|Feb28-12, 03:47 PM||#1|
picking an element
I'm studying a particular system but to ease the understanding let me take a general approach.
1) Suppose I have a system comprised of N=10^10 elements. Assuming now that I deal a damage to a random element for every timestep, I want to figure out how the damages are distributed in the elements. My hypothesis is that this binomial distribution can be approximated very well by an easier poisson distribution.
And this brings me to my first question: Is this a correct hypothesis?
2) Next: Now the above does not really have anything to do with the system I study. Because in my system it is so that the probability of dealing damage to an element is not uniformly distributed but rather something specific to each element (normally distributed).
I want to be able to predict how the damages will be distributed amongst my elements, like in 1) but this time I'm not sure how to approach the problem. Can I predict a distribution and how would I do that?
|Feb28-12, 06:48 PM||#2|
The system is under-specified, even for (1).
If the "damage" is delivered externally (by a hail of bullets, say) then there is a chance that an already damaged element will receive another hit ... i.e. the probability of getting hit does not change with damage, so the more damaged elements there are the less likely a new element will get damaged and so the damage rate slows down. OR - perhaps damaged elements get removed from consideration?
It sounds like only one element can be damaged in a single time step - i.e. unlike, say, radioactive decay where each element has a change of changing state in a given time step so there can be more than one decay per unit time. Or are you just modelling the system on a timescale sufficiently tiny that the probability of more than one decay in one time-step is vanishingly small?
In the case of radioactive decay, where each element has a characteristic probability of changing state from "undamaged" to "damaged", the poisson approximation works quite well.
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