Why isn't the law of entropy applicable in the other direction of time? If we have a system with the total energy concentrated in a few particles as the initial condition, law of entropy says that this energy will get more evenly distributed if the system is allowed to evolve using the laws of physics.
If I take a box of gas right now as my system, the energy of the gas being concentrated in some molecules, I do observe that energy getting more evenly distributed over time.
BUT, let's suppose I change the initial condition a bit. Consider a system identical to the above but with each particle having initial velocity in the opposite direction compared to the above system. Now we have a different initial condition. But the energy is still concentrated in a small number of molecules. After all, I've only reversed the directions of the individual velocities. The $1/2mv^2$ values remain the same, for each molecule.
On one hand, law of entropy says that the energy of this system still gets more evenly distributed, as time passes (as the system initially has a lopsided distribution of energy)
On the other hand, reversing the velocities is, in effect, the same as reversing the flow of time. Watching this system evolve in forward time should look the same as watching the original system evolve in reversed time. So this would mean that the entropy of this system should decrease over time, as that's what we witness when we reverse a video clip. This means the energy of this system should get less evenly distributed as time passes.
Which of these conclusions is correct and why??
 A: To my understanding you start your argument from the wrong end. You set a direction of time, then reverse it by reversing the velocity directions and then come to the conclusion that the entropy decreases.
But you cannot simply claim that you reversed time when reversing velocity vectors. The problem is you do not have any inherent definition of time or the direction in which time flows/passes.
Imagine not knowing anything about the concepts of entropy or time you would just look at these particles and observe that some of them are faster. The next time you check the energy is distributed over all the particles. And as you would observe this every time you measure you make it a law of thermodynamics and state that the entropy always increases in such a system. And based on this fact you actually DEFINE the positive direction in which time passes. It is the direction in which the entropy increases (only in closed systems). And then based on this finding you can also define the concept of causality.

Why isn't the law of entropy applicable in the other direction of time?

So my take on this is that you cannot really ask if entropy is or is not applicable to reversed time because the concept of entropy actually defines the concept of a direction of time. And also keep in mind that the laws of thermodynamics are empirical findings so you cannot really derive them. You can only justify them and integrate them in other theories (which was very succesfull).
