# Can we reduce the entropy of a system arbitrarily by sending out a photon after arbitrary delay?

I am not asking whether this is practically feasible given current technology. Rather I'm asking whether it is possible in principle given current laws of physics.

Suppose we have a system with a distribution $$\rho$$ over the phase-space of the system, and we want to reduce the entropy $$-\int \rho(x)\cdot \log (\rho(x))dx$$ of our system by $$\Delta S=k$$. By Liouville's theorem, we cannot reduce this without increasing the entropy of some other part of the universe. The "standard" way of doing this is by putting the entropy of our system into a heat-bath, which involves placing energy equal to at least $$Tk_b\Delta S$$ into the heat bath (since that is the minimal extra energy needed to increase the heat bath's entropy by $$\Delta S$$, assuming that it was already at maximum-entropy given its energy).

But suppose instead we don't use any heat bath, and want to expel the entropy by emitting a photon (any other particle should also do the job i think, but then it might float back eventually due to gravity, increasing our system's entropy again).

Here is how it might work: Depending on the system's microstate, you expel the photon at a different point in time. In every case, the system's microstate collapses to a desired smaller volume of its phase space. This reduces its entropy, and this entropy is now stored in the position variable of the photon, since if it was emitted earlier, it will have traveled farther.

If this could be done, it would allow for the emission of entropy at arbitrarily low free energy cost (though that also means arbitrarily long delay, so even if possible in principle it wouldn't be practical).

Am I right that this is possible in principle? Or is there a known law of physics that is violated by this idea?