Why is Entropy One-Directional?

Question

If the laws of physics work the same forward and backward in time, why does entropy grow in one direction?

Answer 1

It is actually not, the total entropy can in fact decrease, but it's just highly unlikely.

I will make an analogy to the statistical interpretation of entropy as counting the amount of microstates $\Omega$ according to some configuration in phase space.
$S = k \ln \Omega$
Consider a uniform gas of indistinguishable particles in a box. Each particle has a position in 3D space and a momentum vector, together they form the $6N$-dimensional phase space where each point labels a certain configuration:
We will now compare 2 particular states:

1. The gas is uniformly distributed over the entire box
2. All gas particles suddenly go to one corner of the box and reside there

Clearly, the first state can be realised in many more ways than the second. Therefore, $\Omega_1 \ggg \Omega_2$ and hence $S_1 \ggg S_2$. There is a higher probability to find the gas uniformly distributed than in one corner.
However, does this mean that the second case is unphysical as it would lower the entropy?
No, it is just extremely unlikely! The second law of thermodynamics is not absolute, it's just that processes where the total entropy decreases are highly improbable.

For other examples: look up negative temperatures, the entropy decreases when adding heat.

Answer 2

The question you ask is one of the major open philosophical questions in science today. Why does time appear to move "forward?" Many great minds like Feynman have explored the question (and its dual "what if time doesn't just move forward?") As such, no answer can be completely satisfactory, but some statements can be made and they will hopefully help.

As we see in JulianDeV's answer, entropy and its inevitable statistical increase are baked into the problem statement. They arise when dealing with systems where one can theoretically predict the past and future state of the system perfectly, given enough information, but where "enough" information is unacquirable. For example, the velocity of particles cannot be known perfectly. The best we can do is state some bounds on the velocity.

When you fill in the missing unknowns with randomness, in the truest mathematical sense of the word, your equations exhibit entropy. The measurable "states" and the unmeasurable multitude of "microstates," which appear the same upon measurement but are fundamentally different in the phase space of the systems, inherently lead to entropic like behavior. And in the direction of evolution of the system, we see entropy increasing (on average, as JulianDeV points out). So long as you wish to use random variables to fill in for that which you do not know, this entropic behaviors will emerge as unknown compounds upon unknown.

The great philosophical question is why are these models typically applied to systems with time going in one direction. Is there a fundamental concept of "forward" to the universe, or is it a side effect of the way the H. sapiens mind thinks about time?

And that is where the satisfactory answers must end. When we step beyond "what models exhibit entropic behaviors?" and into "what aspects of reality should be modelable this way?", we quickly get into the fundamental philosophical meaning of time. For an introduction into that question, I highly recommend the SEP article on Time.

Answer 3

Once you define entropy, such as Cort Ammon descibes, add completely symmetric laws of physics (which is the case, as you say), you still need some kind of initial or prior condition to deduce a single emergent arrow of time.

The prior condition is the big bang, and it was "large" enough and lower entropy than today. The laws of physics allow for trajectories in either entropic direction. But if you already have a relatively low entropy state, the most common trajectories will be toward a higher overall entropy state. And thus from one snapshot to the next, all the way to the big bang, entropy has increased overall. And thus a single emergent arrow of time, the second law.

There are other formulations and assumptions that can lead to multiple arrows of time, like shape dynamics of Julian Barbour. They do not assume a low entropy big bang, and one trade off is having multiple arrows of time.

There is much more to the story but I belive this is the bare logical arguement.