I recently thought what if I wrote the $2$'nd law of thermodynamics as:
$$ dS_{\text{universe}} \geq 0 $$
Note when it is written in this form it is time asymmetric. Let, $dS = S(t+ d t) - S(t)=$ where $dt \geq 0$. Dividing both sides of the inequality with $dt$:
$$ \frac{S(t+ d t) - S(t)}{dt} \geq 0 $$
However, if time flows in the opposite direction , $ dS= S(t -dt) - S$, then:
$$ S(t-dt) - S(t) \geq 0$$
Dividing both sides with $-dt$ again:
$$ \frac{(S(t)- S(t-dt)}{dt} \leq 0$$
Hence, if time flows the other direction then one will see entropy decreasing.
However, there is an obvious flaw to this which is the limit will not be continuous unless $\frac{dS}{dt}=0$ (for example a system which has reached maximum entropy.) But all of known physics is essentially continuous and time symmetric with the exception of the collapse of the wave-function. Does this mean the collapse of the wave-function is responsible for the $2$'nd law of thermodynamics?
