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"two point probability" morphology

How do i calculted the second drivative of the "two point probability" of the morphology of microstructures. And how to calculate the "three point probability".?

Any help would be much appreicated

Dear George,

You will have to be more explicit about the type of microstructure that you are talking about.  If you have a two-phase composite where one phase is an inclusion and the other is a matrix then for certain simple geometries you can calculate the correlation functions analytically.  You can find examples in Mark Beran's 1968 classic Statistical Continuum Theories.  I'm sure you will also find examples in Salvatore Torquato's recent (2002) book Random Heterogeneous Materials: Microstructure and Macroscopic Properties

For more complex geometries you will have to compute the correlation functions numerically.  Use a Poisson process to set down a finite number of points on the microstructure.  For 2-point correlation functions you will then have to pick each point and join it to another point.  The correlation function can be computed based on whether the first and second points fall in the same material or in two different materials (and what those materials are).  It can be quite computationally expensive for 3-point correlation functions.  However, the curse of dimensionality while going from 2-D to 3-D is not very pronounced.

For three point probability functions you have to choose any three points and compute how many times each point falls in one material or the other.  You can then compute the probability based on that.  I'm not sure how you would compute a second derivative unless you have a smooth function describing the probability distribution. 

 

Thanks for your help , i really appreciate it.

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