KITP: cluster mass functions

in which we ponder the globular cluster mass function and whether the current mass function really is lognormal and how this came about given that everybody but everybody believes the initial mass function must have been a power law...

wednesday afternoon was a good introduction and lively discussion of theoretical processes affecting globular cluster lifetimes and mass evolution - including some pdf of slides.

Just so we all remember what we are talking about:
log-normal
power law

people in other fields might know these as Somebody's Law, but physical scientists do not deign to name the obvious after themselves or their colleagues...

There is a good guide to these at Mitzenmacher on Generative Models

Getting some intense discussion here on initial conditions and evolutionary processes...
nobody seems to agree on what bumps in log-log plots mean nor how best to plot them.

this is the sort of thing we're bickering about

So here is my personal take on this...

Power laws are ideally scale free, although in practise there must be a cutoff mass either at the low end, high end, or both; lognormal distributions have a built in scale two scales actually, the mean and dispersion.
So, how do you go from a power law to a log-normal?

Naively, one might think there is some process that imprints a scale on the power law and then breaks the power law - that there is selective depletion below the mass scale.

On the other hand, log-normal distributions arise from independent multiplicative processes - ie ln(x) is independent and additive, so a log-normal process should come from many independent processes acting to modify the initial distribution, something like

x_final ← Π_i f_i x_initial

where f_i ∈ [0, 1] is some suitably normalised independent random variable modifying x, through some independent processes i=1...k

In which case, the log-normal scale is just the power law cutoff scale; so if you're cutting off the low end of the power law at some mass scale M_low then the scale ought to be something like

M_scale ≅ M_low/(0.5^k)

where k is the number of independent processes (after normalisation so they are approximately uniformly distributed independent variables on the unit interval as assumed), and similarly if there is an upper mass cutoff instead.

Generically, if this is not true, then we ought not to be seeing a true log-normal distribution since the final distribution would not be arising from independent multiplicative processes.

Note though that Mitzenmacher notes that a power law with a lower cutoff tends to generate an approximate log-normal distribution - which I think is a simpler way of saying the same thing.

Or, the log-normal fit could just be a bad fit to sparse data and there are correlated physical processes with built in scales modifying the initial power law.
Assuming the initial distribution really is a (truncated) power law.