put a clock into the fridge. That clock has to be checked and every time we open the door to check, entropy increases, so outside and inside time match as long as the incoming entropy balances or exceeds the drop in entropy due to the cooling $\endgroup$ – user108787 Oct 29 '16 at 21:49
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    $\begingroup$ note that a refrigerator lowers entropy by removing heat from its interior, but it does not flip the enthropic arrow of time within: energy will still disperse the same way inside as outside $\endgroup$ – Christoph Oct 29 '16 at 22:01
  • $\begingroup$ "if entropy decreases time should run backwards" that is not really true, as it is not entirely true either that increase in entropy equals time running forward (or any other thing time-related). $\endgroup$ – gented Nov 6 '16 at 16:48
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    $\begingroup$ Everyone knows you really need a TARDIS to run time backward! A refigerator? Riduculous! $\endgroup$ – docscience Nov 12 '16 at 0:56
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The short answer is that the entropy in the system decreases only due to the fact that the outflux of entropy is larger than the local growth. But the positive local growth of entropy is what is important for the arrow of time and that is why time is not running backwards in a refrigerator.


This can be demonstrated by using a fluid description of the open system (refrigerator). In fluids, the local increase of entropy can be expressed by the non-conservation of the entropy density $\sigma$: $$\frac{\partial \sigma}{\partial t} + \nabla \cdot (\sigma \vec{v}) \geq 0 $$ where $\sigma\equiv \Delta S /\Delta V$ is the amount of entropy $\Delta S$ in an infinitesimal volume $\Delta V$ at a given point.

You can see that the inequality above is not symmetric with respect to time reversal (which leads to a minus in front of $\partial/\partial t$ and $\vec{v} = d \vec{x}/dt$ ). Its meaning is exactly a local formulation of the law of increase of entropy. I.e., as long as the law above is fulfilled, the arrow of time is running correctly and in the right direction. We will see in the following that a decrease of entropy of a larger open system is not in conflict with this local time arrow.

Let us now study our open system of volume $V$ with a boundary surface $\Sigma$. We integrate the inequality above over this whole volume to obtain $$ -\frac{\partial}{\partial t} \int_V\sigma d V \leq \int_V \nabla \cdot(\sigma \vec{v}) dV $$ The integral of entropy density over the volume of the system is of course simply the total entropy in the system $S_{tot}$. The left-hand side of this new integral inequality is then simply the decrease of total entropy of our system. Also, we can take the right-hand side and use the Gauss (or Divergence) theorem to express it as an integral only over the surface of our open system $$ \int_V \nabla \cdot(\sigma \vec{v}) dV = \int_\Sigma \sigma \vec{v} \cdot d \vec{\Sigma} $$ Physically, $\sigma \vec{v} \cdot d \vec{\Sigma}$ is the flux of entropy outside the system through an infinitesimal element of the boundary surface of the system $\Sigma$. The whole integral is then simply the total flux of entropy out of our open system.

We have thus derived an inequality $$-\frac{\partial S_{tot}}{\partial t} \leq \int_\Sigma \sigma \vec{v} \cdot d \vec{\Sigma} $$ The left-hand side is the total decrease of entropy in our system, and the right-hand side is the total flux of entropy out of the system. Now, you can see explicitly that a decrease of entropy of an open system is entirely consistent with the local law of entropy increase as long as the amount of entropy leaving the system is larger than the amount of entropy decreased. This is also the case of the fridge and any cooling system.

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  • $\begingroup$ I like your answer a lot. It clears up a lot of my confusion. But I have one question left: In case of an ideal fluid (or an adiabatic system) the local production of entropy is zero. Would that mean then that time stops running? Does the rate of entropy production have any relation to the rate at which we experience time flowing? $\endgroup$ – asmaier Nov 12 '16 at 18:53
  • $\begingroup$ @asmaier Perfect fluids are an idealization of fluids where heat exchange and viscosity is neglected. I.e. two streams of a perfect fluid of different temperature travelling in a straight line next to each other would never reach the same temperature. That is obviously unphysical the same as adiabatic processes in thermodynamics, but it can be a useful approximation. $\endgroup$ – Void Nov 12 '16 at 19:23
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    $\begingroup$ @asmaier The arrow of time and the growth of entropy have one and only thing in common (at least in mainstream physics) - their direction. There is no connection in their rates and similar, the rate of time as we talk about in physics is tied to fundamental dynamics governing e.g. pendulum swings or nuclear decays. $\endgroup$ – Void Nov 12 '16 at 19:32
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Entropy always increases in a closed system. The Universe is considered a closed system so the entropy of the Universe is always increasing unless your time arrow is reverted. A refrigerator is exchanging heat with the surroundigs so it is not a closed system. If you check the sum of entropy of inside and outside of the refrigerator it will be increasing over time.

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    $\begingroup$ Not only this: entropy increases in any sub-system, e.g. vegetables inside the fridge do keep increasing their entropy (thus rotting). To flip the arrow of time, entropy should overall on average decrease at any scale I suppose.. $\endgroup$ – JalfredP Nov 4 '16 at 22:36
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The second law can be stated without having to define a notion of entropy, as it primarily emphasizes on the one hand, how energy can be converted into work and on the other hand, the fact that we observe an arrow of time in the macroscopic world, e.g., heat from hot to cold. These two statements are reformulations that Kelvin and Clausius respectively provided for the second law. More precisely:

  • How Kelvin puts it: No thermodynamic process is possible whose only effect is to extract heat and covert it entirely to work.
  • Clausius version: there can be no thermodynamic process capable of solely transferring heat from a cold to a hot reservoir.

The strength of these statements lies in the use of only effect or solely. Apply the second version to your fridge: it says that in order for your fridge to extract the heat inside (cold reservoir) and transfer it to the outside (kitchen) you need to do some work (plug off the electricity cable and it won't work ;). So again what does all this mean? Heat is a flow from hot to cold and never the other way around, and hence the implication of an "arrow" of time.

I hope this convinces you how you can employ the second law when observing real life phenomena without resorting to entropic reformulations. If you want to express all this in terms of change of entropy: please remember that when studying the time evolution of macroscopic systems, a consistent statement about the arrow of time can be made only when you consider the entropy change of the whole universe (in our example that being kitchen+fridge). Moreover, when computing the total change of entropy, you add up the change of entropy of the cold reservoir $\Delta_c$ with that of the hot reservoir $\Delta_h.$ But to compute the latter, the reversible transfer of heat into the system is not only $Q_c$ (heat removed from the cold reservoir) but $Q_c+W,$ with $W$ the supplied work, which is used to compress the working fluid of the fridge. The inclusion of $W$ here is exactly what the statement of Clausius is all about.

Last remark after reading some of the comments: remember that as soon as you limit your system to a completely isolated one (e.g., a thermodynamic cycle comprised of only adiabatic processes), you run into the Clausius inequality, which in words says: the entropy of an isolated system never decreases.

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  • $\begingroup$ Phonon, do I perhaps know you? $\endgroup$ – Žarko Tomičić Nov 12 '16 at 15:26
  • $\begingroup$ @ŽarkoTomičić Hi, I'm not sure, why do you think you know me? :) $\endgroup$ – Phonon Nov 12 '16 at 16:02
  • $\begingroup$ Well your style and your field of interest remind me of my professor. Statistical physics, solid state physics...that and the way in which you answer some of the questions. E.g. you are describing Kelvins and Clausius formulation, which he (my prof.) does in the same way, you are using the phrase "how kelvin puts it..." which is, forgive me, a bit clumsy...so I would say, english is not you mothers tongue. But, maybe I am wrong. Your long and detailed answers on the other hand portrait someone who knows his stuff very well. And he really does. Sometimes he gives interesting analogies $\endgroup$ – Žarko Tomičić Nov 12 '16 at 23:44
  • $\begingroup$ @ŽarkoTomičić haha I see. I'm very glad to find out that you've taken a liking to my answers. But unfortunately I'm not the person you may have in mind, just a struggling young researcher here... :( $\endgroup$ – Phonon Nov 13 '16 at 13:07
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The arrow of time in a thermodynamic system should be thought instead as a statement about time-reversal invariance. For example, in classical mechanics, based solely on the particle motion one cannot tell if time is going forward or backward. In the same way, in an open system one cannot tell which way time is flowing based on the change in entropy alone. In essence, entropy does not define the direction of time in an open system.

However, this is different from asking about why time doesn't run backwards in a refrigerator. If you think of a fridge as something that decreases entropy over time, one can think of its time-reversed partner as a heater which increases entropy over time. Hence, in thinking of the system as a refrigerator, you've already selected a particular time direction. Nobody is stopping you from defining time to flow in the opposite direction, but asking why time doesn't run backwards in a fridge isn't really a well-posed question. Saying time flows backwards is a relative statement; you have to tell me what "backwards" is relative to.

EDIT: My answer seems to be unsatisfactory, so let me try to elaborate.

First, what is meant by time? It is a 1D parameter that controls properties of objects (eg position). You can think of each physical property (object) independently having their own time parameter.

Now, it is interesting to note that the microscopic laws of physics are time-reversal invariant. This means that the form of the equations don't change from $t\rightarrow -t$. However, you'll still get qualitative differences, such as velocity $v \rightarrow -v$ under time-reversal. Hence, a particle moving to the right will be moving to the left under time reversal. Hence, what time-reversal invariance means is that if you look at a clip of a particle moving forward in time vs moving backward in time, you can't tell which one is which. So, if we have don't know which way time should be going, we can just pick one of our choosing.

Then, we can just choose a time direction for every single particle in a collection however we want. But wait a minute; if I watch all the particles move, then clearly their time directions should all be synchronized to my time. Hence, we stumble upon a key property of time that it depends on the observer. The observer is the one who sets the forward flow of time for all particles. Hence, to come back to the question, if I, the observer, call a system a refrigerator, I have already selected a time direction for it, which makes original question itself ill-posed. In other words, the reason why the refrigerator's time and the spoilage of food and the influence of gravity all have the same time direction is because their time direction is assigned by the observer.

Now, the second law states that entropy in a closed system must increase over time. Hence, just as before this defines a time direction for the property of entropy. Now, somehow we must synchronize this time direction with that of all the other time directions, again via an observer. The magical thing, as the article you cite notes, is that for some reason the time direction selected by the second law is always the same as the time direction selected by observers in the physical universe. But, as the article also acknowledges, nobody really knows why. At least, it doesn't seem that this is a derived consequence by any well-accepted physical theory, but rather a postulated coincidence (like the equivalence of inertial and gravitational mass).

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    $\begingroup$ I think this dodges the question OP asked. We've already determined an arrow of time. It's the direction where a clock put in the fridge turns clockwise, and the direction where food ripens and spoils. You have to show why that direction is the same as the direction in which the fridge is a fridge. $\endgroup$ – knzhou Oct 29 '16 at 20:58
  • $\begingroup$ To measure and verify that time is going in the same direction inside and outside, we need to put a clock into the fridge. That clock has to be checked and every time we open the door to check, entropy increases, so outside and inside time match as long as the incoming entropy balances or exceeds the drop in entropy due to the cooling. $\endgroup$ – user108787 Oct 29 '16 at 21:47
  • $\begingroup$ @knzhou is right, only inside, but you need to include the cold air and negative entropy being pumped in, and that it is not energy isolated. When you take it all into account entropy increases, as Mandrill says below. The same is true with lots of things that exchange energy with their environment, e.g., you make a clock with random material gathered: entropy decreases, but you put a lot of work into it. $\endgroup$ – Bob Bee Oct 30 '16 at 1:03
  • $\begingroup$ @knzhou But if the time direction is specified, isn't this question moot? One cannot determine a direction of time based on an open system alone. Note that the external pump is odd under time reversal. Hence, selecting a time direction via a clock selects a direction for the pump to operate. It's not so much the clock turns the same direction as the time direction of the fridge, but rather the clock external to the systems sets the time direction. $\endgroup$ – Aaron Oct 30 '16 at 4:04
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    $\begingroup$ @BruceGreetham As you note, the time direction selected by conscious observers does not seem to be physics question, hence why I stop there and say it seems to be postulated at a physics level. Your points about the initial state of the universe are well-taken, but at the end of the day at most it gives an equivalence between cosmological and thermodynamic arrow. Perhaps there is something to be said about the arrow selected by information entropy. However, OP's question is about the link between perceptual (my watch) and thermodynamic time. Perhaps there is a deeper reason I am not aware of. $\endgroup$ – Aaron Nov 11 '16 at 17:50
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But should that also mean, that no arrow of time can be defined for an open system like a refrigerator? Or must we conclude that the connection between entropy and time is an illusion? If we cannot use entropy to define an arrow of time inside an open system, what is it that makes sure that time doesn't run backwards inside an refrigerator?

Let's take a scientist who lives in our world. After some time experimenting around, he will notice the following thing:

If two bodies are put into physical contact, energy will always spontaneously flow from the hotter body to the colder body, and never in the opposite direction.

Here, the key word is "spontaneously", without any work having been done.

The scientist will then define the "forward" direction in time as the direction in which energy flows spontaneously from an hotter body to a colder body.

Let's now consider a mini-scientist living inside the fridge. He was born inside it, and doesn't know any reality external from the fridge. He will never see heat flow from the inside to the outside, because there is no "outside" for him (let's make the hypothesis that temperature is kept approximately constant inside the fridge). How will the mini-scientist define the "forward" direction in time?

The answer is: in the same way as the scientist who lives outside. Indeed, if two objects at different temperature are put into contact inside the fridge, heat will always flow spontaneously from the hotter body to the colder body (and, like you stated, the pieces of a broken glass will not magically come back together just because we are inside a fridge!).

Yes, an outside observer will see that there is an heat flow from the colder inside of the fridge to the hotter outside environment, but he will also see that the fridge is plugged in and that work is being done, so the process is not spontaneous and the definition of the "forward" direction in time is safe.

So I would say that the problem is only apparent, and that there are no problems when defining the arrow of time inside an open system.

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  • $\begingroup$ "The scientist will then define the direction in time as the direction in which energy flows.." I'm pretty sure that is not how any scientist defines direction of time $\endgroup$ – Bruce Greetham Nov 13 '16 at 16:09
  • $\begingroup$ @BruceGreetham Of course, there are many ways to define an arrow of time. I am considering only the thermodynamic arrow of time to make things simpler. $\endgroup$ – valerio Nov 13 '16 at 16:13
  • $\begingroup$ OK but I would think the fundamental definition is the direction in which an experiment is done and a recorded memory of the result is made. This doesn't make your answer wrong - I just needed to say that to preserve my sanity with all this! $\endgroup$ – Bruce Greetham Nov 13 '16 at 16:22
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Copying from wikipedia:

'The second law of thermodynamics states that the total entropy of an isolated system always increases over time, or remains constant in ideal cases where the system is in a steady state or undergoing a reversible process. The increase in entropy accounts for the irreversibility of natural processes, and the asymmetry between future and past.'

and again:

' The direction of heat transfer is from a region of high temperature to another region of lower temperature, and is governed by the Second Law of Thermodynamics. Heat transfer changes the internal energy of the systems from which and to which the energy is transferred. Heat transfer will occur in a direction that increases the entropy of the collection of systems.'

A refrigerator works as heat-pump:

'Heat pumps are designed to move thermal energy opposite to the direction of spontaneous heat flow by absorbing heat from a cold space and releasing it to a warmer one. A heat pump uses some amount of external power to accomplish the work of transferring energy from the heat source to the heat sink.'

In conclusion there is no natural heat flow from a cold to a hot reservoir but only consumption of electrical energy to do so. Just making a place colder in such way doesn't imply that the arrow of time is reversed.

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