Sunday, May 15, 2005

date rate

So, remember the Date Equation?

Well, it was pointed out to me, confidentially, that this is not a completed problem - having found the number of probable candidates, you then need to estimate the encounter rate, or equivalently, the mean time between dates.

Now, this is trivial: R = n s v

where R is the rate of dates, and 1/R = T is a characteristic time between dates.
as usual, n is the number density (eligible people per unit area); s is your date cross-section in meters, and v is the encounter velocity.

But there are complications;
first of all, n is non-uniform, it is probably poorly averaged by a uniform density,
secondly, cross-section is non-trivial. There is an irreducable s0 = 1 meter, which is where people basically bump into you; but some people have much larger cross-sections, they have a "date charge" which I choose to call charm, which enables long range interactions! So, a rock star or good comedian may have a cross-section of ~ 100 meters, and be in a crowded room with n = 1, whereas a social misfit has s ~ 1 and if they're picky, n may be as low as 10^-8

But, in a crowded region, the velocity is necessarily slow, and the charm coupling is clearly velocity dependent! A highly charmed individual maximises their opportunity at slow speeds, not high speeds as one might naively infer;
so we can approximate s(v) = s0*( sC/( 1 + (vC/v)^2) - of course you recognise this equation, it can't be exact, or there'd be singular charm collapses; but it will do as a first order approximation. vC is clearly of order unity, probably a bit less than one, maybe 0.1 will approximate it.

But, but, we're not done yet! Some couplings are forbidden. Even if you're not picky, and a very charming lecturer...

Further, with clumping and non-trivial topology of the phase space, rigorous estimates will require Monte Carlo simulations.

But, worst of all; you can temporarily enhance your cross-section tremendously through virtual charm coupling; now you might protest that this never leads to bound states, but if the virtual interaction is resonant, then a real bound state may form even if the geometric cross-section, or velocity would imply in the classical limit that no interaction will take place.

Sociology is very complicated.

In the meantime, the rock star; v ~ 10^-4, n ~ 1, s ~ 100 => R ~ 10^-2 s^-1
a truly charming rock star that is just diffusing in a crowd can get a date per minute in the classical limit. In fact if vC is large enough, the rock star is near singular collapse as v -> 0, modulo second order corrections.

Social misfit in a big city - v ~ 1, n ~ 10^-8, s ~ 1 => R ~ 10^-8 - or one date every 3 years or so; sounds about right.

Thus we conclude that the first order classical approximation provides adequate approximation for the extreme limiting cases. Note that if the misfit has very stringent restriction, for example on age ranges, the actual suitable population may change more rapidly than they get dates - so they have a finite probability of never getting a date.
The rock star case is of course self-limiting, for obvious reasons of other physics we have neglected. Think of it as a renormalisation sort of thing, too high an R triggers some new hidden physics which limits the true R, but we can work with an effective theory.

Hm, can we try a realistic mid-range problem: relatively indiscriminating undergraduate at a midsized university.

N = 1000, A = 10^7 m^2, so n ~ 10^-4, s ~ 1, v ~ 0.1 - that's mean progress through ambling along.
So within an order of magnitude, expect a date per day; except there is clumping, which reduces the effective number density, and worse than that, there is screening of the charm charge due to clumping, in fact you may never get to effective range for your charm charge to trigger a resonant interaction. So realistic rates are 1-2 orders of magnitude smaller.
I see the potential for very complicated higher order scatterings if a highly charmed charge individual penetrates the screening radius of a dense clump of datees of the opposite charm.

So we conclude that in dense environment, even for intermediate strength interactions, the second order corrections are important. We note parenthetically that charm screening may be an even more important effect in high schools and work places, mostly from peer clustering, but sometimes due to paper screens or other social constructs.

I think this promises to be a rich field for extensive numerical simulations, particularly since we have not yet estimated the effects of virtual charm cross-sections; nor do we have data on where or how resonances occur.

Now, if you combine this with the "marriage optimisation algorithm", you're almost done. Theoretically.

(yes, there is such a thing, mathematician came up with it; I'll blahg on it later)


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