What do we mean by differential distribution of some particle decaying into its product particles? I have come across this term "Differential distribution" in particle physics but have not been able to clearly understand it. Does it mean how the decay observables change with change in some input parameter? Any small help would be appreciated!
 A: One of the main use of that phrasing I know of relates to the Parton Distribution Functions (PDF). Consider a proton of momentum $p$. What is the probability $dP_u(x)$ to find a quark $u$ with a momentum that is a fraction of $p$ between $x$ and $x+dx$? It can be written as $dP_u(x) = f_u(x)dx$. Then
$$f_u(x) = \frac{dP}{dx}$$
can be called a differential distribution. A mathematician would simply call it a density of probability. 
Note that the concept is more general than that but PDFs provide a nice illustration which is also important in particle physics. Hence my choice. Note also that the same idea underpins the phrasing "differential cross section". 
A: A differential distribution in particle physics is usually a measure of the rate of a process as a function of some parameter. For example, you might have a differential distribution for the cross section of e+ e- > $\mu$+ $\mu$- as a function of the collision energy, $\frac{d\sigma}{d\sqrt{s}}$. 
The idea is to integrate this over some range of the parameter to find an integrated cross section. For instance, you might be interested in the integrated cross section over energies from 7 TeV to 8 TeV. You take the $\frac{d\sigma}{d\sqrt{s}}$ and integrate it from 7 to 8 TeV.
So what if you want to know the cross section at exactly 8 TeV, say? That's unrealistic, because you will always be limited in your ability to measure / set $\sqrt{s}$ precisely. In this case, you'd integrate across some range centered on 8 TeV that represents your measurement uncertainty to find the cross section "at" 8 TeV. That may only be 0.001 TeV to either side, but nevertheless you still technically need to do that integral.
And of course, being a derivative, it is the change of the cross section as a function of, in this case, $\sqrt{s}$, as you guessed.
You can also have "double differential distributions" and so forth. For example, you could have $\frac{d^2\sigma}{d\eta d\sqrt{s}}$, which needs to be integrated over both the collision energy and pseudorapidity $\eta$ to give the cross section in some energy window for the outgoing particles to pass through some pseudorapidity window.
