distscalarfield.nb

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in a scalar field, both analytically and numerically. \nGiven a particle \
density n(x) = dN/dx and a field \[Phi](x), what is the fraction of particles \
that interact with \[Phi]=\[Phi]\[CloseCurlyQuote]?\n\nOne approach is to \
write dN/d\[Phi]=(dN/dx)/(d\[Phi]/dx) as a function of \[Phi] only.\nFor \
example, the particle density follows dN/dx = n(x) = exp(-\[Alpha] x^2) and \
the field is \[Phi](x) = exp(-x^2).\nInverting the relation between \
coordinate and field x = Sqrt[-Log[\[Phi]]]\nThus |\[PartialD]\[Phi]/\
\[PartialD]x| = |-2 x \[Phi]| = 2 \[Phi] Sqrt[-Log[\[Phi]]]\nAlso \
n(x(\[Phi])) = exp(-\[Alpha] (-Log[\[Phi]])) = \[Phi]^\[Alpha]\nFinally dN/d\
\[Phi] = \[Phi]^(\[Alpha]-1) / (2 Sqrt[-Log[\[Phi]]])\n\nAnother approach \
(numerical) might be to calculate the integral dN/d\[Phi](\[Phi]\
\[CloseCurlyQuote]) = \[Integral] \
\[Delta](\[Phi]\[CloseCurlyQuote]-\[Phi](x)) (dN/dx)(x) dx\n\nFinally one can \
sample particle coordinates following  n(x) = exp(-\[Alpha] x^2), calculate \
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