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?
Best Answer
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.